TPTP Problem File: NUM925+4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM925+4 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 192, 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_l192 [Bla11]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.47 v7.5.0, 0.56 v7.4.0, 0.40 v7.3.0, 0.48 v7.1.0, 0.52 v7.0.0, 0.63 v6.4.0, 0.62 v6.2.0, 0.76 v6.1.0, 0.87 v6.0.0, 0.83 v5.5.0, 0.85 v5.4.0, 0.86 v5.3.0
% Syntax : Number of formulae : 5487 (2191 unt; 0 def)
% Number of atoms : 12624 (4153 equ)
% Maximal formula atoms : 25 ( 2 avg)
% Number of connectives : 8191 (1054 ~; 247 |; 914 &)
% ( 938 <=>;5038 =>; 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 : 1084 (1084 usr; 436 con; 0-8 aty)
% Number of variables : 11438 (11189 !; 249 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 18:37:39
% : Encoded with monomorphized guards.
%------------------------------------------------------------------------------
%----Explicit typings (66)
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,hypothesis,
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,axiom,
is_int(m) ).
fof(gsy_v_m1____,axiom,
is_int(m1) ).
fof(gsy_v_r____,axiom,
is_int(r) ).
fof(gsy_v_s1____,axiom,
is_int(s1) ).
fof(gsy_v_s____,axiom,
is_int(s) ).
fof(gsy_v_sa____,axiom,
is_int(sa) ).
fof(gsy_v_t____,axiom,
is_int(t) ).
fof(gsy_v_v____,axiom,
is_int(v) ).
fof(gsy_v_w____,axiom,
is_int(w) ).
fof(gsy_v_x____,axiom,
is_int(x) ).
fof(gsy_v_y____,axiom,
is_int(y) ).
%----Relevant facts (5196)
fof(fact_0_n1pos,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),hAPP_nat_int(semiri1621563631at_int,n)))) ).
fof(fact_1_t1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),t)) ).
fof(fact_2_sum__power2__eq__zero__iff,axiom,
! [Xa,Ya] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Xa),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,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_rat
<=> ( Xa = zero_zero_rat
& Ya = zero_zero_rat ) ) ).
fof(fact_3_sum__power2__eq__zero__iff,axiom,
! [Xa,Ya] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),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))))) = zero_zero_real
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ).
fof(fact_4_sum__power2__eq__zero__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Xa),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,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_int
<=> ( Xa = zero_zero_int
& Ya = zero_zero_int ) ) ) ).
fof(fact_5_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_6_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_7_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_8_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_9_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_10_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_11_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_12_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_13_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_14_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_15_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_16_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_17_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_18_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_19_zero__eq__power2,axiom,
! [A_1] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_rat
<=> A_1 = zero_zero_rat ) ).
fof(fact_20_zero__eq__power2,axiom,
! [A_1] :
( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_complex
<=> A_1 = zero_zero_complex ) ).
fof(fact_21_zero__eq__power2,axiom,
! [A_1] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_real
<=> A_1 = zero_zero_real ) ).
fof(fact_22_zero__eq__power2,axiom,
! [A_1] :
( is_int(A_1)
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int
<=> A_1 = zero_zero_int ) ) ).
fof(fact_23_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_24_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_25_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_26_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_27_add__special_I3_J,axiom,
! [V_25] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_25)),one_one_rat) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),hAPP_int_int(bit1,pls))) ).
fof(fact_28_add__special_I3_J,axiom,
! [V_25] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_25)),one_one_complex) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),hAPP_int_int(bit1,pls))) ).
fof(fact_29_add__special_I3_J,axiom,
! [V_25] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_25)),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),hAPP_int_int(bit1,pls))) ).
fof(fact_30_add__special_I3_J,axiom,
! [V_25] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_25)),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_25),hAPP_int_int(bit1,pls))) ).
fof(fact_31_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_32_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_33_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_34_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_35_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_36_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_37_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_38_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_39_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_40_quartic__square__square,axiom,
! [X] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_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_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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_41_power__0__left__number__of,axiom,
! [W_15] :
( ( hAPP_int_nat(number_number_of_nat,W_15) = 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_15)) = one_one_rat )
& ( hAPP_int_nat(number_number_of_nat,W_15) != 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_15)) = zero_zero_rat ) ) ).
fof(fact_42_power__0__left__number__of,axiom,
! [W_15] :
( ( hAPP_int_nat(number_number_of_nat,W_15) = zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_15)) = one_on1684967323de_int )
& ( hAPP_int_nat(number_number_of_nat,W_15) != zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),hAPP_int_nat(number_number_of_nat,W_15)) = zero_z891286103de_int ) ) ).
fof(fact_43_power__0__left__number__of,axiom,
! [W_15] :
( ( hAPP_int_nat(number_number_of_nat,W_15) = zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_15)) = one_one_complex )
& ( hAPP_int_nat(number_number_of_nat,W_15) != zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),hAPP_int_nat(number_number_of_nat,W_15)) = zero_zero_complex ) ) ).
fof(fact_44_power__0__left__number__of,axiom,
! [W_15] :
( ( hAPP_int_nat(number_number_of_nat,W_15) = zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_15)) = one_on1645066479umeral )
& ( hAPP_int_nat(number_number_of_nat,W_15) != zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),hAPP_int_nat(number_number_of_nat,W_15)) = zero_z126310315umeral ) ) ).
fof(fact_45_power__0__left__number__of,axiom,
! [W_15] :
( ( hAPP_int_nat(number_number_of_nat,W_15) = zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_15)) = one_one_real )
& ( hAPP_int_nat(number_number_of_nat,W_15) != zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),hAPP_int_nat(number_number_of_nat,W_15)) = zero_zero_real ) ) ).
fof(fact_46_power__0__left__number__of,axiom,
! [W_15] :
( ( hAPP_int_nat(number_number_of_nat,W_15) = 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_15)) = one_one_nat )
& ( hAPP_int_nat(number_number_of_nat,W_15) != 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_15)) = zero_zero_nat ) ) ).
fof(fact_47_power__0__left__number__of,axiom,
! [W_15] :
( ( hAPP_int_nat(number_number_of_nat,W_15) = 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_15)) = one_one_int )
& ( hAPP_int_nat(number_number_of_nat,W_15) != 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_15)) = zero_zero_int ) ) ).
fof(fact_48_semiring__norm_I110_J,axiom,
one_one_rat = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls)) ).
fof(fact_49_semiring__norm_I110_J,axiom,
one_one_complex = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)) ).
fof(fact_50_semiring__norm_I110_J,axiom,
one_one_real = hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)) ).
fof(fact_51_semiring__norm_I110_J,axiom,
one_one_int = hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) ).
fof(fact_52_numeral__1__eq__1,axiom,
hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls)) = one_one_rat ).
fof(fact_53_numeral__1__eq__1,axiom,
hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)) = one_one_complex ).
fof(fact_54_numeral__1__eq__1,axiom,
hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)) = one_one_real ).
fof(fact_55_numeral__1__eq__1,axiom,
hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) = one_one_int ).
fof(fact_56_n0,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),n)) ).
fof(fact_57_zless__linear,axiom,
! [X,Y] :
( ( is_int(X)
& is_int(Y) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),Y))
| X = Y
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),X)) ) ) ).
fof(fact_58_less__number__of__int__code,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,K)),hAPP_int_int(number_number_of_int,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) ) ).
fof(fact_59_plus__numeral__code_I9_J,axiom,
! [V_1,W] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_1)),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_1),W)) ).
fof(fact_60_less__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,Xa)),hAPP_int_rat(number_number_of_rat,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ).
fof(fact_61_less__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Xa)),hAPP_int_real(number267125858f_real,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ).
fof(fact_62_less__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,Xa)),hAPP_int_int(number_number_of_int,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ).
fof(fact_63_zero__is__num__zero,axiom,
zero_zero_int = hAPP_int_int(number_number_of_int,pls) ).
fof(fact_64_zpower__int,axiom,
! [M,N] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) ).
fof(fact_65_int__power,axiom,
! [M,N] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M)),N) ).
fof(fact_66_zadd__int__left,axiom,
! [M,N,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)),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))),Z_1) ).
fof(fact_67_zadd__int,axiom,
! [M,N] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
fof(fact_68_int__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
fof(fact_69_nat__number__of__Pls,axiom,
hAPP_int_nat(number_number_of_nat,pls) = zero_zero_nat ).
fof(fact_70_semiring__norm_I113_J,axiom,
zero_zero_nat = hAPP_int_nat(number_number_of_nat,pls) ).
fof(fact_71_int__eq__0__conv,axiom,
! [Na] :
( hAPP_nat_int(semiri1621563631at_int,Na) = zero_zero_int
<=> Na = zero_zero_nat ) ).
fof(fact_72_int__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
fof(fact_73_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_74_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_75_rel__simps_I17_J,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K)),hAPP_int_int(bit1,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) ) ).
fof(fact_76_rel__simps_I2_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),pls)) ).
fof(fact_77_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_78_rel__simps_I14_J,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K)),hAPP_int_int(bit0,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) ) ).
fof(fact_79_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_80_add__nat__number__of,axiom,
! [V_2,V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,V_2) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_2),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,V_1) )
& ( ~ 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_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_1),V_2)) ) ) ) ) ).
fof(fact_81_one__is__num__one,axiom,
one_one_int = hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) ).
fof(fact_82_nat__numeral__1__eq__1,axiom,
hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) = one_one_nat ).
fof(fact_83_Numeral1__eq1__nat,axiom,
one_one_nat = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) ).
fof(fact_84_eq__number__of,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hAPP_int_rat(number_number_of_rat,Xa) = hAPP_int_rat(number_number_of_rat,Ya)
<=> Xa = Ya ) ) ).
fof(fact_85_eq__number__of,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hAPP_int_complex(number528085621omplex,Xa) = hAPP_int_complex(number528085621omplex,Ya)
<=> Xa = Ya ) ) ).
fof(fact_86_eq__number__of,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hAPP_int_real(number267125858f_real,Xa) = hAPP_int_real(number267125858f_real,Ya)
<=> Xa = Ya ) ) ).
fof(fact_87_eq__number__of,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hAPP_int_int(number_number_of_int,Xa) = hAPP_int_int(number_number_of_int,Ya)
<=> Xa = Ya ) ) ).
fof(fact_88_number__of__reorient,axiom,
! [Wa,Xa] :
( hAPP_int_rat(number_number_of_rat,Wa) = Xa
<=> Xa = hAPP_int_rat(number_number_of_rat,Wa) ) ).
fof(fact_89_number__of__reorient,axiom,
! [Wa,Xa] :
( hAPP_i1732201573de_int(number1226105091de_int,Wa) = Xa
<=> Xa = hAPP_i1732201573de_int(number1226105091de_int,Wa) ) ).
fof(fact_90_number__of__reorient,axiom,
! [Wa,Xa] :
( hAPP_int_complex(number528085621omplex,Wa) = Xa
<=> Xa = hAPP_int_complex(number528085621omplex,Wa) ) ).
fof(fact_91_number__of__reorient,axiom,
! [Wa,Xa] :
( hAPP_i769753017umeral(number1443263063umeral,Wa) = Xa
<=> Xa = hAPP_i769753017umeral(number1443263063umeral,Wa) ) ).
fof(fact_92_number__of__reorient,axiom,
! [Wa,Xa] :
( hAPP_int_real(number267125858f_real,Wa) = Xa
<=> Xa = hAPP_int_real(number267125858f_real,Wa) ) ).
fof(fact_93_number__of__reorient,axiom,
! [Wa,Xa] :
( is_int(Xa)
=> ( hAPP_int_int(number_number_of_int,Wa) = Xa
<=> Xa = hAPP_int_int(number_number_of_int,Wa) ) ) ).
fof(fact_94_number__of__reorient,axiom,
! [Wa,Xa] :
( hAPP_int_nat(number_number_of_nat,Wa) = Xa
<=> Xa = hAPP_int_nat(number_number_of_nat,Wa) ) ).
fof(fact_95_rel__simps_I51_J,axiom,
! [K,L] :
( ( is_int(K)
& is_int(L) )
=> ( hAPP_int_int(bit1,K) = hAPP_int_int(bit1,L)
<=> K = L ) ) ).
fof(fact_96_rel__simps_I48_J,axiom,
! [K,L] :
( ( is_int(K)
& is_int(L) )
=> ( hAPP_int_int(bit0,K) = hAPP_int_int(bit0,L)
<=> K = L ) ) ).
fof(fact_97_even__less__0__iff,axiom,
! [A_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),A_1)),zero_zero_rat))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),zero_zero_rat)) ) ).
fof(fact_98_even__less__0__iff,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),zero_zero_real)) ) ).
fof(fact_99_even__less__0__iff,axiom,
! [A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),A_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)) ) ).
fof(fact_100_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_101_zadd__left__commute,axiom,
! [X,Y,Z_1] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),Z_1)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Z_1)) ).
fof(fact_102_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_103_int__int__eq,axiom,
! [Ma,Na] :
( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na)
<=> Ma = Na ) ).
fof(fact_104_less__special_I3_J,axiom,
! [Xa] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,Xa)),zero_zero_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) ) ).
fof(fact_105_less__special_I3_J,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Xa)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) ) ).
fof(fact_106_less__special_I3_J,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,Xa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),pls)) ) ).
fof(fact_107_less__special_I1_J,axiom,
! [Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) ) ).
fof(fact_108_less__special_I1_J,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(number267125858f_real,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) ) ).
fof(fact_109_less__special_I1_J,axiom,
! [Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Ya)) ) ).
fof(fact_110_rel__simps_I12_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).
fof(fact_111_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_112_rel__simps_I16_J,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K)),hAPP_int_int(bit0,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) ) ).
fof(fact_113_rel__simps_I10_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).
fof(fact_114_rel__simps_I4_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit0,K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),K)) ) ).
fof(fact_115_bin__less__0__simps_I4_J,axiom,
! [Wa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,Wa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)) ) ).
fof(fact_116_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),zero_zero_int)) ).
fof(fact_117_bin__less__0__simps_I3_J,axiom,
! [Wa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,Wa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),zero_zero_int)) ) ).
fof(fact_118_int__0__less__1,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
fof(fact_119_zless__add1__eq,axiom,
! [Wa,Z_2] :
( ( is_int(Wa)
& is_int(Z_2) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),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,Wa),Z_2))
| Wa = Z_2 ) ) ) ).
fof(fact_120_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_121_less__special_I4_J,axiom,
! [Xa] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,Xa)),one_one_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),hAPP_int_int(bit1,pls))) ) ).
fof(fact_122_less__special_I4_J,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Xa)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),hAPP_int_int(bit1,pls))) ) ).
fof(fact_123_less__special_I4_J,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,Xa)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),hAPP_int_int(bit1,pls))) ) ).
fof(fact_124_less__special_I2_J,axiom,
! [Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Ya)) ) ).
fof(fact_125_less__special_I2_J,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_int_real(number267125858f_real,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Ya)) ) ).
fof(fact_126_less__special_I2_J,axiom,
! [Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),hAPP_int_int(number_number_of_int,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,pls)),Ya)) ) ).
fof(fact_127_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_128_double__eq__0__iff,axiom,
! [A_1] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),A_1) = zero_zero_rat
<=> A_1 = zero_zero_rat ) ).
fof(fact_129_double__eq__0__iff,axiom,
! [A_1] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1) = zero_zero_real
<=> A_1 = zero_zero_real ) ).
fof(fact_130_double__eq__0__iff,axiom,
! [A_1] :
( is_int(A_1)
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),A_1) = zero_zero_int
<=> A_1 = zero_zero_int ) ) ).
fof(fact_131_rel__simps_I46_J,axiom,
! [K_2] : hAPP_int_int(bit1,K_2) != pls ).
fof(fact_132_rel__simps_I39_J,axiom,
! [L_2] : pls != hAPP_int_int(bit1,L_2) ).
fof(fact_133_rel__simps_I50_J,axiom,
! [K_2,L_2] : hAPP_int_int(bit1,K_2) != hAPP_int_int(bit0,L_2) ).
fof(fact_134_rel__simps_I49_J,axiom,
! [K_2,L_2] : hAPP_int_int(bit0,K_2) != hAPP_int_int(bit1,L_2) ).
fof(fact_135_rel__simps_I44_J,axiom,
! [K] :
( is_int(K)
=> ( hAPP_int_int(bit0,K) = pls
<=> K = pls ) ) ).
fof(fact_136_rel__simps_I38_J,axiom,
! [L] :
( is_int(L)
=> ( pls = hAPP_int_int(bit0,L)
<=> pls = L ) ) ).
fof(fact_137_Bit0__Pls,axiom,
hAPP_int_int(bit0,pls) = pls ).
fof(fact_138_mem__def,axiom,
! [Xa,A] :
( hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,Xa),A))
<=> hBOOL(hAPP_P880235623l_bool(A,Xa)) ) ).
fof(fact_139_mem__def,axiom,
! [Xa,A] :
( hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,Xa),A))
<=> hBOOL(hAPP_P62333565t_bool(A,Xa)) ) ).
fof(fact_140_mem__def,axiom,
! [Xa,A] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),A))
<=> hBOOL(hAPP_real_bool(A,Xa)) ) ).
fof(fact_141_mem__def,axiom,
! [Xa,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),A))
<=> hBOOL(hAPP_nat_bool(A,Xa)) ) ).
fof(fact_142_mem__def,axiom,
! [Xa,A] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,Xa),A))
<=> hBOOL(hAPP_f448129468l_bool(A,Xa)) ) ).
fof(fact_143_mem__def,axiom,
! [Xa,A] :
( hBOOL(hAPP_f115998247l_bool(hAPP_P1876494581l_bool(member180897546at_nat,Xa),A))
<=> hBOOL(hAPP_P1463840893t_bool(A,Xa)) ) ).
fof(fact_144_mem__def,axiom,
! [Xa,A] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A))
<=> hBOOL(hAPP_int_bool(A,Xa)) ) ).
fof(fact_145_Collect__def,axiom,
! [P_1] : collec1979865426at_nat(P_1) = P_1 ).
fof(fact_146_Collect__def,axiom,
! [P_1] : collect_real(P_1) = P_1 ).
fof(fact_147_Collect__def,axiom,
! [P_1] : collec1347809874nt_int(P_1) = P_1 ).
fof(fact_148_Collect__def,axiom,
! [P_1] : collec50511176nt_int(P_1) = P_1 ).
fof(fact_149_Collect__def,axiom,
! [P_1] : collect_nat(P_1) = P_1 ).
fof(fact_150_Collect__def,axiom,
! [P_1] : collect_int(P_1) = P_1 ).
fof(fact_151_Pls__def,axiom,
pls = zero_zero_int ).
fof(fact_152_int__0__neq__1,axiom,
zero_zero_int != one_one_int ).
fof(fact_153_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_154_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_155_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_156_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_157_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_158_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_159_semiring__numeral__0__eq__0,axiom,
hAPP_int_rat(number_number_of_rat,pls) = zero_zero_rat ).
fof(fact_160_semiring__numeral__0__eq__0,axiom,
hAPP_int_complex(number528085621omplex,pls) = zero_zero_complex ).
fof(fact_161_semiring__numeral__0__eq__0,axiom,
hAPP_int_real(number267125858f_real,pls) = zero_zero_real ).
fof(fact_162_semiring__numeral__0__eq__0,axiom,
hAPP_int_nat(number_number_of_nat,pls) = zero_zero_nat ).
fof(fact_163_semiring__numeral__0__eq__0,axiom,
hAPP_int_int(number_number_of_int,pls) = zero_zero_int ).
fof(fact_164_number__of__Pls,axiom,
hAPP_int_rat(number_number_of_rat,pls) = zero_zero_rat ).
fof(fact_165_number__of__Pls,axiom,
hAPP_int_complex(number528085621omplex,pls) = zero_zero_complex ).
fof(fact_166_number__of__Pls,axiom,
hAPP_int_real(number267125858f_real,pls) = zero_zero_real ).
fof(fact_167_number__of__Pls,axiom,
hAPP_int_int(number_number_of_int,pls) = zero_zero_int ).
fof(fact_168_semiring__norm_I112_J,axiom,
zero_zero_rat = hAPP_int_rat(number_number_of_rat,pls) ).
fof(fact_169_semiring__norm_I112_J,axiom,
zero_zero_complex = hAPP_int_complex(number528085621omplex,pls) ).
fof(fact_170_semiring__norm_I112_J,axiom,
zero_zero_real = hAPP_int_real(number267125858f_real,pls) ).
fof(fact_171_semiring__norm_I112_J,axiom,
zero_zero_int = hAPP_int_int(number_number_of_int,pls) ).
fof(fact_172_add__numeral__0,axiom,
! [A_203] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,pls)),A_203) = A_203 ).
fof(fact_173_add__numeral__0,axiom,
! [A_203] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,pls)),A_203) = A_203 ).
fof(fact_174_add__numeral__0,axiom,
! [A_203] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,pls)),A_203) = A_203 ).
fof(fact_175_add__numeral__0,axiom,
! [A_203] :
( is_int(A_203)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,pls)),A_203) = A_203 ) ).
fof(fact_176_add__numeral__0__right,axiom,
! [A_202] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_202),hAPP_int_rat(number_number_of_rat,pls)) = A_202 ).
fof(fact_177_add__numeral__0__right,axiom,
! [A_202] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_202),hAPP_int_complex(number528085621omplex,pls)) = A_202 ).
fof(fact_178_add__numeral__0__right,axiom,
! [A_202] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_202),hAPP_int_real(number267125858f_real,pls)) = A_202 ).
fof(fact_179_add__numeral__0__right,axiom,
! [A_202] :
( is_int(A_202)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_202),hAPP_int_int(number_number_of_int,pls)) = A_202 ) ).
fof(fact_180_power__eq__0__iff__number__of,axiom,
! [A_1,Wa] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_zero_rat
<=> ( A_1 = zero_zero_rat
& hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat ) ) ).
fof(fact_181_power__eq__0__iff__number__of,axiom,
! [A_1,Wa] :
( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_z891286103de_int
<=> ( A_1 = zero_z891286103de_int
& hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat ) ) ).
fof(fact_182_power__eq__0__iff__number__of,axiom,
! [A_1,Wa] :
( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_zero_complex
<=> ( A_1 = zero_zero_complex
& hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat ) ) ).
fof(fact_183_power__eq__0__iff__number__of,axiom,
! [A_1,Wa] :
( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_z126310315umeral
<=> ( A_1 = zero_z126310315umeral
& hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat ) ) ).
fof(fact_184_power__eq__0__iff__number__of,axiom,
! [A_1,Wa] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_zero_real
<=> ( A_1 = zero_zero_real
& hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat ) ) ).
fof(fact_185_power__eq__0__iff__number__of,axiom,
! [A_1,Wa] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_zero_nat
<=> ( A_1 = zero_zero_nat
& hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat ) ) ).
fof(fact_186_power__eq__0__iff__number__of,axiom,
! [A_1,Wa] :
( is_int(A_1)
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),hAPP_int_nat(number_number_of_nat,Wa)) = zero_zero_int
<=> ( A_1 = zero_zero_int
& hAPP_int_nat(number_number_of_nat,Wa) != zero_zero_nat ) ) ) ).
fof(fact_187_add__number__of__left,axiom,
! [V_24,W_14,Z_21] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_24)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,W_14)),Z_21)) = 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_24),W_14))),Z_21) ).
fof(fact_188_add__number__of__left,axiom,
! [V_24,W_14,Z_21] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_24)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,W_14)),Z_21)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_24),W_14))),Z_21) ).
fof(fact_189_add__number__of__left,axiom,
! [V_24,W_14,Z_21] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_24)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,W_14)),Z_21)) = 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_24),W_14))),Z_21) ).
fof(fact_190_add__number__of__left,axiom,
! [V_24,W_14,Z_21] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_24)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,W_14)),Z_21)) = 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_24),W_14))),Z_21) ).
fof(fact_191_add__number__of__eq,axiom,
! [V_23,W_13] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_23)),hAPP_int_rat(number_number_of_rat,W_13)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),W_13)) ).
fof(fact_192_add__number__of__eq,axiom,
! [V_23,W_13] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_23)),hAPP_int_complex(number528085621omplex,W_13)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),W_13)) ).
fof(fact_193_add__number__of__eq,axiom,
! [V_23,W_13] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_23)),hAPP_int_real(number267125858f_real,W_13)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),W_13)) ).
fof(fact_194_add__number__of__eq,axiom,
! [V_23,W_13] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_23)),hAPP_int_int(number_number_of_int,W_13)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_23),W_13)) ).
fof(fact_195_number__of__add,axiom,
! [V_22,W_12] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_22),W_12)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_22)),hAPP_int_rat(number_number_of_rat,W_12)) ).
fof(fact_196_number__of__add,axiom,
! [V_22,W_12] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_22),W_12)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_22)),hAPP_int_complex(number528085621omplex,W_12)) ).
fof(fact_197_number__of__add,axiom,
! [V_22,W_12] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_22),W_12)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_22)),hAPP_int_real(number267125858f_real,W_12)) ).
fof(fact_198_number__of__add,axiom,
! [V_22,W_12] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_22),W_12)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_22)),hAPP_int_int(number_number_of_int,W_12)) ).
fof(fact_199_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_200_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_201_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_202_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_203_number__of__int,axiom,
! [N_58] : hAPP_int_rat(number_number_of_rat,hAPP_nat_int(semiri1621563631at_int,N_58)) = hAPP_nat_rat(semiri151668891at_rat,N_58) ).
fof(fact_204_number__of__int,axiom,
! [N_58] : hAPP_int_complex(number528085621omplex,hAPP_nat_int(semiri1621563631at_int,N_58)) = hAPP_nat_complex(semiri2020571505omplex,N_58) ).
fof(fact_205_number__of__int,axiom,
! [N_58] : hAPP_int_real(number267125858f_real,hAPP_nat_int(semiri1621563631at_int,N_58)) = hAPP_nat_real(semiri132038758t_real,N_58) ).
fof(fact_206_number__of__int,axiom,
! [N_58] : hAPP_int_int(number_number_of_int,hAPP_nat_int(semiri1621563631at_int,N_58)) = hAPP_nat_int(semiri1621563631at_int,N_58) ).
fof(fact_207_number__of__int,axiom,
! [N_58] : hAPP_int_nat(number_number_of_nat,hAPP_nat_int(semiri1621563631at_int,N_58)) = hAPP_nat_nat(semiri984289939at_nat,N_58) ).
fof(fact_208_zero__less__power2,axiom,
! [A_1] :
( 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_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
<=> A_1 != zero_zero_rat ) ).
fof(fact_209_zero__less__power2,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
<=> A_1 != zero_zero_real ) ).
fof(fact_210_zero__less__power2,axiom,
! [A_1] :
( is_int(A_1)
=> ( 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_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))
<=> A_1 != zero_zero_int ) ) ).
fof(fact_211_power2__less__0,axiom,
! [A_201] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_201),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_rat)) ).
fof(fact_212_power2__less__0,axiom,
! [A_201] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_201),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_real)) ).
fof(fact_213_power2__less__0,axiom,
! [A_201] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_201),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),zero_zero_int)) ).
fof(fact_214_sum__power2__gt__zero__iff,axiom,
! [Xa,Ya] :
( 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,Xa),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,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
<=> ( Xa != zero_zero_rat
| Ya != zero_zero_rat ) ) ).
fof(fact_215_sum__power2__gt__zero__iff,axiom,
! [Xa,Ya] :
( 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,Xa),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)))))))
<=> ( Xa != zero_zero_real
| Ya != zero_zero_real ) ) ).
fof(fact_216_sum__power2__gt__zero__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( 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,Xa),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,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))
<=> ( Xa != zero_zero_int
| Ya != zero_zero_int ) ) ) ).
fof(fact_217_not__sum__power2__lt__zero,axiom,
! [X_64,Y_53] : ~ 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_64),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_53),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat)) ).
fof(fact_218_not__sum__power2__lt__zero,axiom,
! [X_64,Y_53] : ~ 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_64),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_53),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_real)) ).
fof(fact_219_not__sum__power2__lt__zero,axiom,
! [X_64,Y_53] : ~ 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_64),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_53),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int)) ).
fof(fact_220_number__of__Bit0,axiom,
! [W_11] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_11)) = 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_11))),hAPP_int_rat(number_number_of_rat,W_11)) ).
fof(fact_221_number__of__Bit0,axiom,
! [W_11] : hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_11)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),hAPP_int_complex(number528085621omplex,W_11))),hAPP_int_complex(number528085621omplex,W_11)) ).
fof(fact_222_number__of__Bit0,axiom,
! [W_11] : hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_11)) = 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_11))),hAPP_int_real(number267125858f_real,W_11)) ).
fof(fact_223_number__of__Bit0,axiom,
! [W_11] : hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_11)) = 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_11))),hAPP_int_int(number_number_of_int,W_11)) ).
fof(fact_224_number__of__Bit1,axiom,
! [W_10] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,W_10)) = 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_10))),hAPP_int_rat(number_number_of_rat,W_10)) ).
fof(fact_225_number__of__Bit1,axiom,
! [W_10] : hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,W_10)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,W_10))),hAPP_int_complex(number528085621omplex,W_10)) ).
fof(fact_226_number__of__Bit1,axiom,
! [W_10] : hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,W_10)) = 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_10))),hAPP_int_real(number267125858f_real,W_10)) ).
fof(fact_227_number__of__Bit1,axiom,
! [W_10] : hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,W_10)) = 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_10))),hAPP_int_int(number_number_of_int,W_10)) ).
fof(fact_228_semiring__numeral__1__eq__1,axiom,
hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls)) = one_one_rat ).
fof(fact_229_semiring__numeral__1__eq__1,axiom,
hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls)) = one_one_complex ).
fof(fact_230_semiring__numeral__1__eq__1,axiom,
hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls)) = one_one_real ).
fof(fact_231_semiring__numeral__1__eq__1,axiom,
hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)) = one_one_nat ).
fof(fact_232_semiring__numeral__1__eq__1,axiom,
hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls)) = one_one_int ).
fof(fact_233_mn,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,m1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))) ).
fof(fact_234_of__nat__less__two__power,axiom,
! [N_57] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,N_57)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),N_57))) ).
fof(fact_235_of__nat__less__two__power,axiom,
! [N_57] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N_57)),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_57))) ).
fof(fact_236_of__nat__less__two__power,axiom,
! [N_57] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,N_57)),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_57))) ).
fof(fact_237_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_238_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_239_realpow__two__sum__zero__iff,axiom,
! [Xa,Ya] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),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))))) = zero_zero_real
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ).
fof(fact_240_of__nat__0__less__iff,axiom,
! [Na] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_241_of__nat__0__less__iff,axiom,
! [Na] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_242_of__nat__0__less__iff,axiom,
! [Na] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_243_of__nat__0__less__iff,axiom,
! [Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_244_of__nat__0__less__iff,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_245_of__nat__0__less__iff,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_246_one__less__power,axiom,
! [N_56,A_200] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_200))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56))
=> 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_200),N_56))) ) ) ).
fof(fact_247_one__less__power,axiom,
! [N_56,A_200] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_200))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_200),N_56))) ) ) ).
fof(fact_248_one__less__power,axiom,
! [N_56,A_200] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_200))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_200),N_56))) ) ) ).
fof(fact_249_one__less__power,axiom,
! [N_56,A_200] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_200))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_200),N_56))) ) ) ).
fof(fact_250_one__less__power,axiom,
! [N_56,A_200] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_200))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56))
=> 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_200),N_56))) ) ) ).
fof(fact_251_one__less__power,axiom,
! [N_56,A_200] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_200))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_56))
=> 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_200),N_56))) ) ) ).
fof(fact_252_power__0__left,axiom,
! [N_55] :
( ( N_55 = zero_zero_nat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_55) = one_one_rat )
& ( N_55 != zero_zero_nat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,zero_zero_rat),N_55) = zero_zero_rat ) ) ).
fof(fact_253_power__0__left,axiom,
! [N_55] :
( ( N_55 = zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_55) = one_on1684967323de_int )
& ( N_55 != zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,zero_z891286103de_int),N_55) = zero_z891286103de_int ) ) ).
fof(fact_254_power__0__left,axiom,
! [N_55] :
( ( N_55 = zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_55) = one_one_complex )
& ( N_55 != zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,zero_zero_complex),N_55) = zero_zero_complex ) ) ).
fof(fact_255_power__0__left,axiom,
! [N_55] :
( ( N_55 = zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_55) = one_on1645066479umeral )
& ( N_55 != zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,zero_z126310315umeral),N_55) = zero_z126310315umeral ) ) ).
fof(fact_256_power__0__left,axiom,
! [N_55] :
( ( N_55 = zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_55) = one_one_real )
& ( N_55 != zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),N_55) = zero_zero_real ) ) ).
fof(fact_257_power__0__left,axiom,
! [N_55] :
( ( N_55 = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_55) = one_one_nat )
& ( N_55 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N_55) = zero_zero_nat ) ) ).
fof(fact_258_power__0__left,axiom,
! [N_55] :
( ( N_55 = zero_zero_nat
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_55) = one_one_int )
& ( N_55 != zero_zero_nat
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,zero_zero_int),N_55) = zero_zero_int ) ) ).
fof(fact_259_power__strict__decreasing,axiom,
! [A_199,N_54,N_53] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_54),N_53))
=> ( 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_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_199),N_53)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_199),N_54))) ) ) ) ).
fof(fact_260_power__strict__decreasing,axiom,
! [A_199,N_54,N_53] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_54),N_53))
=> ( 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_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_199),N_53)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_199),N_54))) ) ) ) ).
fof(fact_261_power__strict__decreasing,axiom,
! [A_199,N_54,N_53] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_54),N_53))
=> ( 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_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_199),N_53)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_199),N_54))) ) ) ) ).
fof(fact_262_power__strict__decreasing,axiom,
! [A_199,N_54,N_53] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_54),N_53))
=> ( 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_nat_real(hAPP_r474017924t_real(power_power_real,A_199),N_53)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_199),N_54))) ) ) ) ).
fof(fact_263_power__strict__decreasing,axiom,
! [A_199,N_54,N_53] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_54),N_53))
=> ( 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(power_power_nat,A_199),N_53)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_199),N_54))) ) ) ) ).
fof(fact_264_power__strict__decreasing,axiom,
! [A_199,N_54,N_53] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_54),N_53))
=> ( 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_nat_int(hAPP_int_fun_nat_int(power_power_int,A_199),N_53)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_199),N_54))) ) ) ) ).
fof(fact_265_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_266_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_267_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_268_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_269_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_270_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_271_int__gr__induct,axiom,
! [P_1,K,I_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,K),one_one_int)))
=> ( ! [I_2] :
( is_int(I_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),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_272_transfer__int__nat__numerals_I1_J,axiom,
zero_zero_int = hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) ).
fof(fact_273_tn0,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),tn)) ).
fof(fact_274_less__zeroE,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_275_real__zero__not__eq__one,axiom,
zero_zero_real != one_one_real ).
fof(fact_276_less__not__refl,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
fof(fact_277_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_278_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_279_number__of__is__id,axiom,
! [K_2] :
( is_int(K_2)
=> hAPP_int_int(number_number_of_int,K_2) = K_2 ) ).
fof(fact_280_nat__neq__iff,axiom,
! [Ma,Na] :
( Ma != Na
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) ) ) ).
fof(fact_281_nat__add__commute,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M) ).
fof(fact_282_nat__add__left__commute,axiom,
! [X,Y,Z_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Z_1)) ).
fof(fact_283_nat__add__assoc,axiom,
! [M,N,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)),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),K_2)) ).
fof(fact_284_nat__add__left__cancel,axiom,
! [K,Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na)
<=> Ma = Na ) ).
fof(fact_285_nat__add__right__cancel,axiom,
! [Ma,K,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Na),K)
<=> Ma = Na ) ).
fof(fact_286_nat__add__left__cancel__less,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_287_linorder__neqE__nat,axiom,
! [X,Y] :
( X != Y
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).
fof(fact_288_less__irrefl__nat,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
fof(fact_289_less__not__refl2,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> M != N ) ).
fof(fact_290_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_291_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_292_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_293_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_294_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_295_less__add__eq__less,axiom,
! [M,N,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)
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
fof(fact_296_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_297_nat__less__cases,axiom,
! [P_1,Ma,Na] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) )
=> ( ( Ma = Na
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Na),Ma)) ) ) ) ).
fof(fact_298_gr0I,axiom,
! [N] :
( N != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_299_gr__implies__not0,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> N != zero_zero_nat ) ).
fof(fact_300_nat__power__less__imp__less,axiom,
! [M,N,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)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
fof(fact_301_less__nat__zero__code,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_302_nat__zero__less__power__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa))
| Na = zero_zero_nat ) ) ).
fof(fact_303_add__gr__0,axiom,
! [Ma,Na] :
( 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),Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ) ).
fof(fact_304_neq0__conv,axiom,
! [Na] :
( Na != zero_zero_nat
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_305_not__less0,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
fof(fact_306_zero__less__power__nat__eq,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),Na)))
<=> ( Na = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) ) ) ).
fof(fact_307_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ).
fof(fact_308_zero__less__power__nat__eq__number__of,axiom,
! [Xa,Wa] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),hAPP_int_nat(number_number_of_nat,Wa))))
<=> ( hAPP_int_nat(number_number_of_nat,Wa) = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Xa)) ) ) ).
fof(fact_309_zless__int,axiom,
! [Ma,Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_310_less__nat__number__of,axiom,
! [Va,V] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_int_nat(number_number_of_nat,V)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V)) )
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Va),V)) ) ) ).
fof(fact_311_linorder__neqE__linordered__idom,axiom,
! [X_63,Y_52] :
( X_63 != Y_52
=> ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_63),Y_52))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_52),X_63)) ) ) ).
fof(fact_312_linorder__neqE__linordered__idom,axiom,
! [X_63,Y_52] :
( X_63 != Y_52
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_63),Y_52))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_52),X_63)) ) ) ).
fof(fact_313_linorder__neqE__linordered__idom,axiom,
! [X_63,Y_52] :
( ( is_int(X_63)
& is_int(Y_52) )
=> ( X_63 != Y_52
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_63),Y_52))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_52),X_63)) ) ) ) ).
fof(fact_314_add__eq__self__zero,axiom,
! [M,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = M
=> N = zero_zero_nat ) ).
fof(fact_315_add__is__0,axiom,
! [Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na) = zero_zero_nat
<=> ( Ma = zero_zero_nat
& Na = zero_zero_nat ) ) ).
fof(fact_316_Nat_Oadd__0__right,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).
fof(fact_317_plus__nat_Oadd__0,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).
fof(fact_318_power__one__right,axiom,
! [A_198] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_198),one_one_nat) = A_198 ).
fof(fact_319_power__one__right,axiom,
! [A_198] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_198),one_one_nat) = A_198 ).
fof(fact_320_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_321_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_322_of__nat__eq__iff,axiom,
! [Ma,Na] :
( hAPP_nat_rat(semiri151668891at_rat,Ma) = hAPP_nat_rat(semiri151668891at_rat,Na)
<=> Ma = Na ) ).
fof(fact_323_of__nat__eq__iff,axiom,
! [Ma,Na] :
( hAPP_nat_complex(semiri2020571505omplex,Ma) = hAPP_nat_complex(semiri2020571505omplex,Na)
<=> Ma = Na ) ).
fof(fact_324_of__nat__eq__iff,axiom,
! [Ma,Na] :
( hAPP_nat_real(semiri132038758t_real,Ma) = hAPP_nat_real(semiri132038758t_real,Na)
<=> Ma = Na ) ).
fof(fact_325_of__nat__eq__iff,axiom,
! [Ma,Na] :
( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_nat_int(semiri1621563631at_int,Na)
<=> Ma = Na ) ).
fof(fact_326_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [Xa,Ya] :
( hAPP_nat_int(semiri1621563631at_int,Xa) = hAPP_nat_int(semiri1621563631at_int,Ya)
<=> Xa = Ya ) ).
fof(fact_327_int__if__cong,axiom,
! [Xa,Ya,P_1] :
( ( hBOOL(P_1)
=> hAPP_nat_int(semiri1621563631at_int,Xa) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),Xa),Ya)) )
& ( ~ hBOOL(P_1)
=> hAPP_nat_int(semiri1621563631at_int,Ya) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,P_1),Xa),Ya)) ) ) ).
fof(fact_328_less__0__number__of,axiom,
! [Va] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,Va)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),Va)) ) ).
fof(fact_329_zero__less__int__conv,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_330_one__neq__zero,axiom,
one_one_rat != zero_zero_rat ).
fof(fact_331_one__neq__zero,axiom,
one_on1684967323de_int != zero_z891286103de_int ).
fof(fact_332_one__neq__zero,axiom,
one_one_complex != zero_zero_complex ).
fof(fact_333_one__neq__zero,axiom,
one_on1645066479umeral != zero_z126310315umeral ).
fof(fact_334_one__neq__zero,axiom,
one_one_real != zero_zero_real ).
fof(fact_335_one__neq__zero,axiom,
one_one_nat != zero_zero_nat ).
fof(fact_336_one__neq__zero,axiom,
one_one_int != zero_zero_int ).
fof(fact_337_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
fof(fact_338_zero__neq__one,axiom,
zero_z891286103de_int != one_on1684967323de_int ).
fof(fact_339_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
fof(fact_340_zero__neq__one,axiom,
zero_z126310315umeral != one_on1645066479umeral ).
fof(fact_341_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
fof(fact_342_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
fof(fact_343_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
fof(fact_344_field__power__not__zero,axiom,
! [N_52,A_197] :
( A_197 != zero_zero_rat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_197),N_52) != zero_zero_rat ) ).
fof(fact_345_field__power__not__zero,axiom,
! [N_52,A_197] :
( A_197 != zero_zero_complex
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_197),N_52) != zero_zero_complex ) ).
fof(fact_346_field__power__not__zero,axiom,
! [N_52,A_197] :
( A_197 != zero_zero_real
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_197),N_52) != zero_zero_real ) ).
fof(fact_347_field__power__not__zero,axiom,
! [N_52,A_197] :
( is_int(A_197)
=> ( A_197 != zero_zero_int
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_197),N_52) != zero_zero_int ) ) ).
fof(fact_348_power__one,axiom,
! [N_51] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,one_one_rat),N_51) = one_one_rat ).
fof(fact_349_power__one,axiom,
! [N_51] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,one_on1684967323de_int),N_51) = one_on1684967323de_int ).
fof(fact_350_power__one,axiom,
! [N_51] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,one_one_complex),N_51) = one_one_complex ).
fof(fact_351_power__one,axiom,
! [N_51] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,one_on1645066479umeral),N_51) = one_on1645066479umeral ).
fof(fact_352_power__one,axiom,
! [N_51] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,one_one_real),N_51) = one_one_real ).
fof(fact_353_power__one,axiom,
! [N_51] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,one_one_nat),N_51) = one_one_nat ).
fof(fact_354_power__one,axiom,
! [N_51] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,one_one_int),N_51) = one_one_int ).
fof(fact_355_of__nat__less__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_356_of__nat__less__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_357_of__nat__less__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_358_of__nat__less__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_359_of__nat__less__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_360_less__imp__of__nat__less,axiom,
! [M_26,N_50] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_26),N_50))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_26)),hAPP_nat_rat(semiri151668891at_rat,N_50))) ) ).
fof(fact_361_less__imp__of__nat__less,axiom,
! [M_26,N_50] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_26),N_50))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_26)),hAPP_n522471361de_int(semiri1424489471de_int,N_50))) ) ).
fof(fact_362_less__imp__of__nat__less,axiom,
! [M_26,N_50] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_26),N_50))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_26)),hAPP_nat_real(semiri132038758t_real,N_50))) ) ).
fof(fact_363_less__imp__of__nat__less,axiom,
! [M_26,N_50] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_26),N_50))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_26)),hAPP_nat_nat(semiri984289939at_nat,N_50))) ) ).
fof(fact_364_less__imp__of__nat__less,axiom,
! [M_26,N_50] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_26),N_50))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_26)),hAPP_nat_int(semiri1621563631at_int,N_50))) ) ).
fof(fact_365_of__nat__less__imp__less,axiom,
! [M_25,N_49] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_25)),hAPP_nat_rat(semiri151668891at_rat,N_49)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_25),N_49)) ) ).
fof(fact_366_of__nat__less__imp__less,axiom,
! [M_25,N_49] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_25)),hAPP_n522471361de_int(semiri1424489471de_int,N_49)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_25),N_49)) ) ).
fof(fact_367_of__nat__less__imp__less,axiom,
! [M_25,N_49] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_25)),hAPP_nat_real(semiri132038758t_real,N_49)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_25),N_49)) ) ).
fof(fact_368_of__nat__less__imp__less,axiom,
! [M_25,N_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_25)),hAPP_nat_nat(semiri984289939at_nat,N_49)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_25),N_49)) ) ).
fof(fact_369_of__nat__less__imp__less,axiom,
! [M_25,N_49] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_25)),hAPP_nat_int(semiri1621563631at_int,N_49)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_25),N_49)) ) ).
fof(fact_370_of__nat__add,axiom,
! [M_24,N_48] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_nat_rat(semiri151668891at_rat,M_24)),hAPP_nat_rat(semiri151668891at_rat,N_48)) ).
fof(fact_371_of__nat__add,axiom,
! [M_24,N_48] : hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_24)),hAPP_n522471361de_int(semiri1424489471de_int,N_48)) ).
fof(fact_372_of__nat__add,axiom,
! [M_24,N_48] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_nat_complex(semiri2020571505omplex,M_24)),hAPP_nat_complex(semiri2020571505omplex,N_48)) ).
fof(fact_373_of__nat__add,axiom,
! [M_24,N_48] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(semiri132038758t_real,M_24)),hAPP_nat_real(semiri132038758t_real,N_48)) ).
fof(fact_374_of__nat__add,axiom,
! [M_24,N_48] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(semiri984289939at_nat,M_24)),hAPP_nat_nat(semiri984289939at_nat,N_48)) ).
fof(fact_375_of__nat__add,axiom,
! [M_24,N_48] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_24),N_48)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,M_24)),hAPP_nat_int(semiri1621563631at_int,N_48)) ).
fof(fact_376_of__nat__1,axiom,
hAPP_nat_rat(semiri151668891at_rat,one_one_nat) = one_one_rat ).
fof(fact_377_of__nat__1,axiom,
hAPP_n522471361de_int(semiri1424489471de_int,one_one_nat) = one_on1684967323de_int ).
fof(fact_378_of__nat__1,axiom,
hAPP_nat_complex(semiri2020571505omplex,one_one_nat) = one_one_complex ).
fof(fact_379_of__nat__1,axiom,
hAPP_n1108039445umeral(semiri1619134803umeral,one_one_nat) = one_on1645066479umeral ).
fof(fact_380_of__nat__1,axiom,
hAPP_nat_real(semiri132038758t_real,one_one_nat) = one_one_real ).
fof(fact_381_of__nat__1,axiom,
hAPP_nat_nat(semiri984289939at_nat,one_one_nat) = one_one_nat ).
fof(fact_382_of__nat__1,axiom,
hAPP_nat_int(semiri1621563631at_int,one_one_nat) = one_one_int ).
fof(fact_383_of__nat__power,axiom,
! [M_23,N_47] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(semiri2020571505omplex,M_23)),N_47) ).
fof(fact_384_of__nat__power,axiom,
! [M_23,N_47] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(semiri132038758t_real,M_23)),N_47) ).
fof(fact_385_of__nat__power,axiom,
! [M_23,N_47] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(semiri984289939at_nat,M_23)),N_47) ).
fof(fact_386_of__nat__power,axiom,
! [M_23,N_47] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,M_23)),N_47) ).
fof(fact_387_of__nat__power,axiom,
! [M_23,N_47] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_23),N_47)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_nat_rat(semiri151668891at_rat,M_23)),N_47) ).
fof(fact_388_transfer__int__nat__numerals_I2_J,axiom,
one_one_int = hAPP_nat_int(semiri1621563631at_int,one_one_nat) ).
fof(fact_389_Nat__Transfer_Otransfer__int__nat__functions_I1_J,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Y)) ).
fof(fact_390_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X,N] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(semiri1621563631at_int,X)),N) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)) ).
fof(fact_391_pos__add__strict,axiom,
! [B_134,C_78,A_196] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_196))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_134),C_78))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_134),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_196),C_78))) ) ) ).
fof(fact_392_pos__add__strict,axiom,
! [B_134,C_78,A_196] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_196))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_134),C_78))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_134),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_196),C_78))) ) ) ).
fof(fact_393_pos__add__strict,axiom,
! [B_134,C_78,A_196] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_196))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_134),C_78))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_134),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_196),C_78))) ) ) ).
fof(fact_394_pos__add__strict,axiom,
! [B_134,C_78,A_196] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_196))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_134),C_78))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_134),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_196),C_78))) ) ) ).
fof(fact_395_pos__add__strict,axiom,
! [B_134,C_78,A_196] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_196))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_134),C_78))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_134),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_196),C_78))) ) ) ).
fof(fact_396_pos__add__strict,axiom,
! [B_134,C_78,A_196] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_196))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_134),C_78))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_134),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_196),C_78))) ) ) ).
fof(fact_397_not__one__less__zero,axiom,
~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),zero_zero_rat)) ).
fof(fact_398_not__one__less__zero,axiom,
~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),zero_z891286103de_int)) ).
fof(fact_399_not__one__less__zero,axiom,
~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),zero_z126310315umeral)) ).
fof(fact_400_not__one__less__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),zero_zero_real)) ).
fof(fact_401_not__one__less__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),zero_zero_nat)) ).
fof(fact_402_not__one__less__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),zero_zero_int)) ).
fof(fact_403_zero__less__one,axiom,
hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),one_one_rat)) ).
fof(fact_404_zero__less__one,axiom,
hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),one_on1684967323de_int)) ).
fof(fact_405_zero__less__one,axiom,
hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),one_on1645066479umeral)) ).
fof(fact_406_zero__less__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),one_one_real)) ).
fof(fact_407_zero__less__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),one_one_nat)) ).
fof(fact_408_zero__less__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),one_one_int)) ).
fof(fact_409_zero__less__power,axiom,
! [N_46,A_195] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_195))
=> 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_195),N_46))) ) ).
fof(fact_410_zero__less__power,axiom,
! [N_46,A_195] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_195))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_195),N_46))) ) ).
fof(fact_411_zero__less__power,axiom,
! [N_46,A_195] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_195))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_195),N_46))) ) ).
fof(fact_412_zero__less__power,axiom,
! [N_46,A_195] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_195))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_195),N_46))) ) ).
fof(fact_413_zero__less__power,axiom,
! [N_46,A_195] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_195))
=> 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_195),N_46))) ) ).
fof(fact_414_zero__less__power,axiom,
! [N_46,A_195] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_195))
=> 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_195),N_46))) ) ).
fof(fact_415_less__add__one,axiom,
! [A_194] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_194),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_194),one_one_rat))) ).
fof(fact_416_less__add__one,axiom,
! [A_194] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_194),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_194),one_on1684967323de_int))) ).
fof(fact_417_less__add__one,axiom,
! [A_194] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_194),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_194),one_on1645066479umeral))) ).
fof(fact_418_less__add__one,axiom,
! [A_194] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_194),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_194),one_one_real))) ).
fof(fact_419_less__add__one,axiom,
! [A_194] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_194),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_194),one_one_nat))) ).
fof(fact_420_less__add__one,axiom,
! [A_194] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_194),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_194),one_one_int))) ).
fof(fact_421_power__inject__exp,axiom,
! [Ma,Na,A_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_1))
=> ( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),Ma) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),Na)
<=> Ma = Na ) ) ).
fof(fact_422_power__inject__exp,axiom,
! [Ma,Na,A_1] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_1))
=> ( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_1),Ma) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_1),Na)
<=> Ma = Na ) ) ).
fof(fact_423_power__inject__exp,axiom,
! [Ma,Na,A_1] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_1))
=> ( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_1),Ma) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_1),Na)
<=> Ma = Na ) ) ).
fof(fact_424_power__inject__exp,axiom,
! [Ma,Na,A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_1))
=> ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),Ma) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),Na)
<=> Ma = Na ) ) ).
fof(fact_425_power__inject__exp,axiom,
! [Ma,Na,A_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_1))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),Na)
<=> Ma = Na ) ) ).
fof(fact_426_power__inject__exp,axiom,
! [Ma,Na,A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1))
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),Ma) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),Na)
<=> Ma = Na ) ) ).
fof(fact_427_power__strict__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B_1))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_1),Xa)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
fof(fact_428_power__strict__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B_1))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_1),Xa)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
fof(fact_429_power__strict__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B_1))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_1),Xa)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
fof(fact_430_power__strict__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_1),Xa)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
fof(fact_431_power__strict__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
fof(fact_432_power__strict__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),Xa)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ) ).
fof(fact_433_power__less__imp__less__exp,axiom,
! [M_22,N_45,A_193] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_193))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_193),M_22)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_193),N_45)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45)) ) ) ).
fof(fact_434_power__less__imp__less__exp,axiom,
! [M_22,N_45,A_193] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_193))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_193),M_22)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_193),N_45)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45)) ) ) ).
fof(fact_435_power__less__imp__less__exp,axiom,
! [M_22,N_45,A_193] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_193))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_193),M_22)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_193),N_45)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45)) ) ) ).
fof(fact_436_power__less__imp__less__exp,axiom,
! [M_22,N_45,A_193] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_193))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_193),M_22)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_193),N_45)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45)) ) ) ).
fof(fact_437_power__less__imp__less__exp,axiom,
! [M_22,N_45,A_193] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_193))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_193),M_22)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_193),N_45)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45)) ) ) ).
fof(fact_438_power__less__imp__less__exp,axiom,
! [M_22,N_45,A_193] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_193))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_193),M_22)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_193),N_45)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_22),N_45)) ) ) ).
fof(fact_439_power__strict__increasing,axiom,
! [A_192,N_44,N_43] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_44),N_43))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_192))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_192),N_44)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_192),N_43))) ) ) ).
fof(fact_440_power__strict__increasing,axiom,
! [A_192,N_44,N_43] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_44),N_43))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_192))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_192),N_44)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_192),N_43))) ) ) ).
fof(fact_441_power__strict__increasing,axiom,
! [A_192,N_44,N_43] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_44),N_43))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_192))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_192),N_44)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_192),N_43))) ) ) ).
fof(fact_442_power__strict__increasing,axiom,
! [A_192,N_44,N_43] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_44),N_43))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_192))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_192),N_44)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_192),N_43))) ) ) ).
fof(fact_443_power__strict__increasing,axiom,
! [A_192,N_44,N_43] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_44),N_43))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_192))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_192),N_44)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_192),N_43))) ) ) ).
fof(fact_444_power__strict__increasing,axiom,
! [A_192,N_44,N_43] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_44),N_43))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_192))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_192),N_44)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_192),N_43))) ) ) ).
fof(fact_445_power__eq__0__iff,axiom,
! [A_1,Na] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_1),Na) = zero_zero_rat
<=> ( A_1 = zero_zero_rat
& Na != zero_zero_nat ) ) ).
fof(fact_446_power__eq__0__iff,axiom,
! [A_1,Na] :
( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_1),Na) = zero_z891286103de_int
<=> ( A_1 = zero_z891286103de_int
& Na != zero_zero_nat ) ) ).
fof(fact_447_power__eq__0__iff,axiom,
! [A_1,Na] :
( hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_1),Na) = zero_zero_complex
<=> ( A_1 = zero_zero_complex
& Na != zero_zero_nat ) ) ).
fof(fact_448_power__eq__0__iff,axiom,
! [A_1,Na] :
( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_1),Na) = zero_z126310315umeral
<=> ( A_1 = zero_z126310315umeral
& Na != zero_zero_nat ) ) ).
fof(fact_449_power__eq__0__iff,axiom,
! [A_1,Na] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_1),Na) = zero_zero_real
<=> ( A_1 = zero_zero_real
& Na != zero_zero_nat ) ) ).
fof(fact_450_power__eq__0__iff,axiom,
! [A_1,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),Na) = zero_zero_nat
<=> ( A_1 = zero_zero_nat
& Na != zero_zero_nat ) ) ).
fof(fact_451_power__eq__0__iff,axiom,
! [A_1,Na] :
( is_int(A_1)
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),Na) = zero_zero_int
<=> ( A_1 = zero_zero_int
& Na != zero_zero_nat ) ) ) ).
fof(fact_452_of__nat__less__0__iff,axiom,
! [M_21] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(semiri151668891at_rat,M_21)),zero_zero_rat)) ).
fof(fact_453_of__nat__less__0__iff,axiom,
! [M_21] : ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_21)),zero_z891286103de_int)) ).
fof(fact_454_of__nat__less__0__iff,axiom,
! [M_21] : ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(semiri1619134803umeral,M_21)),zero_z126310315umeral)) ).
fof(fact_455_of__nat__less__0__iff,axiom,
! [M_21] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(semiri132038758t_real,M_21)),zero_zero_real)) ).
fof(fact_456_of__nat__less__0__iff,axiom,
! [M_21] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(semiri984289939at_nat,M_21)),zero_zero_nat)) ).
fof(fact_457_of__nat__less__0__iff,axiom,
! [M_21] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,M_21)),zero_zero_int)) ).
fof(fact_458_power__0,axiom,
! [A_191] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_191),zero_zero_nat) = one_one_rat ).
fof(fact_459_power__0,axiom,
! [A_191] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_191),zero_zero_nat) = one_on1684967323de_int ).
fof(fact_460_power__0,axiom,
! [A_191] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_191),zero_zero_nat) = one_one_complex ).
fof(fact_461_power__0,axiom,
! [A_191] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_191),zero_zero_nat) = one_on1645066479umeral ).
fof(fact_462_power__0,axiom,
! [A_191] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_191),zero_zero_nat) = one_one_real ).
fof(fact_463_power__0,axiom,
! [A_191] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_191),zero_zero_nat) = one_one_nat ).
fof(fact_464_power__0,axiom,
! [A_191] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_191),zero_zero_nat) = one_one_int ).
fof(fact_465_of__nat__0,axiom,
hAPP_nat_rat(semiri151668891at_rat,zero_zero_nat) = zero_zero_rat ).
fof(fact_466_of__nat__0,axiom,
hAPP_n522471361de_int(semiri1424489471de_int,zero_zero_nat) = zero_z891286103de_int ).
fof(fact_467_of__nat__0,axiom,
hAPP_nat_complex(semiri2020571505omplex,zero_zero_nat) = zero_zero_complex ).
fof(fact_468_of__nat__0,axiom,
hAPP_n1108039445umeral(semiri1619134803umeral,zero_zero_nat) = zero_z126310315umeral ).
fof(fact_469_of__nat__0,axiom,
hAPP_nat_real(semiri132038758t_real,zero_zero_nat) = zero_zero_real ).
fof(fact_470_of__nat__0,axiom,
hAPP_nat_nat(semiri984289939at_nat,zero_zero_nat) = zero_zero_nat ).
fof(fact_471_of__nat__0,axiom,
hAPP_nat_int(semiri1621563631at_int,zero_zero_nat) = zero_zero_int ).
fof(fact_472_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_473_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_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
& K_2 = hAPP_nat_int(semiri1621563631at_int,N_1) ) ) ) ).
fof(fact_474_less__imp__add__positive,axiom,
! [I,J_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ? [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(plus_plus_nat,I),K_1) = J_2 ) ) ).
fof(fact_475_exp__eq__1,axiom,
! [Xa,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),Na) = one_one_nat
<=> ( Xa = one_one_nat
| Na = zero_zero_nat ) ) ).
fof(fact_476_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_62] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_62),zero_zero_nat) = one_one_rat ).
fof(fact_477_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_62] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_62),zero_zero_nat) = one_on1684967323de_int ).
fof(fact_478_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_62] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_62),zero_zero_nat) = one_one_complex ).
fof(fact_479_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_62] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_62),zero_zero_nat) = one_on1645066479umeral ).
fof(fact_480_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_62] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_62),zero_zero_nat) = one_one_real ).
fof(fact_481_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_62] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_62),zero_zero_nat) = one_one_nat ).
fof(fact_482_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_62] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_62),zero_zero_nat) = one_one_int ).
fof(fact_483_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_1] :
( 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_1),A_1)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_1)) ) ).
fof(fact_484_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_1)) ) ).
fof(fact_485_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A_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,A_1),A_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1)) ) ).
fof(fact_486_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),A_1)),zero_zero_rat))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),zero_zero_rat)) ) ).
fof(fact_487_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),zero_zero_real)) ) ).
fof(fact_488_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),A_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)) ) ).
fof(fact_489_add__pos__pos,axiom,
! [B_133,A_190] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_190))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_133))
=> 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_190),B_133))) ) ) ).
fof(fact_490_add__pos__pos,axiom,
! [B_133,A_190] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_190))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_133))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_190),B_133))) ) ) ).
fof(fact_491_add__pos__pos,axiom,
! [B_133,A_190] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_190))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_133))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_190),B_133))) ) ) ).
fof(fact_492_add__pos__pos,axiom,
! [B_133,A_190] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_190))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_133))
=> 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_190),B_133))) ) ) ).
fof(fact_493_add__pos__pos,axiom,
! [B_133,A_190] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_190))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_133))
=> 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_190),B_133))) ) ) ).
fof(fact_494_add__pos__pos,axiom,
! [B_133,A_190] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_190))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_133))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_190),B_133))) ) ) ).
fof(fact_495_add__neg__neg,axiom,
! [B_132,A_189] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_189),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_132),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_189),B_132)),zero_zero_rat)) ) ) ).
fof(fact_496_add__neg__neg,axiom,
! [B_132,A_189] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_189),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_132),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_189),B_132)),zero_z891286103de_int)) ) ) ).
fof(fact_497_add__neg__neg,axiom,
! [B_132,A_189] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_189),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_132),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_189),B_132)),zero_zero_real)) ) ) ).
fof(fact_498_add__neg__neg,axiom,
! [B_132,A_189] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_189),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_132),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_189),B_132)),zero_zero_nat)) ) ) ).
fof(fact_499_add__neg__neg,axiom,
! [B_132,A_189] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_189),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_132),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_189),B_132)),zero_zero_int)) ) ) ).
fof(fact_500_add__neg__neg,axiom,
! [B_132,A_189] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_189),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_132),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_189),B_132)),zero_z126310315umeral)) ) ) ).
fof(fact_501_tpos,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),t)) ).
fof(fact_502_nat__1__add__number__of,axiom,
! [V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_1)) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_1)) ) ) ).
fof(fact_503_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_504_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_505_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_506_zle__refl,axiom,
! [W] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,W),W)) ).
fof(fact_507_le__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,Xa)),hAPP_int_rat(number_number_of_rat,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ).
fof(fact_508_le__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Xa)),hAPP_int_real(number267125858f_real,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ).
fof(fact_509_le__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Xa)),hAPP_int_int(number_number_of_int,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ).
fof(fact_510_of__nat__le__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(semiri151668891at_rat,Ma)),hAPP_nat_rat(semiri151668891at_rat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_511_of__nat__le__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(semiri132038758t_real,Ma)),hAPP_nat_real(semiri132038758t_real,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_512_of__nat__le__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_513_of__nat__le__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(semiri1424489471de_int,Ma)),hAPP_n522471361de_int(semiri1424489471de_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_514_of__nat__le__iff,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(semiri984289939at_nat,Ma)),hAPP_nat_nat(semiri984289939at_nat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_515_add__le__imp__le__left,axiom,
! [C_77,A_188,B_131] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_77),A_188)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_77),B_131)))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_188),B_131)) ) ).
fof(fact_516_add__le__imp__le__left,axiom,
! [C_77,A_188,B_131] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_77),A_188)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_77),B_131)))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_188),B_131)) ) ).
fof(fact_517_add__le__imp__le__left,axiom,
! [C_77,A_188,B_131] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_77),A_188)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_77),B_131)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_188),B_131)) ) ).
fof(fact_518_add__le__imp__le__left,axiom,
! [C_77,A_188,B_131] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_77),A_188)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_77),B_131)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_188),B_131)) ) ).
fof(fact_519_add__le__imp__le__left,axiom,
! [C_77,A_188,B_131] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_77),A_188)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_77),B_131)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_188),B_131)) ) ).
fof(fact_520_add__le__imp__le__right,axiom,
! [A_187,C_76,B_130] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_187),C_76)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_130),C_76)))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_187),B_130)) ) ).
fof(fact_521_add__le__imp__le__right,axiom,
! [A_187,C_76,B_130] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_187),C_76)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_130),C_76)))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_187),B_130)) ) ).
fof(fact_522_add__le__imp__le__right,axiom,
! [A_187,C_76,B_130] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_187),C_76)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_130),C_76)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_187),B_130)) ) ).
fof(fact_523_add__le__imp__le__right,axiom,
! [A_187,C_76,B_130] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_187),C_76)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_130),C_76)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_187),B_130)) ) ).
fof(fact_524_add__le__imp__le__right,axiom,
! [A_187,C_76,B_130] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_187),C_76)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_130),C_76)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_187),B_130)) ) ).
fof(fact_525_add__mono,axiom,
! [C_75,D_22,A_186,B_129] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_186),B_129))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_75),D_22))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_186),C_75)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_129),D_22))) ) ) ).
fof(fact_526_add__mono,axiom,
! [C_75,D_22,A_186,B_129] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_186),B_129))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_75),D_22))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_186),C_75)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_129),D_22))) ) ) ).
fof(fact_527_add__mono,axiom,
! [C_75,D_22,A_186,B_129] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_186),B_129))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_75),D_22))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_186),C_75)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_129),D_22))) ) ) ).
fof(fact_528_add__mono,axiom,
! [C_75,D_22,A_186,B_129] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_186),B_129))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_75),D_22))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_186),C_75)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_129),D_22))) ) ) ).
fof(fact_529_add__mono,axiom,
! [C_75,D_22,A_186,B_129] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_186),B_129))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_75),D_22))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_186),C_75)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_129),D_22))) ) ) ).
fof(fact_530_add__left__mono,axiom,
! [C_74,A_185,B_128] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_185),B_128))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_74),A_185)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_74),B_128))) ) ).
fof(fact_531_add__left__mono,axiom,
! [C_74,A_185,B_128] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_185),B_128))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_74),A_185)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_74),B_128))) ) ).
fof(fact_532_add__left__mono,axiom,
! [C_74,A_185,B_128] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_185),B_128))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_74),A_185)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_74),B_128))) ) ).
fof(fact_533_add__left__mono,axiom,
! [C_74,A_185,B_128] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_185),B_128))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_74),A_185)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_74),B_128))) ) ).
fof(fact_534_add__left__mono,axiom,
! [C_74,A_185,B_128] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_185),B_128))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_74),A_185)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_74),B_128))) ) ).
fof(fact_535_add__right__mono,axiom,
! [C_73,A_184,B_127] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_184),B_127))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_184),C_73)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_127),C_73))) ) ).
fof(fact_536_add__right__mono,axiom,
! [C_73,A_184,B_127] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_184),B_127))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_184),C_73)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_127),C_73))) ) ).
fof(fact_537_add__right__mono,axiom,
! [C_73,A_184,B_127] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_184),B_127))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_184),C_73)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_127),C_73))) ) ).
fof(fact_538_add__right__mono,axiom,
! [C_73,A_184,B_127] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_184),B_127))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_184),C_73)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_127),C_73))) ) ).
fof(fact_539_add__right__mono,axiom,
! [C_73,A_184,B_127] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_184),B_127))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_184),C_73)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_127),C_73))) ) ).
fof(fact_540_add__le__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B_1)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),B_1)) ) ).
fof(fact_541_add__le__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A_1)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B_1)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_1),B_1)) ) ).
fof(fact_542_add__le__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) ) ).
fof(fact_543_add__le__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_1)) ) ).
fof(fact_544_add__le__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_1)) ) ).
fof(fact_545_add__le__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_1),C)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),B_1)) ) ).
fof(fact_546_add__le__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_1),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_1),C)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_1),B_1)) ) ).
fof(fact_547_add__le__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),C)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) ) ).
fof(fact_548_add__le__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),C)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_1)) ) ).
fof(fact_549_add__le__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),C)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_1)) ) ).
fof(fact_550_power__increasing,axiom,
! [A_183,N_42,N_41] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_42),N_41))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_183))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_183),N_42)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_183),N_41))) ) ) ).
fof(fact_551_power__increasing,axiom,
! [A_183,N_42,N_41] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_42),N_41))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_183))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_183),N_42)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_183),N_41))) ) ) ).
fof(fact_552_power__increasing,axiom,
! [A_183,N_42,N_41] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_42),N_41))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_183))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_183),N_42)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_183),N_41))) ) ) ).
fof(fact_553_power__increasing,axiom,
! [A_183,N_42,N_41] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_42),N_41))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_183))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_183),N_42)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_183),N_41))) ) ) ).
fof(fact_554_power__increasing,axiom,
! [A_183,N_42,N_41] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_42),N_41))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_183))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_183),N_42)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_183),N_41))) ) ) ).
fof(fact_555_power__increasing,axiom,
! [A_183,N_42,N_41] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_42),N_41))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_183))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_183),N_42)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_183),N_41))) ) ) ).
fof(fact_556_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_557_rel__simps_I34_J,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K)),hAPP_int_int(bit1,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) ) ).
fof(fact_558_rel__simps_I19_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),pls)) ).
fof(fact_559_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_560_rel__simps_I31_J,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K)),hAPP_int_int(bit0,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) ) ).
fof(fact_561_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_562_zless__le,axiom,
! [Z_2,Wa] :
( ( is_int(Z_2)
& is_int(Wa) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_2),Wa))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_2),Wa))
& Z_2 != Wa ) ) ) ).
fof(fact_563_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_564_less__eq__number__of__int__code,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,K)),hAPP_int_int(number_number_of_int,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) ) ).
fof(fact_565_add__nonpos__nonpos,axiom,
! [B_126,A_182] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_182),zero_zero_rat))
=> ( 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(plus_plus_rat,A_182),B_126)),zero_zero_rat)) ) ) ).
fof(fact_566_add__nonpos__nonpos,axiom,
! [B_126,A_182] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_182),zero_z891286103de_int))
=> ( 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(plus_p1446045655de_int,A_182),B_126)),zero_z891286103de_int)) ) ) ).
fof(fact_567_add__nonpos__nonpos,axiom,
! [B_126,A_182] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_182),zero_zero_real))
=> ( 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(plus_plus_real,A_182),B_126)),zero_zero_real)) ) ) ).
fof(fact_568_add__nonpos__nonpos,axiom,
! [B_126,A_182] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_182),zero_zero_nat))
=> ( 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(plus_plus_nat,A_182),B_126)),zero_zero_nat)) ) ) ).
fof(fact_569_add__nonpos__nonpos,axiom,
! [B_126,A_182] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_182),zero_zero_int))
=> ( 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(plus_plus_int,A_182),B_126)),zero_zero_int)) ) ) ).
fof(fact_570_add__nonpos__nonpos,axiom,
! [B_126,A_182] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_182),zero_z126310315umeral))
=> ( 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(plus_p1627245867umeral,A_182),B_126)),zero_z126310315umeral)) ) ) ).
fof(fact_571_add__increasing2,axiom,
! [B_125,A_181,C_72] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_72))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_125),A_181))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_125),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_181),C_72))) ) ) ).
fof(fact_572_add__increasing2,axiom,
! [B_125,A_181,C_72] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_72))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_125),A_181))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_125),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_181),C_72))) ) ) ).
fof(fact_573_add__increasing2,axiom,
! [B_125,A_181,C_72] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_72))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_125),A_181))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_125),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_181),C_72))) ) ) ).
fof(fact_574_add__increasing2,axiom,
! [B_125,A_181,C_72] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_72))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_125),A_181))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_125),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_181),C_72))) ) ) ).
fof(fact_575_add__increasing2,axiom,
! [B_125,A_181,C_72] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_72))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_125),A_181))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_125),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_181),C_72))) ) ) ).
fof(fact_576_add__increasing2,axiom,
! [B_125,A_181,C_72] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_72))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_125),A_181))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_125),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_181),C_72))) ) ) ).
fof(fact_577_add__increasing,axiom,
! [B_124,C_71,A_180] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_180))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_124),C_71))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_124),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_180),C_71))) ) ) ).
fof(fact_578_add__increasing,axiom,
! [B_124,C_71,A_180] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_180))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_124),C_71))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_124),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_180),C_71))) ) ) ).
fof(fact_579_add__increasing,axiom,
! [B_124,C_71,A_180] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_180))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_124),C_71))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_124),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_180),C_71))) ) ) ).
fof(fact_580_add__increasing,axiom,
! [B_124,C_71,A_180] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_180))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_124),C_71))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_124),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_180),C_71))) ) ) ).
fof(fact_581_add__increasing,axiom,
! [B_124,C_71,A_180] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_180))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_124),C_71))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_124),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_180),C_71))) ) ) ).
fof(fact_582_add__increasing,axiom,
! [B_124,C_71,A_180] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_180))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_124),C_71))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_124),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_180),C_71))) ) ) ).
fof(fact_583_add__nonneg__eq__0__iff,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Xa))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Ya))
=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Xa),Ya) = zero_zero_rat
<=> ( Xa = zero_zero_rat
& Ya = zero_zero_rat ) ) ) ) ).
fof(fact_584_add__nonneg__eq__0__iff,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Xa))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Ya))
=> ( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Xa),Ya) = zero_z891286103de_int
<=> ( Xa = zero_z891286103de_int
& Ya = zero_z891286103de_int ) ) ) ) ).
fof(fact_585_add__nonneg__eq__0__iff,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ya))
=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya) = zero_zero_real
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ) ) ).
fof(fact_586_add__nonneg__eq__0__iff,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Ya))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Xa),Ya) = zero_zero_nat
<=> ( Xa = zero_zero_nat
& Ya = zero_zero_nat ) ) ) ) ).
fof(fact_587_add__nonneg__eq__0__iff,axiom,
! [Ya,Xa] :
( ( is_int(Ya)
& is_int(Xa) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),Ya) = zero_zero_int
<=> ( Xa = zero_zero_int
& Ya = zero_zero_int ) ) ) ) ) ).
fof(fact_588_add__nonneg__eq__0__iff,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Xa))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Ya))
=> ( hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,Xa),Ya) = zero_z126310315umeral
<=> ( Xa = zero_z126310315umeral
& Ya = zero_z126310315umeral ) ) ) ) ).
fof(fact_589_add__nonneg__nonneg,axiom,
! [B_123,A_179] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_179))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_123))
=> 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_179),B_123))) ) ) ).
fof(fact_590_add__nonneg__nonneg,axiom,
! [B_123,A_179] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_179))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_123))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_179),B_123))) ) ) ).
fof(fact_591_add__nonneg__nonneg,axiom,
! [B_123,A_179] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_179))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_123))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_179),B_123))) ) ) ).
fof(fact_592_add__nonneg__nonneg,axiom,
! [B_123,A_179] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_179))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_123))
=> 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_179),B_123))) ) ) ).
fof(fact_593_add__nonneg__nonneg,axiom,
! [B_123,A_179] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_179))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_123))
=> 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_179),B_123))) ) ) ).
fof(fact_594_add__nonneg__nonneg,axiom,
! [B_123,A_179] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_179))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_123))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_179),B_123))) ) ) ).
fof(fact_595_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),A_1)),zero_zero_rat))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),zero_zero_rat)) ) ).
fof(fact_596_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real)) ) ).
fof(fact_597_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),A_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int)) ) ).
fof(fact_598_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_1] :
( 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_1),A_1)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_1)) ) ).
fof(fact_599_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_1)) ) ).
fof(fact_600_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A_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,A_1),A_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1)) ) ).
fof(fact_601_add__le__less__mono,axiom,
! [C_70,D_21,A_178,B_122] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_178),B_122))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_70),D_21))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_178),C_70)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_122),D_21))) ) ) ).
fof(fact_602_add__le__less__mono,axiom,
! [C_70,D_21,A_178,B_122] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_178),B_122))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_70),D_21))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_178),C_70)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_122),D_21))) ) ) ).
fof(fact_603_add__le__less__mono,axiom,
! [C_70,D_21,A_178,B_122] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_178),B_122))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_70),D_21))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_178),C_70)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_122),D_21))) ) ) ).
fof(fact_604_add__le__less__mono,axiom,
! [C_70,D_21,A_178,B_122] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_178),B_122))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_70),D_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_178),C_70)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_122),D_21))) ) ) ).
fof(fact_605_add__le__less__mono,axiom,
! [C_70,D_21,A_178,B_122] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_178),B_122))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_70),D_21))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_178),C_70)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_122),D_21))) ) ) ).
fof(fact_606_add__less__le__mono,axiom,
! [C_69,D_20,A_177,B_121] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_177),B_121))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_69),D_20))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_177),C_69)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_121),D_20))) ) ) ).
fof(fact_607_add__less__le__mono,axiom,
! [C_69,D_20,A_177,B_121] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_177),B_121))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_69),D_20))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_177),C_69)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_121),D_20))) ) ) ).
fof(fact_608_add__less__le__mono,axiom,
! [C_69,D_20,A_177,B_121] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_177),B_121))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_69),D_20))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_177),C_69)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_121),D_20))) ) ) ).
fof(fact_609_add__less__le__mono,axiom,
! [C_69,D_20,A_177,B_121] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_177),B_121))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_69),D_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_177),C_69)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_121),D_20))) ) ) ).
fof(fact_610_add__less__le__mono,axiom,
! [C_69,D_20,A_177,B_121] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_177),B_121))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_69),D_20))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_177),C_69)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_121),D_20))) ) ) ).
fof(fact_611_le__special_I3_J,axiom,
! [Xa] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,Xa)),zero_zero_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) ) ).
fof(fact_612_le__special_I3_J,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Xa)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) ) ).
fof(fact_613_le__special_I3_J,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Xa)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),pls)) ) ).
fof(fact_614_le__special_I1_J,axiom,
! [Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(number_number_of_rat,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)) ) ).
fof(fact_615_le__special_I1_J,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(number267125858f_real,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)) ) ).
fof(fact_616_le__special_I1_J,axiom,
! [Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),Ya)) ) ).
fof(fact_617_power__decreasing,axiom,
! [A_176,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,zero_zero_rat),A_176))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_176),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_176),N_39)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_176),N_40))) ) ) ) ).
fof(fact_618_power__decreasing,axiom,
! [A_176,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,zero_z891286103de_int),A_176))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_176),one_on1684967323de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_176),N_39)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_176),N_40))) ) ) ) ).
fof(fact_619_power__decreasing,axiom,
! [A_176,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,zero_z126310315umeral),A_176))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_176),one_on1645066479umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_176),N_39)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_176),N_40))) ) ) ) ).
fof(fact_620_power__decreasing,axiom,
! [A_176,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,zero_zero_real),A_176))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_176),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_176),N_39)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_176),N_40))) ) ) ) ).
fof(fact_621_power__decreasing,axiom,
! [A_176,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,zero_zero_nat),A_176))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_176),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_176),N_39)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_176),N_40))) ) ) ) ).
fof(fact_622_power__decreasing,axiom,
! [A_176,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,zero_zero_int),A_176))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_176),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_176),N_39)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_176),N_40))) ) ) ) ).
fof(fact_623_power__le__imp__le__exp,axiom,
! [M_20,N_38,A_175] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_175))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_175),M_20)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_175),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38)) ) ) ).
fof(fact_624_power__le__imp__le__exp,axiom,
! [M_20,N_38,A_175] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_175))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_175),M_20)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_175),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38)) ) ) ).
fof(fact_625_power__le__imp__le__exp,axiom,
! [M_20,N_38,A_175] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_175))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_175),M_20)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_175),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38)) ) ) ).
fof(fact_626_power__le__imp__le__exp,axiom,
! [M_20,N_38,A_175] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_175))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_175),M_20)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_175),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38)) ) ) ).
fof(fact_627_power__le__imp__le__exp,axiom,
! [M_20,N_38,A_175] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_175))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_175),M_20)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_175),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38)) ) ) ).
fof(fact_628_power__le__imp__le__exp,axiom,
! [M_20,N_38,A_175] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_175))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_175),M_20)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_175),N_38)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_20),N_38)) ) ) ).
fof(fact_629_power__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),B_1))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_1),Xa)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
fof(fact_630_power__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),B_1))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_1),Xa)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
fof(fact_631_power__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),B_1))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_1),Xa)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
fof(fact_632_power__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),B_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_1),Xa)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
fof(fact_633_power__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),B_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
fof(fact_634_power__increasing__iff,axiom,
! [Xa,Ya,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),Xa)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ) ).
fof(fact_635_le__special_I4_J,axiom,
! [Xa] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,Xa)),one_one_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),hAPP_int_int(bit1,pls))) ) ).
fof(fact_636_le__special_I4_J,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Xa)),one_one_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),hAPP_int_int(bit1,pls))) ) ).
fof(fact_637_le__special_I4_J,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Xa)),one_one_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),hAPP_int_int(bit1,pls))) ) ).
fof(fact_638_le__special_I2_J,axiom,
! [Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Ya)) ) ).
fof(fact_639_le__special_I2_J,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_int_real(number267125858f_real,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Ya)) ) ).
fof(fact_640_le__special_I2_J,axiom,
! [Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(number_number_of_int,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,pls)),Ya)) ) ).
fof(fact_641_add__nonpos__neg,axiom,
! [B_120,A_174] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_174),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_120),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_174),B_120)),zero_zero_rat)) ) ) ).
fof(fact_642_add__nonpos__neg,axiom,
! [B_120,A_174] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_174),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_120),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_174),B_120)),zero_z891286103de_int)) ) ) ).
fof(fact_643_add__nonpos__neg,axiom,
! [B_120,A_174] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_174),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_120),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_174),B_120)),zero_zero_real)) ) ) ).
fof(fact_644_add__nonpos__neg,axiom,
! [B_120,A_174] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_174),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_120),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_174),B_120)),zero_zero_nat)) ) ) ).
fof(fact_645_add__nonpos__neg,axiom,
! [B_120,A_174] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_174),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_120),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_174),B_120)),zero_zero_int)) ) ) ).
fof(fact_646_add__nonpos__neg,axiom,
! [B_120,A_174] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_174),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_120),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_174),B_120)),zero_z126310315umeral)) ) ) ).
fof(fact_647_add__neg__nonpos,axiom,
! [B_119,A_173] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_173),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_119),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_173),B_119)),zero_zero_rat)) ) ) ).
fof(fact_648_add__neg__nonpos,axiom,
! [B_119,A_173] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_173),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_119),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_173),B_119)),zero_z891286103de_int)) ) ) ).
fof(fact_649_add__neg__nonpos,axiom,
! [B_119,A_173] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_173),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_119),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_173),B_119)),zero_zero_real)) ) ) ).
fof(fact_650_add__neg__nonpos,axiom,
! [B_119,A_173] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_173),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_119),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_173),B_119)),zero_zero_nat)) ) ) ).
fof(fact_651_add__neg__nonpos,axiom,
! [B_119,A_173] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_173),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_119),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_173),B_119)),zero_zero_int)) ) ) ).
fof(fact_652_add__neg__nonpos,axiom,
! [B_119,A_173] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_173),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_119),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_173),B_119)),zero_z126310315umeral)) ) ) ).
fof(fact_653_add__strict__increasing2,axiom,
! [B_118,C_68,A_172] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_172))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_118),C_68))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_118),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_172),C_68))) ) ) ).
fof(fact_654_add__strict__increasing2,axiom,
! [B_118,C_68,A_172] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_172))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_118),C_68))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_118),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_172),C_68))) ) ) ).
fof(fact_655_add__strict__increasing2,axiom,
! [B_118,C_68,A_172] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_172))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_118),C_68))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_118),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_172),C_68))) ) ) ).
fof(fact_656_add__strict__increasing2,axiom,
! [B_118,C_68,A_172] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_172))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_118),C_68))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_118),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_172),C_68))) ) ) ).
fof(fact_657_add__strict__increasing2,axiom,
! [B_118,C_68,A_172] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_172))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_118),C_68))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_118),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_172),C_68))) ) ) ).
fof(fact_658_add__strict__increasing2,axiom,
! [B_118,C_68,A_172] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_172))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_118),C_68))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_118),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_172),C_68))) ) ) ).
fof(fact_659_add__strict__increasing,axiom,
! [B_117,C_67,A_171] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_171))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_117),C_67))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_117),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_171),C_67))) ) ) ).
fof(fact_660_add__strict__increasing,axiom,
! [B_117,C_67,A_171] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_171))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_117),C_67))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_117),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_171),C_67))) ) ) ).
fof(fact_661_add__strict__increasing,axiom,
! [B_117,C_67,A_171] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_171))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_117),C_67))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_117),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_171),C_67))) ) ) ).
fof(fact_662_add__strict__increasing,axiom,
! [B_117,C_67,A_171] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_171))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_117),C_67))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_117),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_171),C_67))) ) ) ).
fof(fact_663_add__strict__increasing,axiom,
! [B_117,C_67,A_171] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_171))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_117),C_67))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_117),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_171),C_67))) ) ) ).
fof(fact_664_add__strict__increasing,axiom,
! [B_117,C_67,A_171] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_171))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_117),C_67))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_117),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_171),C_67))) ) ) ).
fof(fact_665_add__nonneg__pos,axiom,
! [B_116,A_170] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_170))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_116))
=> 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_170),B_116))) ) ) ).
fof(fact_666_add__nonneg__pos,axiom,
! [B_116,A_170] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_170))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_116))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_170),B_116))) ) ) ).
fof(fact_667_add__nonneg__pos,axiom,
! [B_116,A_170] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_170))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_116))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_170),B_116))) ) ) ).
fof(fact_668_add__nonneg__pos,axiom,
! [B_116,A_170] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_170))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_116))
=> 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_170),B_116))) ) ) ).
fof(fact_669_add__nonneg__pos,axiom,
! [B_116,A_170] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_170))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_116))
=> 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_170),B_116))) ) ) ).
fof(fact_670_add__nonneg__pos,axiom,
! [B_116,A_170] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_170))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_116))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_170),B_116))) ) ) ).
fof(fact_671_add__pos__nonneg,axiom,
! [B_115,A_169] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_169))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_115))
=> 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_169),B_115))) ) ) ).
fof(fact_672_add__pos__nonneg,axiom,
! [B_115,A_169] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_169))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_115))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_169),B_115))) ) ) ).
fof(fact_673_add__pos__nonneg,axiom,
! [B_115,A_169] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_169))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_115))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_169),B_115))) ) ) ).
fof(fact_674_add__pos__nonneg,axiom,
! [B_115,A_169] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_169))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_115))
=> 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_169),B_115))) ) ) ).
fof(fact_675_add__pos__nonneg,axiom,
! [B_115,A_169] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_169))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_115))
=> 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_169),B_115))) ) ) ).
fof(fact_676_add__pos__nonneg,axiom,
! [B_115,A_169] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_169))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_115))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_169),B_115))) ) ) ).
fof(fact_677_zero__le__one,axiom,
hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),one_one_rat)) ).
fof(fact_678_zero__le__one,axiom,
hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),one_on1684967323de_int)) ).
fof(fact_679_zero__le__one,axiom,
hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),one_on1645066479umeral)) ).
fof(fact_680_zero__le__one,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),one_one_real)) ).
fof(fact_681_zero__le__one,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),one_one_nat)) ).
fof(fact_682_zero__le__one,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),one_one_int)) ).
fof(fact_683_not__one__le__zero,axiom,
~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),zero_zero_rat)) ).
fof(fact_684_not__one__le__zero,axiom,
~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),zero_z891286103de_int)) ).
fof(fact_685_not__one__le__zero,axiom,
~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),zero_z126310315umeral)) ).
fof(fact_686_not__one__le__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),zero_zero_real)) ).
fof(fact_687_not__one__le__zero,axiom,
~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),zero_zero_nat)) ).
fof(fact_688_not__one__le__zero,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),zero_zero_int)) ).
fof(fact_689_le__number__of__eq__not__less,axiom,
! [Va,Wa] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(number_number_of_rat,Va)),hAPP_int_rat(number_number_of_rat,Wa)))
<=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(number_number_of_rat,Wa)),hAPP_int_rat(number_number_of_rat,Va))) ) ).
fof(fact_690_le__number__of__eq__not__less,axiom,
! [Va,Wa] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_i1732201573de_int(number1226105091de_int,Va)),hAPP_i1732201573de_int(number1226105091de_int,Wa)))
<=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_i1732201573de_int(number1226105091de_int,Wa)),hAPP_i1732201573de_int(number1226105091de_int,Va))) ) ).
fof(fact_691_le__number__of__eq__not__less,axiom,
! [Va,Wa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Va)),hAPP_int_real(number267125858f_real,Wa)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Wa)),hAPP_int_real(number267125858f_real,Va))) ) ).
fof(fact_692_le__number__of__eq__not__less,axiom,
! [Va,Wa] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_int_nat(number_number_of_nat,Wa)))
<=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,Wa)),hAPP_int_nat(number_number_of_nat,Va))) ) ).
fof(fact_693_le__number__of__eq__not__less,axiom,
! [Va,Wa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Va)),hAPP_int_int(number_number_of_int,Wa)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,Wa)),hAPP_int_int(number_number_of_int,Va))) ) ).
fof(fact_694_le__number__of__eq__not__less,axiom,
! [Va,Wa] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_i769753017umeral(number1443263063umeral,Va)),hAPP_i769753017umeral(number1443263063umeral,Wa)))
<=> ~ hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_i769753017umeral(number1443263063umeral,Wa)),hAPP_i769753017umeral(number1443263063umeral,Va))) ) ).
fof(fact_695_power__mono,axiom,
! [N_37,A_168,B_114] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_168),B_114))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_168))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_168),N_37)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_114),N_37))) ) ) ).
fof(fact_696_power__mono,axiom,
! [N_37,A_168,B_114] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_168),B_114))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_168))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_168),N_37)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_114),N_37))) ) ) ).
fof(fact_697_power__mono,axiom,
! [N_37,A_168,B_114] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_168),B_114))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_168))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_168),N_37)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_114),N_37))) ) ) ).
fof(fact_698_power__mono,axiom,
! [N_37,A_168,B_114] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_168),B_114))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_168))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_168),N_37)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_114),N_37))) ) ) ).
fof(fact_699_power__mono,axiom,
! [N_37,A_168,B_114] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_168),B_114))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_168))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_168),N_37)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_114),N_37))) ) ) ).
fof(fact_700_power__mono,axiom,
! [N_37,A_168,B_114] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_168),B_114))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_168))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_168),N_37)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_114),N_37))) ) ) ).
fof(fact_701_zero__le__power,axiom,
! [N_36,A_167] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_167))
=> 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_167),N_36))) ) ).
fof(fact_702_zero__le__power,axiom,
! [N_36,A_167] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_167))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_167),N_36))) ) ).
fof(fact_703_zero__le__power,axiom,
! [N_36,A_167] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_167))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_167),N_36))) ) ).
fof(fact_704_zero__le__power,axiom,
! [N_36,A_167] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_167))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_167),N_36))) ) ).
fof(fact_705_zero__le__power,axiom,
! [N_36,A_167] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_167))
=> 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_167),N_36))) ) ).
fof(fact_706_zero__le__power,axiom,
! [N_36,A_167] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_167))
=> 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_167),N_36))) ) ).
fof(fact_707_one__le__power,axiom,
! [N_35,A_166] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),A_166))
=> 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_166),N_35))) ) ).
fof(fact_708_one__le__power,axiom,
! [N_35,A_166] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),A_166))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,one_on1684967323de_int),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_166),N_35))) ) ).
fof(fact_709_one__le__power,axiom,
! [N_35,A_166] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),A_166))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,one_on1645066479umeral),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_166),N_35))) ) ).
fof(fact_710_one__le__power,axiom,
! [N_35,A_166] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),A_166))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_166),N_35))) ) ).
fof(fact_711_one__le__power,axiom,
! [N_35,A_166] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),A_166))
=> 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_166),N_35))) ) ).
fof(fact_712_one__le__power,axiom,
! [N_35,A_166] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),A_166))
=> 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_166),N_35))) ) ).
fof(fact_713_of__nat__0__le__iff,axiom,
! [N_34] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,N_34))) ).
fof(fact_714_of__nat__0__le__iff,axiom,
! [N_34] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,N_34))) ).
fof(fact_715_of__nat__0__le__iff,axiom,
! [N_34] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,N_34))) ).
fof(fact_716_of__nat__0__le__iff,axiom,
! [N_34] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,N_34))) ).
fof(fact_717_of__nat__0__le__iff,axiom,
! [N_34] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,N_34))) ).
fof(fact_718_of__nat__0__le__iff,axiom,
! [N_34] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N_34))) ).
fof(fact_719_zero__le__imp__of__nat,axiom,
! [M_19] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_nat_rat(semiri151668891at_rat,M_19))) ).
fof(fact_720_zero__le__imp__of__nat,axiom,
! [M_19] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_n522471361de_int(semiri1424489471de_int,M_19))) ).
fof(fact_721_zero__le__imp__of__nat,axiom,
! [M_19] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_n1108039445umeral(semiri1619134803umeral,M_19))) ).
fof(fact_722_zero__le__imp__of__nat,axiom,
! [M_19] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(semiri132038758t_real,M_19))) ).
fof(fact_723_zero__le__imp__of__nat,axiom,
! [M_19] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(semiri984289939at_nat,M_19))) ).
fof(fact_724_zero__le__imp__of__nat,axiom,
! [M_19] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,M_19))) ).
fof(fact_725_rel__simps_I22_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit1,K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).
fof(fact_726_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_727_rel__simps_I32_J,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K)),hAPP_int_int(bit1,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) ) ).
fof(fact_728_rel__simps_I27_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),pls)) ) ).
fof(fact_729_rel__simps_I21_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),hAPP_int_int(bit0,K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).
fof(fact_730_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_731_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,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),Y))
=> 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),Y))) ) ) ).
fof(fact_732_zadd__zless__mono,axiom,
! [Z_4,Z_1,W_9,W] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,W_9),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_9),Z_4)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,W),Z_1))) ) ) ).
fof(fact_733_zero__zle__int,axiom,
! [N] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_nat_int(semiri1621563631at_int,N))) ).
fof(fact_734_transfer__int__nat__quantifiers_I1_J,axiom,
! [P_1] :
( ! [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(P_1,X_1)) ) )
<=> ! [X_1] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_1))) ) ).
fof(fact_735_transfer__int__nat__quantifiers_I2_J,axiom,
! [P_1] :
( ? [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(P_1,X_1)) )
<=> ? [X_1] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,X_1))) ) ).
fof(fact_736_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_737_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N,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_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) ) ).
fof(fact_738_zle__iff__zadd,axiom,
! [Wa,Z_2] :
( is_int(Z_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_2))
<=> ? [N_1] : Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Wa),hAPP_nat_int(semiri1621563631at_int,N_1)) ) ) ).
fof(fact_739_semiring__number__of__add__1,axiom,
! [V_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_21)),one_one_rat) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(succ,V_21)) ) ).
fof(fact_740_semiring__number__of__add__1,axiom,
! [V_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
=> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_21)),one_one_complex) = hAPP_int_complex(number528085621omplex,hAPP_int_int(succ,V_21)) ) ).
fof(fact_741_semiring__number__of__add__1,axiom,
! [V_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_21)),one_one_real) = hAPP_int_real(number267125858f_real,hAPP_int_int(succ,V_21)) ) ).
fof(fact_742_semiring__number__of__add__1,axiom,
! [V_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_21)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_21)) ) ).
fof(fact_743_semiring__number__of__add__1,axiom,
! [V_21] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_21))
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_21)),one_one_int) = hAPP_int_int(number_number_of_int,hAPP_int_int(succ,V_21)) ) ).
fof(fact_744_semiring__1__add__number__of,axiom,
! [V_20] :
( 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,one_one_rat),hAPP_int_rat(number_number_of_rat,V_20)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(succ,V_20)) ) ).
fof(fact_745_semiring__1__add__number__of,axiom,
! [V_20] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20))
=> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,V_20)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(succ,V_20)) ) ).
fof(fact_746_semiring__1__add__number__of,axiom,
! [V_20] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_20))
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,V_20)) = hAPP_int_real(number267125858f_real,hAPP_int_int(succ,V_20)) ) ).
fof(fact_747_semiring__1__add__number__of,axiom,
! [V_20] :
( 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,one_one_nat),hAPP_int_nat(number_number_of_nat,V_20)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_20)) ) ).
fof(fact_748_semiring__1__add__number__of,axiom,
! [V_20] :
( 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,one_one_int),hAPP_int_int(number_number_of_int,V_20)) = hAPP_int_int(number_number_of_int,hAPP_int_int(succ,V_20)) ) ).
fof(fact_749_power__less__imp__less__base,axiom,
! [A_165,N_33,B_113] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_165),N_33)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_113),N_33)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_113))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_165),B_113)) ) ) ).
fof(fact_750_power__less__imp__less__base,axiom,
! [A_165,N_33,B_113] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_165),N_33)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_113),N_33)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_113))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_165),B_113)) ) ) ).
fof(fact_751_power__less__imp__less__base,axiom,
! [A_165,N_33,B_113] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_165),N_33)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_113),N_33)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_113))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_165),B_113)) ) ) ).
fof(fact_752_power__less__imp__less__base,axiom,
! [A_165,N_33,B_113] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_165),N_33)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_113),N_33)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_113))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_165),B_113)) ) ) ).
fof(fact_753_power__less__imp__less__base,axiom,
! [A_165,N_33,B_113] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_165),N_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_113),N_33)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_113))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_165),B_113)) ) ) ).
fof(fact_754_power__less__imp__less__base,axiom,
! [A_165,N_33,B_113] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_165),N_33)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_113),N_33)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_113))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_165),B_113)) ) ) ).
fof(fact_755_eq__number__of__0,axiom,
! [Va] :
( hAPP_int_nat(number_number_of_nat,Va) = zero_zero_nat
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) ) ).
fof(fact_756_eq__0__number__of,axiom,
! [Va] :
( zero_zero_nat = hAPP_int_nat(number_number_of_nat,Va)
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) ) ).
fof(fact_757_rel__simps_I29_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K)),pls))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),pls)) ) ).
fof(fact_758_rel__simps_I5_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),hAPP_int_int(bit1,K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),K)) ) ).
fof(fact_759_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_760_rel__simps_I33_J,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K)),hAPP_int_int(bit0,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L)) ) ).
fof(fact_761_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_762_rel__simps_I15_J,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K)),hAPP_int_int(bit1,L)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L)) ) ).
fof(fact_763_zero__reorient,axiom,
! [Xa] :
( zero_zero_rat = Xa
<=> Xa = zero_zero_rat ) ).
fof(fact_764_zero__reorient,axiom,
! [Xa] :
( zero_z891286103de_int = Xa
<=> Xa = zero_z891286103de_int ) ).
fof(fact_765_zero__reorient,axiom,
! [Xa] :
( zero_zero_complex = Xa
<=> Xa = zero_zero_complex ) ).
fof(fact_766_zero__reorient,axiom,
! [Xa] :
( zero_z126310315umeral = Xa
<=> Xa = zero_z126310315umeral ) ).
fof(fact_767_zero__reorient,axiom,
! [Xa] :
( zero_zero_real = Xa
<=> Xa = zero_zero_real ) ).
fof(fact_768_zero__reorient,axiom,
! [Xa] :
( zero_zero_nat = Xa
<=> Xa = zero_zero_nat ) ).
fof(fact_769_zero__reorient,axiom,
! [Xa] :
( is_int(Xa)
=> ( zero_zero_int = Xa
<=> Xa = zero_zero_int ) ) ).
fof(fact_770_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_771_int__le__0__conv,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),zero_zero_int))
<=> Na = zero_zero_nat ) ).
fof(fact_772_succ__Pls,axiom,
hAPP_int_int(succ,pls) = hAPP_int_int(bit1,pls) ).
fof(fact_773_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_774_succ__Bit0,axiom,
! [K_2] : hAPP_int_int(succ,hAPP_int_int(bit0,K_2)) = hAPP_int_int(bit1,K_2) ).
fof(fact_775_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_776_add1__zle__eq,axiom,
! [Wa,Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Wa),one_one_int)),Z_2))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_2)) ) ).
fof(fact_777_zle__add1__eq__le,axiom,
! [Wa,Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),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,Wa),Z_2)) ) ).
fof(fact_778_add__right__imp__eq,axiom,
! [B_112,A_164,C_66] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_112),A_164) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_66),A_164)
=> B_112 = C_66 ) ).
fof(fact_779_add__right__imp__eq,axiom,
! [B_112,A_164,C_66] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_112),A_164) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_66),A_164)
=> B_112 = C_66 ) ).
fof(fact_780_add__right__imp__eq,axiom,
! [B_112,A_164,C_66] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_112),A_164) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_66),A_164)
=> B_112 = C_66 ) ).
fof(fact_781_add__right__imp__eq,axiom,
! [B_112,A_164,C_66] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_112),A_164) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_66),A_164)
=> B_112 = C_66 ) ).
fof(fact_782_add__right__imp__eq,axiom,
! [B_112,A_164,C_66] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_112),A_164) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_66),A_164)
=> B_112 = C_66 ) ).
fof(fact_783_add__right__imp__eq,axiom,
! [B_112,A_164,C_66] :
( ( is_int(B_112)
& is_int(C_66) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_112),A_164) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_66),A_164)
=> B_112 = C_66 ) ) ).
fof(fact_784_add__imp__eq,axiom,
! [A_163,B_111,C_65] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_163),B_111) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_163),C_65)
=> B_111 = C_65 ) ).
fof(fact_785_add__imp__eq,axiom,
! [A_163,B_111,C_65] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_163),B_111) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_163),C_65)
=> B_111 = C_65 ) ).
fof(fact_786_add__imp__eq,axiom,
! [A_163,B_111,C_65] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_163),B_111) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_163),C_65)
=> B_111 = C_65 ) ).
fof(fact_787_add__imp__eq,axiom,
! [A_163,B_111,C_65] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_163),B_111) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_163),C_65)
=> B_111 = C_65 ) ).
fof(fact_788_add__imp__eq,axiom,
! [A_163,B_111,C_65] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_163),B_111) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_163),C_65)
=> B_111 = C_65 ) ).
fof(fact_789_add__imp__eq,axiom,
! [A_163,B_111,C_65] :
( ( is_int(B_111)
& is_int(C_65) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_163),B_111) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_163),C_65)
=> B_111 = C_65 ) ) ).
fof(fact_790_add__left__imp__eq,axiom,
! [A_162,B_110,C_64] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_162),B_110) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_162),C_64)
=> B_110 = C_64 ) ).
fof(fact_791_add__left__imp__eq,axiom,
! [A_162,B_110,C_64] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_162),B_110) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_162),C_64)
=> B_110 = C_64 ) ).
fof(fact_792_add__left__imp__eq,axiom,
! [A_162,B_110,C_64] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_162),B_110) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_162),C_64)
=> B_110 = C_64 ) ).
fof(fact_793_add__left__imp__eq,axiom,
! [A_162,B_110,C_64] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_162),B_110) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_162),C_64)
=> B_110 = C_64 ) ).
fof(fact_794_add__left__imp__eq,axiom,
! [A_162,B_110,C_64] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_162),B_110) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_162),C_64)
=> B_110 = C_64 ) ).
fof(fact_795_add__left__imp__eq,axiom,
! [A_162,B_110,C_64] :
( ( is_int(B_110)
& is_int(C_64) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_162),B_110) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_162),C_64)
=> B_110 = C_64 ) ) ).
fof(fact_796_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_161,B_109,C_63,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_161),B_109)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_63),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_161),C_63)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_109),D_19)) ).
fof(fact_797_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_161,B_109,C_63,D_19] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_161),B_109)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_63),D_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_161),C_63)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_109),D_19)) ).
fof(fact_798_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_161,B_109,C_63,D_19] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_161),B_109)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_63),D_19)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_161),C_63)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_109),D_19)) ).
fof(fact_799_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_161,B_109,C_63,D_19] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_161),B_109)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_63),D_19)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_161),C_63)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_109),D_19)) ).
fof(fact_800_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_161,B_109,C_63,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_161),B_109)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_63),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_161),C_63)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_109),D_19)) ).
fof(fact_801_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_161,B_109,C_63,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_161),B_109)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_63),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_161),C_63)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_109),D_19)) ).
fof(fact_802_add__right__cancel,axiom,
! [B_1,A_1,C] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_1),A_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A_1)
<=> B_1 = C ) ).
fof(fact_803_add__right__cancel,axiom,
! [B_1,A_1,C] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_1),A_1) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A_1)
<=> B_1 = C ) ).
fof(fact_804_add__right__cancel,axiom,
! [B_1,A_1,C] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_1),A_1) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C),A_1)
<=> B_1 = C ) ).
fof(fact_805_add__right__cancel,axiom,
! [B_1,A_1,C] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),A_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A_1)
<=> B_1 = C ) ).
fof(fact_806_add__right__cancel,axiom,
! [B_1,A_1,C] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),A_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A_1)
<=> B_1 = C ) ).
fof(fact_807_add__right__cancel,axiom,
! [B_1,A_1,C] :
( ( is_int(B_1)
& is_int(C) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),A_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A_1)
<=> B_1 = C ) ) ).
fof(fact_808_add__left__cancel,axiom,
! [A_1,B_1,C] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),B_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),C)
<=> B_1 = C ) ).
fof(fact_809_add__left__cancel,axiom,
! [A_1,B_1,C] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_1),B_1) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_1),C)
<=> B_1 = C ) ).
fof(fact_810_add__left__cancel,axiom,
! [A_1,B_1,C] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_1),B_1) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_1),C)
<=> B_1 = C ) ).
fof(fact_811_add__left__cancel,axiom,
! [A_1,B_1,C] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),B_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),C)
<=> B_1 = C ) ).
fof(fact_812_add__left__cancel,axiom,
! [A_1,B_1,C] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),B_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),C)
<=> B_1 = C ) ).
fof(fact_813_add__left__cancel,axiom,
! [A_1,B_1,C] :
( ( is_int(B_1)
& is_int(C) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),C)
<=> B_1 = C ) ) ).
fof(fact_814_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_160,B_108,C_62] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_160),B_108)),C_62) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_160),C_62)),B_108) ).
fof(fact_815_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_160,B_108,C_62] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_160),B_108)),C_62) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_160),C_62)),B_108) ).
fof(fact_816_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_160,B_108,C_62] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_160),B_108)),C_62) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_160),C_62)),B_108) ).
fof(fact_817_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_160,B_108,C_62] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_160),B_108)),C_62) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_160),C_62)),B_108) ).
fof(fact_818_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_160,B_108,C_62] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_160),B_108)),C_62) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_160),C_62)),B_108) ).
fof(fact_819_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_160,B_108,C_62] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_160),B_108)),C_62) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_160),C_62)),B_108) ).
fof(fact_820_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_159,B_107,C_61] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_159),B_107)),C_61) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_159),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_107),C_61)) ).
fof(fact_821_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_159,B_107,C_61] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_159),B_107)),C_61) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_159),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_107),C_61)) ).
fof(fact_822_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_159,B_107,C_61] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_159),B_107)),C_61) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_159),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_107),C_61)) ).
fof(fact_823_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_159,B_107,C_61] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_159),B_107)),C_61) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_159),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_107),C_61)) ).
fof(fact_824_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_159,B_107,C_61] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_159),B_107)),C_61) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_159),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_107),C_61)) ).
fof(fact_825_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_159,B_107,C_61] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_159),B_107)),C_61) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_159),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_107),C_61)) ).
fof(fact_826_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_158,B_106,C_60] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_158),B_106)),C_60) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_158),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_106),C_60)) ).
fof(fact_827_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_158,B_106,C_60] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_158),B_106)),C_60) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_158),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_106),C_60)) ).
fof(fact_828_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_158,B_106,C_60] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_158),B_106)),C_60) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_158),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_106),C_60)) ).
fof(fact_829_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_158,B_106,C_60] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_158),B_106)),C_60) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_158),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_106),C_60)) ).
fof(fact_830_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_158,B_106,C_60] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_158),B_106)),C_60) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_158),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_106),C_60)) ).
fof(fact_831_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_158,B_106,C_60] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_158),B_106)),C_60) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_158),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_106),C_60)) ).
fof(fact_832_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_157,C_59,D_18] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_157),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_59),D_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_157),C_59)),D_18) ).
fof(fact_833_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_157,C_59,D_18] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_157),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_59),D_18)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_157),C_59)),D_18) ).
fof(fact_834_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_157,C_59,D_18] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_157),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_59),D_18)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_157),C_59)),D_18) ).
fof(fact_835_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_157,C_59,D_18] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_157),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_59),D_18)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_157),C_59)),D_18) ).
fof(fact_836_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_157,C_59,D_18] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_157),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_59),D_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_157),C_59)),D_18) ).
fof(fact_837_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_157,C_59,D_18] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_157),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_59),D_18)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_157),C_59)),D_18) ).
fof(fact_838_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_156,C_58,D_17] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_156),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_58),D_17)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_58),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_156),D_17)) ).
fof(fact_839_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_156,C_58,D_17] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_156),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_58),D_17)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_58),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_156),D_17)) ).
fof(fact_840_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_156,C_58,D_17] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_156),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_58),D_17)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_58),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_156),D_17)) ).
fof(fact_841_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_156,C_58,D_17] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_156),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_58),D_17)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_58),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_156),D_17)) ).
fof(fact_842_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_156,C_58,D_17] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_156),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_58),D_17)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_58),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_156),D_17)) ).
fof(fact_843_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_156,C_58,D_17] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_156),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_58),D_17)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_58),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_156),D_17)) ).
fof(fact_844_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_155,C_57] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_155),C_57) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_57),A_155) ).
fof(fact_845_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_155,C_57] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_155),C_57) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_57),A_155) ).
fof(fact_846_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_155,C_57] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_155),C_57) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,C_57),A_155) ).
fof(fact_847_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_155,C_57] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_155),C_57) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_57),A_155) ).
fof(fact_848_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_155,C_57] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_155),C_57) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_57),A_155) ).
fof(fact_849_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_155,C_57] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_155),C_57) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_57),A_155) ).
fof(fact_850_one__reorient,axiom,
! [Xa] :
( one_one_rat = Xa
<=> Xa = one_one_rat ) ).
fof(fact_851_one__reorient,axiom,
! [Xa] :
( one_on1684967323de_int = Xa
<=> Xa = one_on1684967323de_int ) ).
fof(fact_852_one__reorient,axiom,
! [Xa] :
( one_one_complex = Xa
<=> Xa = one_one_complex ) ).
fof(fact_853_one__reorient,axiom,
! [Xa] :
( one_on1645066479umeral = Xa
<=> Xa = one_on1645066479umeral ) ).
fof(fact_854_one__reorient,axiom,
! [Xa] :
( one_one_real = Xa
<=> Xa = one_one_real ) ).
fof(fact_855_one__reorient,axiom,
! [Xa] :
( one_one_nat = Xa
<=> Xa = one_one_nat ) ).
fof(fact_856_one__reorient,axiom,
! [Xa] :
( is_int(Xa)
=> ( one_one_int = Xa
<=> Xa = one_one_int ) ) ).
fof(fact_857_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_858_power__eq__imp__eq__base,axiom,
! [A_154,N_32,B_105] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_154),N_32) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_105),N_32)
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_154))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_105))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
=> A_154 = B_105 ) ) ) ) ).
fof(fact_859_power__eq__imp__eq__base,axiom,
! [A_154,N_32,B_105] :
( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_154),N_32) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_105),N_32)
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_154))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_105))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
=> A_154 = B_105 ) ) ) ) ).
fof(fact_860_power__eq__imp__eq__base,axiom,
! [A_154,N_32,B_105] :
( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_154),N_32) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_105),N_32)
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_154))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_105))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
=> A_154 = B_105 ) ) ) ) ).
fof(fact_861_power__eq__imp__eq__base,axiom,
! [A_154,N_32,B_105] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_154),N_32) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_105),N_32)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_154))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_105))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
=> A_154 = B_105 ) ) ) ) ).
fof(fact_862_power__eq__imp__eq__base,axiom,
! [A_154,N_32,B_105] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_154),N_32) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_105),N_32)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_154))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_105))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
=> A_154 = B_105 ) ) ) ) ).
fof(fact_863_power__eq__imp__eq__base,axiom,
! [A_154,N_32,B_105] :
( ( is_int(A_154)
& is_int(B_105) )
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_154),N_32) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_105),N_32)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_154))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_105))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_32))
=> A_154 = B_105 ) ) ) ) ) ).
fof(fact_864_semiring__add__number__of,axiom,
! [V_19,V_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_18)),hAPP_int_rat(number_number_of_rat,V_19)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_18),V_19)) ) ) ).
fof(fact_865_semiring__add__number__of,axiom,
! [V_19,V_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_18)),hAPP_int_complex(number528085621omplex,V_19)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_18),V_19)) ) ) ).
fof(fact_866_semiring__add__number__of,axiom,
! [V_19,V_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_18)),hAPP_int_real(number267125858f_real,V_19)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_18),V_19)) ) ) ).
fof(fact_867_semiring__add__number__of,axiom,
! [V_19,V_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_18)),hAPP_int_nat(number_number_of_nat,V_19)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_18),V_19)) ) ) ).
fof(fact_868_semiring__add__number__of,axiom,
! [V_19,V_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_19))
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_18)),hAPP_int_int(number_number_of_int,V_19)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_18),V_19)) ) ) ).
fof(fact_869_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_870_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_871_number__of__succ,axiom,
! [K_8] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(succ,K_8)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,one_one_rat),hAPP_int_rat(number_number_of_rat,K_8)) ).
fof(fact_872_number__of__succ,axiom,
! [K_8] : hAPP_int_complex(number528085621omplex,hAPP_int_int(succ,K_8)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),hAPP_int_complex(number528085621omplex,K_8)) ).
fof(fact_873_number__of__succ,axiom,
! [K_8] : hAPP_int_real(number267125858f_real,hAPP_int_int(succ,K_8)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),hAPP_int_real(number267125858f_real,K_8)) ).
fof(fact_874_number__of__succ,axiom,
! [K_8] : hAPP_int_int(number_number_of_int,hAPP_int_int(succ,K_8)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_int_int(number_number_of_int,K_8)) ).
fof(fact_875_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_876_power__strict__mono,axiom,
! [N_31,A_153,B_104] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_153),B_104))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_153))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_153),N_31)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_104),N_31))) ) ) ) ).
fof(fact_877_power__strict__mono,axiom,
! [N_31,A_153,B_104] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_153),B_104))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_153))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_153),N_31)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_104),N_31))) ) ) ) ).
fof(fact_878_power__strict__mono,axiom,
! [N_31,A_153,B_104] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_153),B_104))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_153))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_153),N_31)),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,B_104),N_31))) ) ) ) ).
fof(fact_879_power__strict__mono,axiom,
! [N_31,A_153,B_104] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_153),B_104))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_153))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_153),N_31)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_104),N_31))) ) ) ) ).
fof(fact_880_power__strict__mono,axiom,
! [N_31,A_153,B_104] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_153),B_104))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_153))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_153),N_31)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_104),N_31))) ) ) ) ).
fof(fact_881_power__strict__mono,axiom,
! [N_31,A_153,B_104] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_153),B_104))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_153))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_31))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_153),N_31)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_104),N_31))) ) ) ) ).
fof(fact_882_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_883_power2__ge__self,axiom,
! [X] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X),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)))))) ).
fof(fact_884_power2__eq__imp__eq,axiom,
! [X_61,Y_51] :
( hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_61),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_51),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_61))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_51))
=> X_61 = Y_51 ) ) ) ).
fof(fact_885_power2__eq__imp__eq,axiom,
! [X_61,Y_51] :
( hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_61),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_51),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_61))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),Y_51))
=> X_61 = Y_51 ) ) ) ).
fof(fact_886_power2__eq__imp__eq,axiom,
! [X_61,Y_51] :
( hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_61),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_51),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_61))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),Y_51))
=> X_61 = Y_51 ) ) ) ).
fof(fact_887_power2__eq__imp__eq,axiom,
! [X_61,Y_51] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_61),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_51),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_61))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_51))
=> X_61 = Y_51 ) ) ) ).
fof(fact_888_power2__eq__imp__eq,axiom,
! [X_61,Y_51] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_61),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_51),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_61))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_51))
=> X_61 = Y_51 ) ) ) ).
fof(fact_889_power2__eq__imp__eq,axiom,
! [X_61,Y_51] :
( ( is_int(X_61)
& is_int(Y_51) )
=> ( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_61),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_51),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_61))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_51))
=> X_61 = Y_51 ) ) ) ) ).
fof(fact_890_power2__le__imp__le,axiom,
! [X_60,Y_50] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_60),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_50),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_50))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_60),Y_50)) ) ) ).
fof(fact_891_power2__le__imp__le,axiom,
! [X_60,Y_50] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_60),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_50),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_50))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_60),Y_50)) ) ) ).
fof(fact_892_power2__le__imp__le,axiom,
! [X_60,Y_50] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_60),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_50),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_50))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,X_60),Y_50)) ) ) ).
fof(fact_893_power2__le__imp__le,axiom,
! [X_60,Y_50] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_60),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_50),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_50))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_60),Y_50)) ) ) ).
fof(fact_894_power2__le__imp__le,axiom,
! [X_60,Y_50] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_60),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_50),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_50))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_60),Y_50)) ) ) ).
fof(fact_895_power2__le__imp__le,axiom,
! [X_60,Y_50] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_60),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_50),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_50))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_60),Y_50)) ) ) ).
fof(fact_896_zero__le__power2,axiom,
! [A_152] : 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_152),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_897_zero__le__power2,axiom,
! [A_152] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_152),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_898_zero__le__power2,axiom,
! [A_152] : 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_152),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_899_of__nat__number__of__lemma,axiom,
! [V_17] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17)))
=> hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_17)) = hAPP_int_rat(number_number_of_rat,V_17) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17)))
=> hAPP_nat_rat(semiri151668891at_rat,hAPP_int_nat(number_number_of_nat,V_17)) = zero_zero_rat ) ) ).
fof(fact_900_of__nat__number__of__lemma,axiom,
! [V_17] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17)))
=> hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_17)) = hAPP_int_complex(number528085621omplex,V_17) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17)))
=> hAPP_nat_complex(semiri2020571505omplex,hAPP_int_nat(number_number_of_nat,V_17)) = zero_zero_complex ) ) ).
fof(fact_901_of__nat__number__of__lemma,axiom,
! [V_17] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17)))
=> hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_17)) = hAPP_int_real(number267125858f_real,V_17) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17)))
=> hAPP_nat_real(semiri132038758t_real,hAPP_int_nat(number_number_of_nat,V_17)) = zero_zero_real ) ) ).
fof(fact_902_of__nat__number__of__lemma,axiom,
! [V_17] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17)))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_17)) = hAPP_int_int(number_number_of_int,V_17) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_17)))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_17)) = zero_zero_int ) ) ).
fof(fact_903_add__0__iff,axiom,
! [B_1,A_1] :
( B_1 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_1),A_1)
<=> A_1 = zero_zero_rat ) ).
fof(fact_904_add__0__iff,axiom,
! [B_1,A_1] :
( B_1 = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_1),A_1)
<=> A_1 = zero_zero_complex ) ).
fof(fact_905_add__0__iff,axiom,
! [B_1,A_1] :
( B_1 = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),A_1)
<=> A_1 = zero_zero_real ) ).
fof(fact_906_add__0__iff,axiom,
! [B_1,A_1] :
( B_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),A_1)
<=> A_1 = zero_zero_nat ) ).
fof(fact_907_add__0__iff,axiom,
! [B_1,A_1] :
( ( is_int(B_1)
& is_int(A_1) )
=> ( B_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),A_1)
<=> A_1 = zero_zero_int ) ) ).
fof(fact_908_add_Ocomm__neutral,axiom,
! [A_151] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_151),zero_zero_rat) = A_151 ).
fof(fact_909_add_Ocomm__neutral,axiom,
! [A_151] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_151),zero_z891286103de_int) = A_151 ).
fof(fact_910_add_Ocomm__neutral,axiom,
! [A_151] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_151),zero_zero_complex) = A_151 ).
fof(fact_911_add_Ocomm__neutral,axiom,
! [A_151] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_151),zero_zero_real) = A_151 ).
fof(fact_912_add_Ocomm__neutral,axiom,
! [A_151] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_151),zero_zero_nat) = A_151 ).
fof(fact_913_add_Ocomm__neutral,axiom,
! [A_151] :
( is_int(A_151)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_151),zero_zero_int) = A_151 ) ).
fof(fact_914_add_Ocomm__neutral,axiom,
! [A_151] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_151),zero_z126310315umeral) = A_151 ).
fof(fact_915_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_150] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_150),zero_zero_rat) = A_150 ).
fof(fact_916_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_150] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_150),zero_z891286103de_int) = A_150 ).
fof(fact_917_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_150] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_150),zero_zero_complex) = A_150 ).
fof(fact_918_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_150] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_150),zero_zero_real) = A_150 ).
fof(fact_919_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_150] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_150),zero_zero_nat) = A_150 ).
fof(fact_920_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_150] :
( is_int(A_150)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_150),zero_zero_int) = A_150 ) ).
fof(fact_921_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_150] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_150),zero_z126310315umeral) = A_150 ).
fof(fact_922_add__0__right,axiom,
! [A_149] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_149),zero_zero_rat) = A_149 ).
fof(fact_923_add__0__right,axiom,
! [A_149] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_149),zero_z891286103de_int) = A_149 ).
fof(fact_924_add__0__right,axiom,
! [A_149] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_149),zero_zero_complex) = A_149 ).
fof(fact_925_add__0__right,axiom,
! [A_149] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_149),zero_zero_real) = A_149 ).
fof(fact_926_add__0__right,axiom,
! [A_149] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_149),zero_zero_nat) = A_149 ).
fof(fact_927_add__0__right,axiom,
! [A_149] :
( is_int(A_149)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_149),zero_zero_int) = A_149 ) ).
fof(fact_928_add__0__right,axiom,
! [A_149] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_149),zero_z126310315umeral) = A_149 ).
fof(fact_929_double__zero__sym,axiom,
! [A_1] :
( zero_zero_rat = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),A_1)
<=> A_1 = zero_zero_rat ) ).
fof(fact_930_double__zero__sym,axiom,
! [A_1] :
( zero_zero_real = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),A_1)
<=> A_1 = zero_zero_real ) ).
fof(fact_931_double__zero__sym,axiom,
! [A_1] :
( is_int(A_1)
=> ( zero_zero_int = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),A_1)
<=> A_1 = zero_zero_int ) ) ).
fof(fact_932_add__0,axiom,
! [A_148] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_148) = A_148 ).
fof(fact_933_add__0,axiom,
! [A_148] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_148) = A_148 ).
fof(fact_934_add__0,axiom,
! [A_148] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_148) = A_148 ).
fof(fact_935_add__0,axiom,
! [A_148] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_148) = A_148 ).
fof(fact_936_add__0,axiom,
! [A_148] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_148) = A_148 ).
fof(fact_937_add__0,axiom,
! [A_148] :
( is_int(A_148)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_148) = A_148 ) ).
fof(fact_938_add__0,axiom,
! [A_148] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_148) = A_148 ).
fof(fact_939_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_147] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_147) = A_147 ).
fof(fact_940_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_147] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_147) = A_147 ).
fof(fact_941_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_147] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_147) = A_147 ).
fof(fact_942_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_147] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_147) = A_147 ).
fof(fact_943_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_147] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_147) = A_147 ).
fof(fact_944_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_147] :
( is_int(A_147)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_147) = A_147 ) ).
fof(fact_945_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_147] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_147) = A_147 ).
fof(fact_946_add__0__left,axiom,
! [A_146] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A_146) = A_146 ).
fof(fact_947_add__0__left,axiom,
! [A_146] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,zero_z891286103de_int),A_146) = A_146 ).
fof(fact_948_add__0__left,axiom,
! [A_146] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,zero_zero_complex),A_146) = A_146 ).
fof(fact_949_add__0__left,axiom,
! [A_146] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,zero_zero_real),A_146) = A_146 ).
fof(fact_950_add__0__left,axiom,
! [A_146] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_146) = A_146 ).
fof(fact_951_add__0__left,axiom,
! [A_146] :
( is_int(A_146)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,zero_zero_int),A_146) = A_146 ) ).
fof(fact_952_add__0__left,axiom,
! [A_146] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,zero_z126310315umeral),A_146) = A_146 ).
fof(fact_953_add__less__imp__less__left,axiom,
! [C_56,A_145,B_103] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_56),A_145)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_56),B_103)))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_145),B_103)) ) ).
fof(fact_954_add__less__imp__less__left,axiom,
! [C_56,A_145,B_103] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_56),A_145)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_56),B_103)))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_145),B_103)) ) ).
fof(fact_955_add__less__imp__less__left,axiom,
! [C_56,A_145,B_103] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_56),A_145)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_56),B_103)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_145),B_103)) ) ).
fof(fact_956_add__less__imp__less__left,axiom,
! [C_56,A_145,B_103] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_56),A_145)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_56),B_103)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_145),B_103)) ) ).
fof(fact_957_add__less__imp__less__left,axiom,
! [C_56,A_145,B_103] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_56),A_145)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_56),B_103)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_145),B_103)) ) ).
fof(fact_958_add__less__imp__less__right,axiom,
! [A_144,C_55,B_102] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_144),C_55)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_102),C_55)))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_144),B_102)) ) ).
fof(fact_959_add__less__imp__less__right,axiom,
! [A_144,C_55,B_102] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_144),C_55)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_102),C_55)))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_144),B_102)) ) ).
fof(fact_960_add__less__imp__less__right,axiom,
! [A_144,C_55,B_102] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_144),C_55)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_102),C_55)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_144),B_102)) ) ).
fof(fact_961_add__less__imp__less__right,axiom,
! [A_144,C_55,B_102] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_144),C_55)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_102),C_55)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_144),B_102)) ) ).
fof(fact_962_add__less__imp__less__right,axiom,
! [A_144,C_55,B_102] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_144),C_55)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_102),C_55)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_144),B_102)) ) ).
fof(fact_963_add__strict__mono,axiom,
! [C_54,D_16,A_143,B_101] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_143),B_101))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_54),D_16))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_143),C_54)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_101),D_16))) ) ) ).
fof(fact_964_add__strict__mono,axiom,
! [C_54,D_16,A_143,B_101] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_143),B_101))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_54),D_16))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_143),C_54)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_101),D_16))) ) ) ).
fof(fact_965_add__strict__mono,axiom,
! [C_54,D_16,A_143,B_101] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_143),B_101))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_54),D_16))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_143),C_54)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_101),D_16))) ) ) ).
fof(fact_966_add__strict__mono,axiom,
! [C_54,D_16,A_143,B_101] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_143),B_101))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_54),D_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_143),C_54)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_101),D_16))) ) ) ).
fof(fact_967_add__strict__mono,axiom,
! [C_54,D_16,A_143,B_101] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_143),B_101))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_54),D_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_143),C_54)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_101),D_16))) ) ) ).
fof(fact_968_add__strict__left__mono,axiom,
! [C_53,A_142,B_100] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_142),B_100))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_53),A_142)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_53),B_100))) ) ).
fof(fact_969_add__strict__left__mono,axiom,
! [C_53,A_142,B_100] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_142),B_100))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_53),A_142)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C_53),B_100))) ) ).
fof(fact_970_add__strict__left__mono,axiom,
! [C_53,A_142,B_100] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_142),B_100))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_53),A_142)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_53),B_100))) ) ).
fof(fact_971_add__strict__left__mono,axiom,
! [C_53,A_142,B_100] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_142),B_100))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_53),A_142)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_53),B_100))) ) ).
fof(fact_972_add__strict__left__mono,axiom,
! [C_53,A_142,B_100] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_142),B_100))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_53),A_142)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_53),B_100))) ) ).
fof(fact_973_add__strict__right__mono,axiom,
! [C_52,A_141,B_99] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_141),B_99))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_141),C_52)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_99),C_52))) ) ).
fof(fact_974_add__strict__right__mono,axiom,
! [C_52,A_141,B_99] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_141),B_99))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_141),C_52)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_99),C_52))) ) ).
fof(fact_975_add__strict__right__mono,axiom,
! [C_52,A_141,B_99] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_141),B_99))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_141),C_52)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_99),C_52))) ) ).
fof(fact_976_add__strict__right__mono,axiom,
! [C_52,A_141,B_99] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_141),B_99))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_141),C_52)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_99),C_52))) ) ).
fof(fact_977_add__strict__right__mono,axiom,
! [C_52,A_141,B_99] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_141),B_99))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_141),C_52)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_99),C_52))) ) ).
fof(fact_978_add__less__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),A_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C),B_1)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),B_1)) ) ).
fof(fact_979_add__less__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),A_1)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,C),B_1)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_1),B_1)) ) ).
fof(fact_980_add__less__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),A_1)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C),B_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1)) ) ).
fof(fact_981_add__less__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),A_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C),B_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1)) ) ).
fof(fact_982_add__less__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),A_1)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) ) ).
fof(fact_983_add__less__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_1),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_1),C)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),B_1)) ) ).
fof(fact_984_add__less__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_1),C)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_1),C)))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_1),B_1)) ) ).
fof(fact_985_add__less__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_1),C)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1)) ) ).
fof(fact_986_add__less__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),C)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1)) ) ).
fof(fact_987_add__less__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),C)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_1),C)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) ) ).
fof(fact_988_power2__less__imp__less,axiom,
! [X_59,Y_49] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_59),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_49),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_49))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_59),Y_49)) ) ) ).
fof(fact_989_power2__less__imp__less,axiom,
! [X_59,Y_49] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_59),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_49),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_49))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_59),Y_49)) ) ) ).
fof(fact_990_power2__less__imp__less,axiom,
! [X_59,Y_49] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_59),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,Y_49),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_49))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,X_59),Y_49)) ) ) ).
fof(fact_991_power2__less__imp__less,axiom,
! [X_59,Y_49] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_59),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_49),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_49))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_59),Y_49)) ) ) ).
fof(fact_992_power2__less__imp__less,axiom,
! [X_59,Y_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_59),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_49),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_49))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_59),Y_49)) ) ) ).
fof(fact_993_power2__less__imp__less,axiom,
! [X_59,Y_49] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_59),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_49),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_49))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_59),Y_49)) ) ) ).
fof(fact_994_sum__power2__le__zero__iff,axiom,
! [Xa,Ya] :
( 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,Xa),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,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_rat))
<=> ( Xa = zero_zero_rat
& Ya = zero_zero_rat ) ) ).
fof(fact_995_sum__power2__le__zero__iff,axiom,
! [Xa,Ya] :
( 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,Xa),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)))))),zero_zero_real))
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ).
fof(fact_996_sum__power2__le__zero__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( 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,Xa),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,Ya),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),zero_zero_int))
<=> ( Xa = zero_zero_int
& Ya = zero_zero_int ) ) ) ).
fof(fact_997_sum__power2__ge__zero,axiom,
! [X_58,Y_48] : 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_58),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_48),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_998_sum__power2__ge__zero,axiom,
! [X_58,Y_48] : 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_58),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_48),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_999_sum__power2__ge__zero,axiom,
! [X_58,Y_48] : 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_58),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_48),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_1000_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_57] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_57),one_one_nat) = X_57 ).
fof(fact_1001_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_57] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_57),one_one_nat) = X_57 ).
fof(fact_1002_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_57] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_57),one_one_nat) = X_57 ).
fof(fact_1003_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_57] :
( is_int(X_57)
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_57),one_one_nat) = X_57 ) ).
fof(fact_1004_int__pos__lt__two__imp__zero__or__one,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(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
=> ( X = zero_zero_int
| X = one_one_int ) ) ) ) ).
fof(fact_1005_nat__power__eq__0__iff,axiom,
! [Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ma),Na) = zero_zero_nat
<=> ( Na != zero_zero_nat
& Ma = zero_zero_nat ) ) ).
fof(fact_1006_nat__number__of__add__1,axiom,
! [V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),one_one_nat) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_1)) ) ) ).
fof(fact_1007_number__of1,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,N)))
=> ( 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))))
& 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)))) ) ) ).
fof(fact_1008_realpow__pos__nth__unique,axiom,
! [A_3,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),A_3))
=> ? [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,X_1),N) = A_3
& ! [Y_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y_1))
& hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_1),N) = A_3 )
=> Y_1 = X_1 ) ) ) ) ).
fof(fact_1009_realpow__pos__nth,axiom,
! [A_3,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),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) = A_3 ) ) ) ).
fof(fact_1010_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_1011_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_1012_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_1] : K_2 != hAPP_nat_int(semiri1621563631at_int,N_1) ) ) ).
fof(fact_1013_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_1] : K_2 = hAPP_nat_int(semiri1621563631at_int,N_1) ) ) ).
fof(fact_1014_int__ge__induct,axiom,
! [P_1,K,I_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I_1))
=> ( hBOOL(hAPP_int_bool(P_1,K))
=> ( ! [I_2] :
( is_int(I_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),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_1015_order__refl,axiom,
! [X_56] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_56),X_56)) ).
fof(fact_1016_order__refl,axiom,
! [Xa] : hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Xa)) ).
fof(fact_1017_order__refl,axiom,
! [Xa] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Xa)) ).
fof(fact_1018_order__refl,axiom,
! [X_56] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_56),X_56)) ).
fof(fact_1019_order__refl,axiom,
! [X_56] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_56),X_56)) ).
fof(fact_1020_order__refl,axiom,
! [X_56] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_56),X_56)) ).
fof(fact_1021_order__refl,axiom,
! [X_56] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_56),X_56)) ).
fof(fact_1022_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_1023_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_1024_le0,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
fof(fact_1025_real__less__def,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_1026_less__eq__real__def,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_1027_le__antisym,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> M = N ) ) ).
fof(fact_1028_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_1029_eq__imp__le,axiom,
! [M,N] :
( M = N
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_1030_nat__le__linear,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).
fof(fact_1031_le__refl,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).
fof(fact_1032_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_140,B_98] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_140),B_98) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_98),A_140) ).
fof(fact_1033_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_140,B_98] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_140),B_98) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_98),A_140) ).
fof(fact_1034_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_140,B_98] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_140),B_98) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_98),A_140) ).
fof(fact_1035_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_140,B_98] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_140),B_98) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_98),A_140) ).
fof(fact_1036_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_140,B_98] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_140),B_98) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_98),A_140) ).
fof(fact_1037_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_140,B_98] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_140),B_98) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_98),A_140) ).
fof(fact_1038_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),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,Rx_6),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_6),Ry_4)) ).
fof(fact_1039_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_6),Ry_4)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_6),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_6),Ry_4)) ).
fof(fact_1040_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_6),Ry_4)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_6),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_6),Ry_4)) ).
fof(fact_1041_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_6),Ry_4)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_6),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_6),Ry_4)) ).
fof(fact_1042_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),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,Rx_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_6),Ry_4)) ).
fof(fact_1043_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),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,Rx_6),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_6),Ry_4)) ).
fof(fact_1044_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),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,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx_5),Rx_5)),Ry_3) ).
fof(fact_1045_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_5),Ry_3)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_5),Rx_5)),Ry_3) ).
fof(fact_1046_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_5),Ry_3)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_5),Rx_5)),Ry_3) ).
fof(fact_1047_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_5),Ry_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_5),Rx_5)),Ry_3) ).
fof(fact_1048_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx_5),Rx_5)),Ry_3) ).
fof(fact_1049_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),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,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx_5),Rx_5)),Ry_3) ).
fof(fact_1050_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_139,B_97,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_139),B_97)),C_51) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_139),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_97),C_51)) ).
fof(fact_1051_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_139,B_97,C_51] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_139),B_97)),C_51) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_139),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_97),C_51)) ).
fof(fact_1052_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_139,B_97,C_51] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_139),B_97)),C_51) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_139),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_97),C_51)) ).
fof(fact_1053_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_139,B_97,C_51] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_139),B_97)),C_51) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_139),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_97),C_51)) ).
fof(fact_1054_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_139,B_97,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_139),B_97)),C_51) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_139),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_97),C_51)) ).
fof(fact_1055_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_139,B_97,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_139),B_97)),C_51) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_139),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_97),C_51)) ).
fof(fact_1056_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : 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_4)),Rx_4) = 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_4),Rx_4)) ).
fof(fact_1057_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),Ly_4)),Rx_4) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_4),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_4),Rx_4)) ).
fof(fact_1058_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),Ly_4)),Rx_4) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_4),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_4),Rx_4)) ).
fof(fact_1059_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),Ly_4)),Rx_4) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_4),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_4),Rx_4)) ).
fof(fact_1060_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : 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_4)),Rx_4) = 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_4),Rx_4)) ).
fof(fact_1061_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : 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_4)),Rx_4) = 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_4),Rx_4)) ).
fof(fact_1062_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,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_3)),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_3) ).
fof(fact_1063_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_3),Ly_3)),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_3) ).
fof(fact_1064_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Ly_3)),Rx_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_3),Rx_3)),Ly_3) ).
fof(fact_1065_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_3),Ly_3)),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_3) ).
fof(fact_1066_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,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_3)),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_3) ).
fof(fact_1067_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,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_3)),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_3) ).
fof(fact_1068_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,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_2),Ly_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_2),Ry_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_2),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx_2),Ry_2))) ).
fof(fact_1069_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),Ly_2)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_2),Ry_2)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly_2),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_2),Ry_2))) ).
fof(fact_1070_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),Ly_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_2),Ry_2)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly_2),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_2),Ry_2))) ).
fof(fact_1071_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),Ly_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_2),Ry_2)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly_2),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_2),Ry_2))) ).
fof(fact_1072_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,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_2),Ly_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_2),Ry_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_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx_2),Ry_2))) ).
fof(fact_1073_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,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_2),Ly_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_2),Ry_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_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx_2),Ry_2))) ).
fof(fact_1074_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,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),Ly_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,Rx_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),Ly_1)),Ry_1)) ).
fof(fact_1075_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),Ly_1)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_1),Ry_1)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx_1),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx_1),Ly_1)),Ry_1)) ).
fof(fact_1076_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),Ly_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_1),Ry_1)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx_1),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx_1),Ly_1)),Ry_1)) ).
fof(fact_1077_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),Ly_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_1),Ry_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx_1),Ly_1)),Ry_1)) ).
fof(fact_1078_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,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),Ly_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,Rx_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),Ly_1)),Ry_1)) ).
fof(fact_1079_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,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),Ly_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,Rx_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),Ly_1)),Ry_1)) ).
fof(fact_1080_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),Ly)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Rx),Ry)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Lx),Rx)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ly),Ry)) ).
fof(fact_1081_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),Ly)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Rx),Ry)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Lx),Rx)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,Ly),Ry)) ).
fof(fact_1082_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),Ly)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Rx),Ry)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Lx),Rx)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Ly),Ry)) ).
fof(fact_1083_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),Ly)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Rx),Ry)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Lx),Rx)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ly),Ry)) ).
fof(fact_1084_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Ly)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Rx),Ry)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Lx),Rx)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ly),Ry)) ).
fof(fact_1085_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Ly)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Rx),Ry)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Lx),Rx)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ly),Ry)) ).
fof(fact_1086_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_1087_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_1088_mult__number__of__left,axiom,
! [V_16,W_8,Z_20] : 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_8)),Z_20)) = 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_8))),Z_20) ).
fof(fact_1089_mult__number__of__left,axiom,
! [V_16,W_8,Z_20] : 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_8)),Z_20)) = 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_8))),Z_20) ).
fof(fact_1090_mult__number__of__left,axiom,
! [V_16,W_8,Z_20] : 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_8)),Z_20)) = 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_8))),Z_20) ).
fof(fact_1091_mult__number__of__left,axiom,
! [V_16,W_8,Z_20] : 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_8)),Z_20)) = 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_8))),Z_20) ).
fof(fact_1092_arith__simps_I32_J,axiom,
! [V_15,W_7] : 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_7)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_7)) ).
fof(fact_1093_arith__simps_I32_J,axiom,
! [V_15,W_7] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_15)),hAPP_int_complex(number528085621omplex,W_7)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_7)) ).
fof(fact_1094_arith__simps_I32_J,axiom,
! [V_15,W_7] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_15)),hAPP_int_real(number267125858f_real,W_7)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_7)) ).
fof(fact_1095_arith__simps_I32_J,axiom,
! [V_15,W_7] : 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_7)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_15),W_7)) ).
fof(fact_1096_number__of__mult,axiom,
! [V_14,W_6] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_6)) = 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_6)) ).
fof(fact_1097_number__of__mult,axiom,
! [V_14,W_6] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_6)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_14)),hAPP_int_complex(number528085621omplex,W_6)) ).
fof(fact_1098_number__of__mult,axiom,
! [V_14,W_6] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_6)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_14)),hAPP_int_real(number267125858f_real,W_6)) ).
fof(fact_1099_number__of__mult,axiom,
! [V_14,W_6] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_14),W_6)) = 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_6)) ).
fof(fact_1100_of__nat__mult,axiom,
! [M_18,N_30] : hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(semiri151668891at_rat,M_18)),hAPP_nat_rat(semiri151668891at_rat,N_30)) ).
fof(fact_1101_of__nat__mult,axiom,
! [M_18,N_30] : hAPP_n522471361de_int(semiri1424489471de_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(semiri1424489471de_int,M_18)),hAPP_n522471361de_int(semiri1424489471de_int,N_30)) ).
fof(fact_1102_of__nat__mult,axiom,
! [M_18,N_30] : hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(semiri2020571505omplex,M_18)),hAPP_nat_complex(semiri2020571505omplex,N_30)) ).
fof(fact_1103_of__nat__mult,axiom,
! [M_18,N_30] : hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(semiri132038758t_real,M_18)),hAPP_nat_real(semiri132038758t_real,N_30)) ).
fof(fact_1104_of__nat__mult,axiom,
! [M_18,N_30] : hAPP_nat_nat(semiri984289939at_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(semiri984289939at_nat,M_18)),hAPP_nat_nat(semiri984289939at_nat,N_30)) ).
fof(fact_1105_of__nat__mult,axiom,
! [M_18,N_30] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_18),N_30)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M_18)),hAPP_nat_int(semiri1621563631at_int,N_30)) ).
fof(fact_1106_mult__zero__left,axiom,
! [A_138] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_138) = zero_zero_rat ).
fof(fact_1107_mult__zero__left,axiom,
! [A_138] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_138) = zero_z891286103de_int ).
fof(fact_1108_mult__zero__left,axiom,
! [A_138] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_138) = zero_zero_complex ).
fof(fact_1109_mult__zero__left,axiom,
! [A_138] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_138) = zero_z126310315umeral ).
fof(fact_1110_mult__zero__left,axiom,
! [A_138] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_138) = zero_zero_real ).
fof(fact_1111_mult__zero__left,axiom,
! [A_138] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_138) = zero_zero_nat ).
fof(fact_1112_mult__zero__left,axiom,
! [A_138] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_138) = zero_zero_int ).
fof(fact_1113_mult__zero__right,axiom,
! [A_137] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_137),zero_zero_rat) = zero_zero_rat ).
fof(fact_1114_mult__zero__right,axiom,
! [A_137] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_137),zero_z891286103de_int) = zero_z891286103de_int ).
fof(fact_1115_mult__zero__right,axiom,
! [A_137] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_137),zero_zero_complex) = zero_zero_complex ).
fof(fact_1116_mult__zero__right,axiom,
! [A_137] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_137),zero_z126310315umeral) = zero_z126310315umeral ).
fof(fact_1117_mult__zero__right,axiom,
! [A_137] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_137),zero_zero_real) = zero_zero_real ).
fof(fact_1118_mult__zero__right,axiom,
! [A_137] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_137),zero_zero_nat) = zero_zero_nat ).
fof(fact_1119_mult__zero__right,axiom,
! [A_137] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_137),zero_zero_int) = zero_zero_int ).
fof(fact_1120_mult__eq__0__iff,axiom,
! [A_1,B_1] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),B_1) = zero_zero_rat
<=> ( A_1 = zero_zero_rat
| B_1 = zero_zero_rat ) ) ).
fof(fact_1121_mult__eq__0__iff,axiom,
! [A_1,B_1] :
( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),B_1) = zero_zero_complex
<=> ( A_1 = zero_zero_complex
| B_1 = zero_zero_complex ) ) ).
fof(fact_1122_mult__eq__0__iff,axiom,
! [A_1,B_1] :
( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),B_1) = zero_zero_real
<=> ( A_1 = zero_zero_real
| B_1 = zero_zero_real ) ) ).
fof(fact_1123_mult__eq__0__iff,axiom,
! [A_1,B_1] :
( ( is_int(A_1)
& is_int(B_1) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),B_1) = zero_zero_int
<=> ( A_1 = zero_zero_int
| B_1 = zero_zero_int ) ) ) ).
fof(fact_1124_no__zero__divisors,axiom,
! [B_96,A_136] :
( A_136 != zero_zero_rat
=> ( B_96 != zero_zero_rat
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_136),B_96) != zero_zero_rat ) ) ).
fof(fact_1125_no__zero__divisors,axiom,
! [B_96,A_136] :
( A_136 != zero_z891286103de_int
=> ( B_96 != zero_z891286103de_int
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_136),B_96) != zero_z891286103de_int ) ) ).
fof(fact_1126_no__zero__divisors,axiom,
! [B_96,A_136] :
( A_136 != zero_zero_complex
=> ( B_96 != zero_zero_complex
=> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_136),B_96) != zero_zero_complex ) ) ).
fof(fact_1127_no__zero__divisors,axiom,
! [B_96,A_136] :
( A_136 != zero_z126310315umeral
=> ( B_96 != zero_z126310315umeral
=> hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_136),B_96) != zero_z126310315umeral ) ) ).
fof(fact_1128_no__zero__divisors,axiom,
! [B_96,A_136] :
( A_136 != zero_zero_real
=> ( B_96 != zero_zero_real
=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_136),B_96) != zero_zero_real ) ) ).
fof(fact_1129_no__zero__divisors,axiom,
! [B_96,A_136] :
( A_136 != zero_zero_nat
=> ( B_96 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_136),B_96) != zero_zero_nat ) ) ).
fof(fact_1130_no__zero__divisors,axiom,
! [B_96,A_136] :
( ( is_int(B_96)
& is_int(A_136) )
=> ( A_136 != zero_zero_int
=> ( B_96 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_136),B_96) != zero_zero_int ) ) ) ).
fof(fact_1131_divisors__zero,axiom,
! [A_135,B_95] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_135),B_95) = zero_zero_rat
=> ( A_135 = zero_zero_rat
| B_95 = zero_zero_rat ) ) ).
fof(fact_1132_divisors__zero,axiom,
! [A_135,B_95] :
( hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_135),B_95) = zero_z891286103de_int
=> ( A_135 = zero_z891286103de_int
| B_95 = zero_z891286103de_int ) ) ).
fof(fact_1133_divisors__zero,axiom,
! [A_135,B_95] :
( hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_135),B_95) = zero_zero_complex
=> ( A_135 = zero_zero_complex
| B_95 = zero_zero_complex ) ) ).
fof(fact_1134_divisors__zero,axiom,
! [A_135,B_95] :
( hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_135),B_95) = zero_z126310315umeral
=> ( A_135 = zero_z126310315umeral
| B_95 = zero_z126310315umeral ) ) ).
fof(fact_1135_divisors__zero,axiom,
! [A_135,B_95] :
( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_135),B_95) = zero_zero_real
=> ( A_135 = zero_zero_real
| B_95 = zero_zero_real ) ) ).
fof(fact_1136_divisors__zero,axiom,
! [A_135,B_95] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_135),B_95) = zero_zero_nat
=> ( A_135 = zero_zero_nat
| B_95 = zero_zero_nat ) ) ).
fof(fact_1137_divisors__zero,axiom,
! [A_135,B_95] :
( ( is_int(A_135)
& is_int(B_95) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_135),B_95) = zero_zero_int
=> ( A_135 = zero_zero_int
| B_95 = zero_zero_int ) ) ) ).
fof(fact_1138_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_134] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A_134) = zero_zero_rat ).
fof(fact_1139_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_134] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,zero_z891286103de_int),A_134) = zero_z891286103de_int ).
fof(fact_1140_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_134] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,zero_zero_complex),A_134) = zero_zero_complex ).
fof(fact_1141_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_134] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,zero_z126310315umeral),A_134) = zero_z126310315umeral ).
fof(fact_1142_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_134] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,zero_zero_real),A_134) = zero_zero_real ).
fof(fact_1143_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_134] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A_134) = zero_zero_nat ).
fof(fact_1144_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A_134] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,zero_zero_int),A_134) = zero_zero_int ).
fof(fact_1145_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_133] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_133),zero_zero_rat) = zero_zero_rat ).
fof(fact_1146_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_133] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_133),zero_z891286103de_int) = zero_z891286103de_int ).
fof(fact_1147_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_133] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_133),zero_zero_complex) = zero_zero_complex ).
fof(fact_1148_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_133] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_133),zero_z126310315umeral) = zero_z126310315umeral ).
fof(fact_1149_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_133] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_133),zero_zero_real) = zero_zero_real ).
fof(fact_1150_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_133] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_133),zero_zero_nat) = zero_zero_nat ).
fof(fact_1151_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A_133] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_133),zero_zero_int) = zero_zero_int ).
fof(fact_1152_comm__semiring__class_Odistrib,axiom,
! [A_132,B_94,C_50] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_132),B_94)),C_50) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_132),C_50)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_94),C_50)) ).
fof(fact_1153_comm__semiring__class_Odistrib,axiom,
! [A_132,B_94,C_50] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_132),B_94)),C_50) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_132),C_50)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_94),C_50)) ).
fof(fact_1154_comm__semiring__class_Odistrib,axiom,
! [A_132,B_94,C_50] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_132),B_94)),C_50) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_132),C_50)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_94),C_50)) ).
fof(fact_1155_comm__semiring__class_Odistrib,axiom,
! [A_132,B_94,C_50] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_132),B_94)),C_50) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_132),C_50)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_94),C_50)) ).
fof(fact_1156_comm__semiring__class_Odistrib,axiom,
! [A_132,B_94,C_50] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_132),B_94)),C_50) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_132),C_50)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_94),C_50)) ).
fof(fact_1157_comm__semiring__class_Odistrib,axiom,
! [A_132,B_94,C_50] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_132),B_94)),C_50) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_132),C_50)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_94),C_50)) ).
fof(fact_1158_combine__common__factor,axiom,
! [A_131,E_3,B_93,C_49] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_131),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_93),E_3)),C_49)) = 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_131),B_93)),E_3)),C_49) ).
fof(fact_1159_combine__common__factor,axiom,
! [A_131,E_3,B_93,C_49] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_131),E_3)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_93),E_3)),C_49)) = 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_131),B_93)),E_3)),C_49) ).
fof(fact_1160_combine__common__factor,axiom,
! [A_131,E_3,B_93,C_49] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_131),E_3)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_93),E_3)),C_49)) = 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_131),B_93)),E_3)),C_49) ).
fof(fact_1161_combine__common__factor,axiom,
! [A_131,E_3,B_93,C_49] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_131),E_3)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_93),E_3)),C_49)) = 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_131),B_93)),E_3)),C_49) ).
fof(fact_1162_combine__common__factor,axiom,
! [A_131,E_3,B_93,C_49] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_131),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_93),E_3)),C_49)) = 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_131),B_93)),E_3)),C_49) ).
fof(fact_1163_combine__common__factor,axiom,
! [A_131,E_3,B_93,C_49] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_131),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_93),E_3)),C_49)) = 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_131),B_93)),E_3)),C_49) ).
fof(fact_1164_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_55,Y_47,Z_19] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_55),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Y_47),Z_19)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_55),Y_47)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_55),Z_19)) ).
fof(fact_1165_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_55,Y_47,Z_19] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_55),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,Y_47),Z_19)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_55),Y_47)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_55),Z_19)) ).
fof(fact_1166_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_55,Y_47,Z_19] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_55),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Y_47),Z_19)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_55),Y_47)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_55),Z_19)) ).
fof(fact_1167_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_55,Y_47,Z_19] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_55),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y_47),Z_19)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_55),Y_47)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_55),Z_19)) ).
fof(fact_1168_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_55,Y_47,Z_19] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_55),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_47),Z_19)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_55),Y_47)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_55),Z_19)) ).
fof(fact_1169_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_55,Y_47,Z_19] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_55),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Y_47),Z_19)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_55),Y_47)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_55),Z_19)) ).
fof(fact_1170_crossproduct__noteq,axiom,
! [C,D_1,A_1,B_1] :
( ( A_1 != B_1
& C != D_1 )
<=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_1),D_1)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),D_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_1),C)) ) ).
fof(fact_1171_crossproduct__noteq,axiom,
! [C,D_1,A_1,B_1] :
( ( A_1 != B_1
& C != D_1 )
<=> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_1),D_1)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),D_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_1),C)) ) ).
fof(fact_1172_crossproduct__noteq,axiom,
! [C,D_1,A_1,B_1] :
( ( A_1 != B_1
& C != D_1 )
<=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),D_1)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),D_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),C)) ) ).
fof(fact_1173_crossproduct__noteq,axiom,
! [C,D_1,A_1,B_1] :
( ( A_1 != B_1
& C != D_1 )
<=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),C)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),D_1)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),D_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),C)) ) ).
fof(fact_1174_crossproduct__noteq,axiom,
! [C,D_1,A_1,B_1] :
( ( is_int(C)
& is_int(D_1)
& is_int(A_1)
& is_int(B_1) )
=> ( ( A_1 != B_1
& C != D_1 )
<=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),D_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),C)) ) ) ).
fof(fact_1175_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_130,B_92,C_48] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_130),B_92)),C_48) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_130),C_48)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_92),C_48)) ).
fof(fact_1176_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_130,B_92,C_48] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_130),B_92)),C_48) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_130),C_48)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_92),C_48)) ).
fof(fact_1177_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_130,B_92,C_48] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_130),B_92)),C_48) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_130),C_48)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_92),C_48)) ).
fof(fact_1178_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_130,B_92,C_48] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_130),B_92)),C_48) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_130),C_48)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_92),C_48)) ).
fof(fact_1179_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_130,B_92,C_48] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_130),B_92)),C_48) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_130),C_48)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_92),C_48)) ).
fof(fact_1180_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_130,B_92,C_48] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_130),B_92)),C_48) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_130),C_48)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_92),C_48)) ).
fof(fact_1181_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_129,M_17,B_91] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_129),M_17)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_91),M_17)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_129),B_91)),M_17) ).
fof(fact_1182_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_129,M_17,B_91] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_129),M_17)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_91),M_17)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_129),B_91)),M_17) ).
fof(fact_1183_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_129,M_17,B_91] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_129),M_17)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_91),M_17)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_129),B_91)),M_17) ).
fof(fact_1184_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_129,M_17,B_91] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_129),M_17)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_91),M_17)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_129),B_91)),M_17) ).
fof(fact_1185_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_129,M_17,B_91] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_129),M_17)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_91),M_17)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_129),B_91)),M_17) ).
fof(fact_1186_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_129,M_17,B_91] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_129),M_17)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_91),M_17)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_129),B_91)),M_17) ).
fof(fact_1187_crossproduct__eq,axiom,
! [Wa,Ya,Xa,Z_2] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Wa),Ya)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Xa),Z_2)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Wa),Z_2)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Xa),Ya))
<=> ( Wa = Xa
| Ya = Z_2 ) ) ).
fof(fact_1188_crossproduct__eq,axiom,
! [Wa,Ya,Xa,Z_2] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Wa),Ya)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa),Z_2)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Wa),Z_2)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Xa),Ya))
<=> ( Wa = Xa
| Ya = Z_2 ) ) ).
fof(fact_1189_crossproduct__eq,axiom,
! [Wa,Ya,Xa,Z_2] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Wa),Ya)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Z_2)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Wa),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Ya))
<=> ( Wa = Xa
| Ya = Z_2 ) ) ).
fof(fact_1190_crossproduct__eq,axiom,
! [Wa,Ya,Xa,Z_2] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Wa),Ya)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Xa),Z_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Wa),Z_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Xa),Ya))
<=> ( Wa = Xa
| Ya = Z_2 ) ) ).
fof(fact_1191_crossproduct__eq,axiom,
! [Wa,Ya,Xa,Z_2] :
( ( is_int(Wa)
& is_int(Ya)
& is_int(Xa)
& 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,Wa),Ya)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Z_2)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Wa),Z_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya))
<=> ( Wa = Xa
| Ya = Z_2 ) ) ) ).
fof(fact_1192_mult__1__left,axiom,
! [A_128] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_128) = A_128 ).
fof(fact_1193_mult__1__left,axiom,
! [A_128] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_128) = A_128 ).
fof(fact_1194_mult__1__left,axiom,
! [A_128] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_128) = A_128 ).
fof(fact_1195_mult__1__left,axiom,
! [A_128] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_128) = A_128 ).
fof(fact_1196_mult__1__left,axiom,
! [A_128] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_128) = A_128 ).
fof(fact_1197_mult__1__left,axiom,
! [A_128] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_128) = A_128 ).
fof(fact_1198_mult__1__left,axiom,
! [A_128] :
( is_int(A_128)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_128) = A_128 ) ).
fof(fact_1199_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_127] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_127) = A_127 ).
fof(fact_1200_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_127] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_127) = A_127 ).
fof(fact_1201_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_127] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_127) = A_127 ).
fof(fact_1202_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_127] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_127) = A_127 ).
fof(fact_1203_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_127] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_127) = A_127 ).
fof(fact_1204_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_127] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_127) = A_127 ).
fof(fact_1205_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A_127] :
( is_int(A_127)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_127) = A_127 ) ).
fof(fact_1206_mult__1,axiom,
! [A_126] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,one_one_rat),A_126) = A_126 ).
fof(fact_1207_mult__1,axiom,
! [A_126] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,one_on1684967323de_int),A_126) = A_126 ).
fof(fact_1208_mult__1,axiom,
! [A_126] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,one_one_complex),A_126) = A_126 ).
fof(fact_1209_mult__1,axiom,
! [A_126] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,one_on1645066479umeral),A_126) = A_126 ).
fof(fact_1210_mult__1,axiom,
! [A_126] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),A_126) = A_126 ).
fof(fact_1211_mult__1,axiom,
! [A_126] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),A_126) = A_126 ).
fof(fact_1212_mult__1,axiom,
! [A_126] :
( is_int(A_126)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,one_one_int),A_126) = A_126 ) ).
fof(fact_1213_mult__1__right,axiom,
! [A_125] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_125),one_one_rat) = A_125 ).
fof(fact_1214_mult__1__right,axiom,
! [A_125] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_125),one_on1684967323de_int) = A_125 ).
fof(fact_1215_mult__1__right,axiom,
! [A_125] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_125),one_one_complex) = A_125 ).
fof(fact_1216_mult__1__right,axiom,
! [A_125] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_125),one_on1645066479umeral) = A_125 ).
fof(fact_1217_mult__1__right,axiom,
! [A_125] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_125),one_one_real) = A_125 ).
fof(fact_1218_mult__1__right,axiom,
! [A_125] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_125),one_one_nat) = A_125 ).
fof(fact_1219_mult__1__right,axiom,
! [A_125] :
( is_int(A_125)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_125),one_one_int) = A_125 ) ).
fof(fact_1220_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_124] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_124),one_one_rat) = A_124 ).
fof(fact_1221_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_124] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_124),one_on1684967323de_int) = A_124 ).
fof(fact_1222_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_124] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_124),one_one_complex) = A_124 ).
fof(fact_1223_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_124] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_124),one_on1645066479umeral) = A_124 ).
fof(fact_1224_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_124] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_124),one_one_real) = A_124 ).
fof(fact_1225_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_124] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_124),one_one_nat) = A_124 ).
fof(fact_1226_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A_124] :
( is_int(A_124)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_124),one_one_int) = A_124 ) ).
fof(fact_1227_mult_Ocomm__neutral,axiom,
! [A_123] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_123),one_one_rat) = A_123 ).
fof(fact_1228_mult_Ocomm__neutral,axiom,
! [A_123] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_123),one_on1684967323de_int) = A_123 ).
fof(fact_1229_mult_Ocomm__neutral,axiom,
! [A_123] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_123),one_one_complex) = A_123 ).
fof(fact_1230_mult_Ocomm__neutral,axiom,
! [A_123] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_123),one_on1645066479umeral) = A_123 ).
fof(fact_1231_mult_Ocomm__neutral,axiom,
! [A_123] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_123),one_one_real) = A_123 ).
fof(fact_1232_mult_Ocomm__neutral,axiom,
! [A_123] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_123),one_one_nat) = A_123 ).
fof(fact_1233_mult_Ocomm__neutral,axiom,
! [A_123] :
( is_int(A_123)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_123),one_one_int) = A_123 ) ).
fof(fact_1234_power__commutes,axiom,
! [A_122,N_29] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_122),N_29)),A_122) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_122),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_122),N_29)) ).
fof(fact_1235_power__commutes,axiom,
! [A_122,N_29] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_122),N_29)),A_122) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_122),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_122),N_29)) ).
fof(fact_1236_power__commutes,axiom,
! [A_122,N_29] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_122),N_29)),A_122) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_122),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_122),N_29)) ).
fof(fact_1237_power__commutes,axiom,
! [A_122,N_29] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_122),N_29)),A_122) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_122),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_122),N_29)) ).
fof(fact_1238_power__commutes,axiom,
! [A_122,N_29] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_122),N_29)),A_122) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_122),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_122),N_29)) ).
fof(fact_1239_power__commutes,axiom,
! [A_122,N_29] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_122),N_29)),A_122) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_122),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_122),N_29)) ).
fof(fact_1240_power__mult__distrib,axiom,
! [A_121,B_90,N_28] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_121),B_90)),N_28) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_121),N_28)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,B_90),N_28)) ).
fof(fact_1241_power__mult__distrib,axiom,
! [A_121,B_90,N_28] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_121),B_90)),N_28) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_121),N_28)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,B_90),N_28)) ).
fof(fact_1242_power__mult__distrib,axiom,
! [A_121,B_90,N_28] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_121),B_90)),N_28) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_121),N_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_90),N_28)) ).
fof(fact_1243_power__mult__distrib,axiom,
! [A_121,B_90,N_28] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_121),B_90)),N_28) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_121),N_28)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_90),N_28)) ).
fof(fact_1244_power__mult__distrib,axiom,
! [A_121,B_90,N_28] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_121),B_90)),N_28) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_121),N_28)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,B_90),N_28)) ).
fof(fact_1245_power__mult__distrib,axiom,
! [A_121,B_90,N_28] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_121),B_90)),N_28) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_121),N_28)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,B_90),N_28)) ).
fof(fact_1246_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_54,Y_46,Q_9] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_54),Y_46)),Q_9) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_54),Q_9)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_46),Q_9)) ).
fof(fact_1247_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_54,Y_46,Q_9] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_54),Y_46)),Q_9) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_54),Q_9)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_46),Q_9)) ).
fof(fact_1248_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_54,Y_46,Q_9] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_54),Y_46)),Q_9) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_54),Q_9)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_46),Q_9)) ).
fof(fact_1249_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_54,Y_46,Q_9] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_54),Y_46)),Q_9) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_54),Q_9)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_46),Q_9)) ).
fof(fact_1250_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_54,Y_46,Q_9] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_54),Y_46)),Q_9) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_54),Q_9)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,Y_46),Q_9)) ).
fof(fact_1251_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_54,Y_46,Q_9] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_54),Y_46)),Q_9) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_54),Q_9)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,Y_46),Q_9)) ).
fof(fact_1252_le__0__eq,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),zero_zero_nat))
<=> Na = zero_zero_nat ) ).
fof(fact_1253_less__eq__nat_Osimps_I1_J,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
fof(fact_1254_mult__Pls,axiom,
! [W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,pls),W) = pls ).
fof(fact_1255_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_1256_nat__less__le,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
& Ma != Na ) ) ).
fof(fact_1257_le__eq__less__or__eq,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
| Ma = Na ) ) ).
fof(fact_1258_less__imp__le__nat,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_1259_le__neq__implies__less,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( M != N
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
fof(fact_1260_less__or__eq__imp__le,axiom,
! [M,N] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
| M = N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_1261_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_1262_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_1263_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_1264_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_1265_le__add2,axiom,
! [N,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) ).
fof(fact_1266_le__add1,axiom,
! [N,M] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M))) ).
fof(fact_1267_le__iff__add,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
<=> ? [K_1] : Na = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1) ) ).
fof(fact_1268_nat__add__left__cancel__le,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_1269_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_1270_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_1271_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_1272_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_1273_add__leD2,axiom,
! [M,K_2,N] :
( 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))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)) ) ).
fof(fact_1274_add__leD1,axiom,
! [M,K_2,N] :
( 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))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_1275_add__leE,axiom,
! [M,K_2,N] :
( 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))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)) ) ) ).
fof(fact_1276_times__numeral__code_I5_J,axiom,
! [V_1,W] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_1)),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_1),W)) ).
fof(fact_1277_real__add__left__mono,axiom,
! [Z_1,X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_1),Y))) ) ).
fof(fact_1278_semiring__mult__number__of,axiom,
! [V_13,V_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_12)),hAPP_int_rat(number_number_of_rat,V_13)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_12),V_13)) ) ) ).
fof(fact_1279_semiring__mult__number__of,axiom,
! [V_13,V_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_12)),hAPP_int_complex(number528085621omplex,V_13)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_12),V_13)) ) ) ).
fof(fact_1280_semiring__mult__number__of,axiom,
! [V_13,V_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_12)),hAPP_int_real(number267125858f_real,V_13)) = hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_12),V_13)) ) ) ).
fof(fact_1281_semiring__mult__number__of,axiom,
! [V_13,V_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_12)),hAPP_int_nat(number_number_of_nat,V_13)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_12),V_13)) ) ) ).
fof(fact_1282_semiring__mult__number__of,axiom,
! [V_13,V_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),V_13))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_12)),hAPP_int_int(number_number_of_int,V_13)) = hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_12),V_13)) ) ) ).
fof(fact_1283_zero__le__square,axiom,
! [A_120] : 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_120),A_120))) ).
fof(fact_1284_zero__le__square,axiom,
! [A_120] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_120),A_120))) ).
fof(fact_1285_zero__le__square,axiom,
! [A_120] : 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_120),A_120))) ).
fof(fact_1286_zero__le__mult__iff,axiom,
! [A_1,B_1] :
( 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_1),B_1)))
<=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_1))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_1)) )
| ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),zero_zero_rat))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_1),zero_zero_rat)) ) ) ) ).
fof(fact_1287_zero__le__mult__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),B_1)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_1)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_1),zero_zero_real)) ) ) ) ).
fof(fact_1288_zero__le__mult__iff,axiom,
! [A_1,B_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,A_1),B_1)))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) ) ) ) ).
fof(fact_1289_mult__le__0__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),B_1)),zero_zero_rat))
<=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_1))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_1),zero_zero_rat)) )
| ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),zero_zero_rat))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_1)) ) ) ) ).
fof(fact_1290_mult__le__0__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),B_1)),zero_zero_real))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_1),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_1)) ) ) ) ).
fof(fact_1291_mult__le__0__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),B_1)),zero_zero_int))
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_1)) ) ) ) ).
fof(fact_1292_mult__nonneg__nonneg,axiom,
! [B_89,A_119] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_119))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_89))
=> 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_119),B_89))) ) ) ).
fof(fact_1293_mult__nonneg__nonneg,axiom,
! [B_89,A_119] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_119))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_89))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_119),B_89))) ) ) ).
fof(fact_1294_mult__nonneg__nonneg,axiom,
! [B_89,A_119] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_119))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_89))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_119),B_89))) ) ) ).
fof(fact_1295_mult__nonneg__nonneg,axiom,
! [B_89,A_119] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_119))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_89))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_119),B_89))) ) ) ).
fof(fact_1296_mult__nonneg__nonneg,axiom,
! [B_89,A_119] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_119))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_89))
=> 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_119),B_89))) ) ) ).
fof(fact_1297_mult__nonneg__nonneg,axiom,
! [B_89,A_119] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_119))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_89))
=> 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_119),B_89))) ) ) ).
fof(fact_1298_mult__nonneg__nonpos,axiom,
! [B_88,A_118] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_118))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_88),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_118),B_88)),zero_zero_rat)) ) ) ).
fof(fact_1299_mult__nonneg__nonpos,axiom,
! [B_88,A_118] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_118))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_88),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_118),B_88)),zero_z891286103de_int)) ) ) ).
fof(fact_1300_mult__nonneg__nonpos,axiom,
! [B_88,A_118] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_118))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_88),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_118),B_88)),zero_z126310315umeral)) ) ) ).
fof(fact_1301_mult__nonneg__nonpos,axiom,
! [B_88,A_118] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_118))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_88),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_118),B_88)),zero_zero_real)) ) ) ).
fof(fact_1302_mult__nonneg__nonpos,axiom,
! [B_88,A_118] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_118))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_88),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_118),B_88)),zero_zero_nat)) ) ) ).
fof(fact_1303_mult__nonneg__nonpos,axiom,
! [B_88,A_118] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_118))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_88),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_118),B_88)),zero_zero_int)) ) ) ).
fof(fact_1304_mult__nonneg__nonpos2,axiom,
! [B_87,A_117] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_117))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_87),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_87),A_117)),zero_zero_rat)) ) ) ).
fof(fact_1305_mult__nonneg__nonpos2,axiom,
! [B_87,A_117] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_117))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_87),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_87),A_117)),zero_z891286103de_int)) ) ) ).
fof(fact_1306_mult__nonneg__nonpos2,axiom,
! [B_87,A_117] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_117))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_87),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_87),A_117)),zero_z126310315umeral)) ) ) ).
fof(fact_1307_mult__nonneg__nonpos2,axiom,
! [B_87,A_117] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_117))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_87),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_87),A_117)),zero_zero_real)) ) ) ).
fof(fact_1308_mult__nonneg__nonpos2,axiom,
! [B_87,A_117] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_117))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_87),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_87),A_117)),zero_zero_nat)) ) ) ).
fof(fact_1309_mult__nonneg__nonpos2,axiom,
! [B_87,A_117] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_117))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_87),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_87),A_117)),zero_zero_int)) ) ) ).
fof(fact_1310_mult__nonpos__nonneg,axiom,
! [B_86,A_116] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_116),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_86))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_116),B_86)),zero_zero_rat)) ) ) ).
fof(fact_1311_mult__nonpos__nonneg,axiom,
! [B_86,A_116] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_116),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_86))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_116),B_86)),zero_z891286103de_int)) ) ) ).
fof(fact_1312_mult__nonpos__nonneg,axiom,
! [B_86,A_116] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_116),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_86))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_116),B_86)),zero_z126310315umeral)) ) ) ).
fof(fact_1313_mult__nonpos__nonneg,axiom,
! [B_86,A_116] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_116),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_86))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_116),B_86)),zero_zero_real)) ) ) ).
fof(fact_1314_mult__nonpos__nonneg,axiom,
! [B_86,A_116] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_116),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_86))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_116),B_86)),zero_zero_nat)) ) ) ).
fof(fact_1315_mult__nonpos__nonneg,axiom,
! [B_86,A_116] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_116),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_86))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_116),B_86)),zero_zero_int)) ) ) ).
fof(fact_1316_mult__nonpos__nonpos,axiom,
! [B_85,A_115] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_115),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_85),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_115),B_85))) ) ) ).
fof(fact_1317_mult__nonpos__nonpos,axiom,
! [B_85,A_115] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_115),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_85),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_115),B_85))) ) ) ).
fof(fact_1318_mult__nonpos__nonpos,axiom,
! [B_85,A_115] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_115),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_85),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_115),B_85))) ) ) ).
fof(fact_1319_mult__right__mono,axiom,
! [C_47,A_114,B_84] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_114),B_84))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_47))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_114),C_47)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_84),C_47))) ) ) ).
fof(fact_1320_mult__right__mono,axiom,
! [C_47,A_114,B_84] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_114),B_84))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_47))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_114),C_47)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_84),C_47))) ) ) ).
fof(fact_1321_mult__right__mono,axiom,
! [C_47,A_114,B_84] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_114),B_84))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_47))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_114),C_47)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_84),C_47))) ) ) ).
fof(fact_1322_mult__right__mono,axiom,
! [C_47,A_114,B_84] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_114),B_84))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_47))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_114),C_47)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_84),C_47))) ) ) ).
fof(fact_1323_mult__right__mono,axiom,
! [C_47,A_114,B_84] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_114),B_84))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_47))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_114),C_47)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_84),C_47))) ) ) ).
fof(fact_1324_mult__right__mono,axiom,
! [C_47,A_114,B_84] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_114),B_84))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_47))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_114),C_47)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_84),C_47))) ) ) ).
fof(fact_1325_mult__left__mono,axiom,
! [C_46,A_113,B_83] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_113),B_83))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_46))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_46),A_113)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_46),B_83))) ) ) ).
fof(fact_1326_mult__left__mono,axiom,
! [C_46,A_113,B_83] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_113),B_83))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_46))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_46),A_113)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_46),B_83))) ) ) ).
fof(fact_1327_mult__left__mono,axiom,
! [C_46,A_113,B_83] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_113),B_83))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_46))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_46),A_113)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_46),B_83))) ) ) ).
fof(fact_1328_mult__left__mono,axiom,
! [C_46,A_113,B_83] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_113),B_83))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_46))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_46),A_113)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_46),B_83))) ) ) ).
fof(fact_1329_mult__left__mono,axiom,
! [C_46,A_113,B_83] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_113),B_83))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_46))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_46),A_113)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_46),B_83))) ) ) ).
fof(fact_1330_mult__left__mono,axiom,
! [C_46,A_113,B_83] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_113),B_83))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_46))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_46),A_113)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_46),B_83))) ) ) ).
fof(fact_1331_comm__mult__left__mono,axiom,
! [C_45,A_112,B_82] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_112),B_82))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_45))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_45),A_112)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_45),B_82))) ) ) ).
fof(fact_1332_comm__mult__left__mono,axiom,
! [C_45,A_112,B_82] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_112),B_82))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_45))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_45),A_112)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_45),B_82))) ) ) ).
fof(fact_1333_comm__mult__left__mono,axiom,
! [C_45,A_112,B_82] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_112),B_82))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_45))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_45),A_112)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_45),B_82))) ) ) ).
fof(fact_1334_comm__mult__left__mono,axiom,
! [C_45,A_112,B_82] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_112),B_82))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_45))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_45),A_112)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_45),B_82))) ) ) ).
fof(fact_1335_comm__mult__left__mono,axiom,
! [C_45,A_112,B_82] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_112),B_82))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_45))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_45),A_112)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_45),B_82))) ) ) ).
fof(fact_1336_comm__mult__left__mono,axiom,
! [C_45,A_112,B_82] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_112),B_82))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_45))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_45),A_112)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_45),B_82))) ) ) ).
fof(fact_1337_mult__right__mono__neg,axiom,
! [C_44,B_81,A_111] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_81),A_111))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_44),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_111),C_44)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_81),C_44))) ) ) ).
fof(fact_1338_mult__right__mono__neg,axiom,
! [C_44,B_81,A_111] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_81),A_111))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_44),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_111),C_44)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_81),C_44))) ) ) ).
fof(fact_1339_mult__right__mono__neg,axiom,
! [C_44,B_81,A_111] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_81),A_111))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_44),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_111),C_44)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_81),C_44))) ) ) ).
fof(fact_1340_mult__left__mono__neg,axiom,
! [C_43,B_80,A_110] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_80),A_110))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_43),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_43),A_110)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_43),B_80))) ) ) ).
fof(fact_1341_mult__left__mono__neg,axiom,
! [C_43,B_80,A_110] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_80),A_110))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_43),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_43),A_110)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_43),B_80))) ) ) ).
fof(fact_1342_mult__left__mono__neg,axiom,
! [C_43,B_80,A_110] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_80),A_110))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_43),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_43),A_110)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_43),B_80))) ) ) ).
fof(fact_1343_mult__mono_H,axiom,
! [C_42,D_15,A_109,B_79] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_109),B_79))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_42),D_15))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_109))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_42))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_109),C_42)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_79),D_15))) ) ) ) ) ).
fof(fact_1344_mult__mono_H,axiom,
! [C_42,D_15,A_109,B_79] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_109),B_79))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_42),D_15))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_109))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_42))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_109),C_42)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_79),D_15))) ) ) ) ) ).
fof(fact_1345_mult__mono_H,axiom,
! [C_42,D_15,A_109,B_79] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_109),B_79))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_42),D_15))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_109))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_42))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_109),C_42)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_79),D_15))) ) ) ) ) ).
fof(fact_1346_mult__mono_H,axiom,
! [C_42,D_15,A_109,B_79] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_109),B_79))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_42),D_15))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_109))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_42))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_109),C_42)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_79),D_15))) ) ) ) ) ).
fof(fact_1347_mult__mono_H,axiom,
! [C_42,D_15,A_109,B_79] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_109),B_79))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_42),D_15))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_109))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_42))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_109),C_42)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_79),D_15))) ) ) ) ) ).
fof(fact_1348_mult__mono_H,axiom,
! [C_42,D_15,A_109,B_79] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_109),B_79))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_42),D_15))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_109))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_42))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_109),C_42)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_79),D_15))) ) ) ) ) ).
fof(fact_1349_mult__mono,axiom,
! [C_41,D_14,A_108,B_78] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_108),B_78))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_41),D_14))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_78))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_41))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_108),C_41)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_78),D_14))) ) ) ) ) ).
fof(fact_1350_mult__mono,axiom,
! [C_41,D_14,A_108,B_78] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_108),B_78))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_41),D_14))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_78))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_41))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_108),C_41)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_78),D_14))) ) ) ) ) ).
fof(fact_1351_mult__mono,axiom,
! [C_41,D_14,A_108,B_78] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_108),B_78))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_41),D_14))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_78))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_41))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_108),C_41)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_78),D_14))) ) ) ) ) ).
fof(fact_1352_mult__mono,axiom,
! [C_41,D_14,A_108,B_78] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_108),B_78))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_41),D_14))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_78))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_41))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_108),C_41)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_78),D_14))) ) ) ) ) ).
fof(fact_1353_mult__mono,axiom,
! [C_41,D_14,A_108,B_78] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_108),B_78))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_41),D_14))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_78))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_41))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_108),C_41)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_78),D_14))) ) ) ) ) ).
fof(fact_1354_mult__mono,axiom,
! [C_41,D_14,A_108,B_78] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_108),B_78))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_41),D_14))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_78))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_41))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_108),C_41)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_78),D_14))) ) ) ) ) ).
fof(fact_1355_split__mult__pos__le,axiom,
! [B_77,A_107] :
( ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_107))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_77)) )
| ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_107),zero_zero_rat))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_77),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_107),B_77))) ) ).
fof(fact_1356_split__mult__pos__le,axiom,
! [B_77,A_107] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_107))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_77)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_107),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_77),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_107),B_77))) ) ).
fof(fact_1357_split__mult__pos__le,axiom,
! [B_77,A_107] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_107))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_77)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_107),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_77),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_107),B_77))) ) ).
fof(fact_1358_split__mult__neg__le,axiom,
! [B_76,A_106] :
( ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_106))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_76),zero_zero_rat)) )
| ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_106),zero_zero_rat))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_76)) ) )
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_106),B_76)),zero_zero_rat)) ) ).
fof(fact_1359_split__mult__neg__le,axiom,
! [B_76,A_106] :
( ( ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_106))
& hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_76),zero_z891286103de_int)) )
| ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_106),zero_z891286103de_int))
& hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),B_76)) ) )
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_106),B_76)),zero_z891286103de_int)) ) ).
fof(fact_1360_split__mult__neg__le,axiom,
! [B_76,A_106] :
( ( ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_106))
& hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,B_76),zero_z126310315umeral)) )
| ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_106),zero_z126310315umeral))
& hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),B_76)) ) )
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_106),B_76)),zero_z126310315umeral)) ) ).
fof(fact_1361_split__mult__neg__le,axiom,
! [B_76,A_106] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_106))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_76),zero_zero_real)) )
| ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_106),zero_zero_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_76)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_106),B_76)),zero_zero_real)) ) ).
fof(fact_1362_split__mult__neg__le,axiom,
! [B_76,A_106] :
( ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_106))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_76),zero_zero_nat)) )
| ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_106),zero_zero_nat))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_76)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_106),B_76)),zero_zero_nat)) ) ).
fof(fact_1363_split__mult__neg__le,axiom,
! [B_76,A_106] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_106))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_76),zero_zero_int)) )
| ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_106),zero_zero_int))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_76)) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_106),B_76)),zero_zero_int)) ) ).
fof(fact_1364_not__square__less__zero,axiom,
! [A_105] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_105),A_105)),zero_zero_rat)) ).
fof(fact_1365_not__square__less__zero,axiom,
! [A_105] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_105),A_105)),zero_zero_real)) ).
fof(fact_1366_not__square__less__zero,axiom,
! [A_105] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_105),A_105)),zero_zero_int)) ).
fof(fact_1367_mult__less__cancel__right__disj,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_1),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_1),B_1)) )
| ( 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_1),A_1)) ) ) ) ).
fof(fact_1368_mult__less__cancel__right__disj,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),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_1),B_1)) )
| ( 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_1),A_1)) ) ) ) ).
fof(fact_1369_mult__less__cancel__right__disj,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),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_1),B_1)) )
| ( 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_1),A_1)) ) ) ) ).
fof(fact_1370_mult__less__cancel__left__disj,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B_1)))
<=> ( ( 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_1),B_1)) )
| ( 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_1),A_1)) ) ) ) ).
fof(fact_1371_mult__less__cancel__left__disj,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1)))
<=> ( ( 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_1),B_1)) )
| ( 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_1),A_1)) ) ) ) ).
fof(fact_1372_mult__less__cancel__left__disj,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),A_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)))
<=> ( ( 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_1),B_1)) )
| ( 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_1),A_1)) ) ) ) ).
fof(fact_1373_mult__less__cancel__left__pos,axiom,
! [A_1,B_1,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_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B_1)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),B_1)) ) ) ).
fof(fact_1374_mult__less__cancel__left__pos,axiom,
! [A_1,B_1,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_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1)) ) ) ).
fof(fact_1375_mult__less__cancel__left__pos,axiom,
! [A_1,B_1,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_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) ) ) ).
fof(fact_1376_mult__pos__pos,axiom,
! [B_75,A_104] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_104))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_75))
=> 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_104),B_75))) ) ) ).
fof(fact_1377_mult__pos__pos,axiom,
! [B_75,A_104] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_104))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_75))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_104),B_75))) ) ) ).
fof(fact_1378_mult__pos__pos,axiom,
! [B_75,A_104] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_104))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_75))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_104),B_75))) ) ) ).
fof(fact_1379_mult__pos__pos,axiom,
! [B_75,A_104] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_104))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_75))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_104),B_75))) ) ) ).
fof(fact_1380_mult__pos__pos,axiom,
! [B_75,A_104] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_104))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_75))
=> 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_104),B_75))) ) ) ).
fof(fact_1381_mult__pos__pos,axiom,
! [B_75,A_104] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_104))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_75))
=> 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_104),B_75))) ) ) ).
fof(fact_1382_mult__pos__neg,axiom,
! [B_74,A_103] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_103))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_74),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_103),B_74)),zero_zero_rat)) ) ) ).
fof(fact_1383_mult__pos__neg,axiom,
! [B_74,A_103] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_103))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_74),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_103),B_74)),zero_z891286103de_int)) ) ) ).
fof(fact_1384_mult__pos__neg,axiom,
! [B_74,A_103] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_103))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_74),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_103),B_74)),zero_z126310315umeral)) ) ) ).
fof(fact_1385_mult__pos__neg,axiom,
! [B_74,A_103] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_103))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_74),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_103),B_74)),zero_zero_real)) ) ) ).
fof(fact_1386_mult__pos__neg,axiom,
! [B_74,A_103] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_103))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_74),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_103),B_74)),zero_zero_nat)) ) ) ).
fof(fact_1387_mult__pos__neg,axiom,
! [B_74,A_103] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_103))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_74),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_103),B_74)),zero_zero_int)) ) ) ).
fof(fact_1388_mult__pos__neg2,axiom,
! [B_73,A_102] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_102))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_73),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_73),A_102)),zero_zero_rat)) ) ) ).
fof(fact_1389_mult__pos__neg2,axiom,
! [B_73,A_102] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_102))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_73),zero_z891286103de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_73),A_102)),zero_z891286103de_int)) ) ) ).
fof(fact_1390_mult__pos__neg2,axiom,
! [B_73,A_102] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_102))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,B_73),zero_z126310315umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_73),A_102)),zero_z126310315umeral)) ) ) ).
fof(fact_1391_mult__pos__neg2,axiom,
! [B_73,A_102] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_102))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_73),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_73),A_102)),zero_zero_real)) ) ) ).
fof(fact_1392_mult__pos__neg2,axiom,
! [B_73,A_102] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_102))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_73),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_73),A_102)),zero_zero_nat)) ) ) ).
fof(fact_1393_mult__pos__neg2,axiom,
! [B_73,A_102] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_102))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_73),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_73),A_102)),zero_zero_int)) ) ) ).
fof(fact_1394_zero__less__mult__pos,axiom,
! [A_101,B_72] :
( 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_101),B_72)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_101))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_72)) ) ) ).
fof(fact_1395_zero__less__mult__pos,axiom,
! [A_101,B_72] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_101),B_72)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_101))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_72)) ) ) ).
fof(fact_1396_zero__less__mult__pos,axiom,
! [A_101,B_72] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_101),B_72)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_101))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_72)) ) ) ).
fof(fact_1397_zero__less__mult__pos,axiom,
! [A_101,B_72] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_101),B_72)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_101))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_72)) ) ) ).
fof(fact_1398_zero__less__mult__pos,axiom,
! [A_101,B_72] :
( 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_101),B_72)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_101))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_72)) ) ) ).
fof(fact_1399_zero__less__mult__pos,axiom,
! [A_101,B_72] :
( 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_101),B_72)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_101))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_72)) ) ) ).
fof(fact_1400_zero__less__mult__pos2,axiom,
! [B_71,A_100] :
( 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_71),A_100)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_100))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_71)) ) ) ).
fof(fact_1401_zero__less__mult__pos2,axiom,
! [B_71,A_100] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_71),A_100)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_100))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_71)) ) ) ).
fof(fact_1402_zero__less__mult__pos2,axiom,
! [B_71,A_100] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_71),A_100)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_100))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_71)) ) ) ).
fof(fact_1403_zero__less__mult__pos2,axiom,
! [B_71,A_100] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_71),A_100)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_100))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_71)) ) ) ).
fof(fact_1404_zero__less__mult__pos2,axiom,
! [B_71,A_100] :
( 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_71),A_100)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_100))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_71)) ) ) ).
fof(fact_1405_zero__less__mult__pos2,axiom,
! [B_71,A_100] :
( 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_71),A_100)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_100))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_71)) ) ) ).
fof(fact_1406_mult__less__cancel__left__neg,axiom,
! [A_1,B_1,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_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B_1)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_1),A_1)) ) ) ).
fof(fact_1407_mult__less__cancel__left__neg,axiom,
! [A_1,B_1,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_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_1),A_1)) ) ) ).
fof(fact_1408_mult__less__cancel__left__neg,axiom,
! [A_1,B_1,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_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A_1)) ) ) ).
fof(fact_1409_mult__neg__pos,axiom,
! [B_70,A_99] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_99),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_70))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_99),B_70)),zero_zero_rat)) ) ) ).
fof(fact_1410_mult__neg__pos,axiom,
! [B_70,A_99] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_99),zero_z891286103de_int))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_70))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_99),B_70)),zero_z891286103de_int)) ) ) ).
fof(fact_1411_mult__neg__pos,axiom,
! [B_70,A_99] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_99),zero_z126310315umeral))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_70))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_99),B_70)),zero_z126310315umeral)) ) ) ).
fof(fact_1412_mult__neg__pos,axiom,
! [B_70,A_99] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_99),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_70))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_99),B_70)),zero_zero_real)) ) ) ).
fof(fact_1413_mult__neg__pos,axiom,
! [B_70,A_99] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_99),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_70))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_99),B_70)),zero_zero_nat)) ) ) ).
fof(fact_1414_mult__neg__pos,axiom,
! [B_70,A_99] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_99),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_70))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_99),B_70)),zero_zero_int)) ) ) ).
fof(fact_1415_mult__neg__neg,axiom,
! [B_69,A_98] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_98),zero_zero_rat))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_69),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_98),B_69))) ) ) ).
fof(fact_1416_mult__neg__neg,axiom,
! [B_69,A_98] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_98),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_69),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_98),B_69))) ) ) ).
fof(fact_1417_mult__neg__neg,axiom,
! [B_69,A_98] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_98),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_69),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_98),B_69))) ) ) ).
fof(fact_1418_mult__strict__right__mono,axiom,
! [C_40,A_97,B_68] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_97),B_68))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_40))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_97),C_40)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_68),C_40))) ) ) ).
fof(fact_1419_mult__strict__right__mono,axiom,
! [C_40,A_97,B_68] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_97),B_68))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_40))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_97),C_40)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_68),C_40))) ) ) ).
fof(fact_1420_mult__strict__right__mono,axiom,
! [C_40,A_97,B_68] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_97),B_68))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_40))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_97),C_40)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_68),C_40))) ) ) ).
fof(fact_1421_mult__strict__right__mono,axiom,
! [C_40,A_97,B_68] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_97),B_68))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_40))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_97),C_40)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_68),C_40))) ) ) ).
fof(fact_1422_mult__strict__right__mono,axiom,
! [C_40,A_97,B_68] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_97),B_68))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_40))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_97),C_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_68),C_40))) ) ) ).
fof(fact_1423_mult__strict__right__mono,axiom,
! [C_40,A_97,B_68] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_97),B_68))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_40))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_97),C_40)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_68),C_40))) ) ) ).
fof(fact_1424_mult__strict__left__mono,axiom,
! [C_39,A_96,B_67] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_96),B_67))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_39))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_39),A_96)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_39),B_67))) ) ) ).
fof(fact_1425_mult__strict__left__mono,axiom,
! [C_39,A_96,B_67] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_96),B_67))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_39))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_39),A_96)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_39),B_67))) ) ) ).
fof(fact_1426_mult__strict__left__mono,axiom,
! [C_39,A_96,B_67] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_96),B_67))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_39))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_39),A_96)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_39),B_67))) ) ) ).
fof(fact_1427_mult__strict__left__mono,axiom,
! [C_39,A_96,B_67] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_96),B_67))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_39))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_39),A_96)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_39),B_67))) ) ) ).
fof(fact_1428_mult__strict__left__mono,axiom,
! [C_39,A_96,B_67] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_96),B_67))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_39))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_39),A_96)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_39),B_67))) ) ) ).
fof(fact_1429_mult__strict__left__mono,axiom,
! [C_39,A_96,B_67] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_96),B_67))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_39))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_39),A_96)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_39),B_67))) ) ) ).
fof(fact_1430_comm__mult__strict__left__mono,axiom,
! [C_38,A_95,B_66] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_95),B_66))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_38))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_38),A_95)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_38),B_66))) ) ) ).
fof(fact_1431_comm__mult__strict__left__mono,axiom,
! [C_38,A_95,B_66] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_95),B_66))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_38))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_38),A_95)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_38),B_66))) ) ) ).
fof(fact_1432_comm__mult__strict__left__mono,axiom,
! [C_38,A_95,B_66] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_95),B_66))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_38))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_38),A_95)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_38),B_66))) ) ) ).
fof(fact_1433_comm__mult__strict__left__mono,axiom,
! [C_38,A_95,B_66] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_95),B_66))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_38))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_38),A_95)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_38),B_66))) ) ) ).
fof(fact_1434_comm__mult__strict__left__mono,axiom,
! [C_38,A_95,B_66] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_95),B_66))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_38))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_38),A_95)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_38),B_66))) ) ) ).
fof(fact_1435_comm__mult__strict__left__mono,axiom,
! [C_38,A_95,B_66] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_95),B_66))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_38))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_38),A_95)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_38),B_66))) ) ) ).
fof(fact_1436_mult__strict__right__mono__neg,axiom,
! [C_37,B_65,A_94] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_65),A_94))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_37),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_94),C_37)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_65),C_37))) ) ) ).
fof(fact_1437_mult__strict__right__mono__neg,axiom,
! [C_37,B_65,A_94] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_65),A_94))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_37),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_94),C_37)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_65),C_37))) ) ) ).
fof(fact_1438_mult__strict__right__mono__neg,axiom,
! [C_37,B_65,A_94] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_65),A_94))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_37),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_94),C_37)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_65),C_37))) ) ) ).
fof(fact_1439_mult__strict__left__mono__neg,axiom,
! [C_36,B_64,A_93] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_64),A_93))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_36),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_36),A_93)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_36),B_64))) ) ) ).
fof(fact_1440_mult__strict__left__mono__neg,axiom,
! [C_36,B_64,A_93] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_64),A_93))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_36),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_36),A_93)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_36),B_64))) ) ) ).
fof(fact_1441_mult__strict__left__mono__neg,axiom,
! [C_36,B_64,A_93] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_64),A_93))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_36),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_36),A_93)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_36),B_64))) ) ) ).
fof(fact_1442_sum__squares__eq__zero__iff,axiom,
! [Xa,Ya] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Xa),Xa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ya),Ya)) = zero_zero_rat
<=> ( Xa = zero_zero_rat
& Ya = zero_zero_rat ) ) ).
fof(fact_1443_sum__squares__eq__zero__iff,axiom,
! [Xa,Ya] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Xa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Ya)) = zero_zero_real
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ).
fof(fact_1444_sum__squares__eq__zero__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Xa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ya),Ya)) = zero_zero_int
<=> ( Xa = zero_zero_int
& Ya = zero_zero_int ) ) ) ).
fof(fact_1445_add__scale__eq__noteq,axiom,
! [C_35,D_13,A_92,B_63,R_6] :
( R_6 != zero_zero_rat
=> ( ( A_92 = B_63
& C_35 != D_13 )
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_92),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_6),C_35)) != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_63),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,R_6),D_13)) ) ) ).
fof(fact_1446_add__scale__eq__noteq,axiom,
! [C_35,D_13,A_92,B_63,R_6] :
( R_6 != zero_zero_complex
=> ( ( A_92 = B_63
& C_35 != D_13 )
=> hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_92),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_6),C_35)) != hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_63),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,R_6),D_13)) ) ) ).
fof(fact_1447_add__scale__eq__noteq,axiom,
! [C_35,D_13,A_92,B_63,R_6] :
( R_6 != zero_zero_real
=> ( ( A_92 = B_63
& C_35 != D_13 )
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_92),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_6),C_35)) != hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_63),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_6),D_13)) ) ) ).
fof(fact_1448_add__scale__eq__noteq,axiom,
! [C_35,D_13,A_92,B_63,R_6] :
( R_6 != zero_zero_nat
=> ( ( A_92 = B_63
& C_35 != D_13 )
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_92),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_6),C_35)) != hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_63),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,R_6),D_13)) ) ) ).
fof(fact_1449_add__scale__eq__noteq,axiom,
! [C_35,D_13,A_92,B_63,R_6] :
( ( is_int(C_35)
& is_int(D_13)
& is_int(R_6) )
=> ( R_6 != zero_zero_int
=> ( ( A_92 = B_63
& C_35 != D_13 )
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_92),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_6),C_35)) != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_63),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_6),D_13)) ) ) ) ).
fof(fact_1450_less__1__mult,axiom,
! [N_27,M_16] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),M_16))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),N_27))
=> 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_16),N_27))) ) ) ).
fof(fact_1451_less__1__mult,axiom,
! [N_27,M_16] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),M_16))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),N_27))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_16),N_27))) ) ) ).
fof(fact_1452_less__1__mult,axiom,
! [N_27,M_16] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),M_16))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),N_27))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_16),N_27))) ) ) ).
fof(fact_1453_less__1__mult,axiom,
! [N_27,M_16] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),M_16))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),N_27))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_16),N_27))) ) ) ).
fof(fact_1454_less__1__mult,axiom,
! [N_27,M_16] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),M_16))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N_27))
=> 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_16),N_27))) ) ) ).
fof(fact_1455_less__1__mult,axiom,
! [N_27,M_16] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),M_16))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),N_27))
=> 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_16),N_27))) ) ) ).
fof(fact_1456_right__distrib__number__of,axiom,
! [V_11,B_62,C_34] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_11)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_62),C_34)) = 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_11)),B_62)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_11)),C_34)) ).
fof(fact_1457_right__distrib__number__of,axiom,
! [V_11,B_62,C_34] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_11)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_62),C_34)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_11)),B_62)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_i1732201573de_int(number1226105091de_int,V_11)),C_34)) ).
fof(fact_1458_right__distrib__number__of,axiom,
! [V_11,B_62,C_34] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_11)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_62),C_34)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_11)),B_62)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_11)),C_34)) ).
fof(fact_1459_right__distrib__number__of,axiom,
! [V_11,B_62,C_34] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_11)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_62),C_34)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_11)),B_62)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_11)),C_34)) ).
fof(fact_1460_right__distrib__number__of,axiom,
! [V_11,B_62,C_34] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_11)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_62),C_34)) = 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_11)),B_62)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_11)),C_34)) ).
fof(fact_1461_right__distrib__number__of,axiom,
! [V_11,B_62,C_34] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_11)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_62),C_34)) = 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_11)),B_62)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_11)),C_34)) ).
fof(fact_1462_right__distrib__number__of,axiom,
! [V_11,B_62,C_34] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_11)),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,B_62),C_34)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_11)),B_62)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_i769753017umeral(number1443263063umeral,V_11)),C_34)) ).
fof(fact_1463_left__distrib__number__of,axiom,
! [A_91,B_61,V_10] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_91),B_61)),hAPP_int_rat(number_number_of_rat,V_10)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_91),hAPP_int_rat(number_number_of_rat,V_10))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_61),hAPP_int_rat(number_number_of_rat,V_10))) ).
fof(fact_1464_left__distrib__number__of,axiom,
! [A_91,B_61,V_10] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_91),B_61)),hAPP_i1732201573de_int(number1226105091de_int,V_10)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_91),hAPP_i1732201573de_int(number1226105091de_int,V_10))),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_61),hAPP_i1732201573de_int(number1226105091de_int,V_10))) ).
fof(fact_1465_left__distrib__number__of,axiom,
! [A_91,B_61,V_10] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_91),B_61)),hAPP_int_complex(number528085621omplex,V_10)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_91),hAPP_int_complex(number528085621omplex,V_10))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_61),hAPP_int_complex(number528085621omplex,V_10))) ).
fof(fact_1466_left__distrib__number__of,axiom,
! [A_91,B_61,V_10] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_91),B_61)),hAPP_int_real(number267125858f_real,V_10)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_91),hAPP_int_real(number267125858f_real,V_10))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_61),hAPP_int_real(number267125858f_real,V_10))) ).
fof(fact_1467_left__distrib__number__of,axiom,
! [A_91,B_61,V_10] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_91),B_61)),hAPP_int_nat(number_number_of_nat,V_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_91),hAPP_int_nat(number_number_of_nat,V_10))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_61),hAPP_int_nat(number_number_of_nat,V_10))) ).
fof(fact_1468_left__distrib__number__of,axiom,
! [A_91,B_61,V_10] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_91),B_61)),hAPP_int_int(number_number_of_int,V_10)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_91),hAPP_int_int(number_number_of_int,V_10))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_61),hAPP_int_int(number_number_of_int,V_10))) ).
fof(fact_1469_left__distrib__number__of,axiom,
! [A_91,B_61,V_10] : hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_91),B_61)),hAPP_i769753017umeral(number1443263063umeral,V_10)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_91),hAPP_i769753017umeral(number1443263063umeral,V_10))),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_61),hAPP_i769753017umeral(number1443263063umeral,V_10))) ).
fof(fact_1470_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_15] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_15),M_15) = 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_15) ).
fof(fact_1471_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_15] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_15),M_15) = 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_15) ).
fof(fact_1472_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_15] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_15),M_15) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,one_one_complex),one_one_complex)),M_15) ).
fof(fact_1473_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_15] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_15),M_15) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,one_on1645066479umeral),one_on1645066479umeral)),M_15) ).
fof(fact_1474_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_15] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_15),M_15) = 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_15) ).
fof(fact_1475_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_15] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_15),M_15) = 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_15) ).
fof(fact_1476_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [M_15] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_15),M_15) = 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_15) ).
fof(fact_1477_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_14,A_90] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,M_14),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_90),M_14)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_90),one_one_rat)),M_14) ).
fof(fact_1478_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_14,A_90] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,M_14),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_90),M_14)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_90),one_on1684967323de_int)),M_14) ).
fof(fact_1479_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_14,A_90] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,M_14),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_90),M_14)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_90),one_one_complex)),M_14) ).
fof(fact_1480_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_14,A_90] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,M_14),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_90),M_14)) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_90),one_on1645066479umeral)),M_14) ).
fof(fact_1481_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_14,A_90] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,M_14),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_90),M_14)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_90),one_one_real)),M_14) ).
fof(fact_1482_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_14,A_90] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_90),M_14)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_90),one_one_nat)),M_14) ).
fof(fact_1483_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [M_14,A_90] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M_14),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_90),M_14)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_90),one_one_int)),M_14) ).
fof(fact_1484_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_89,M_13] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_89),M_13)),M_13) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_89),one_one_rat)),M_13) ).
fof(fact_1485_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_89,M_13] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_89),M_13)),M_13) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,A_89),one_on1684967323de_int)),M_13) ).
fof(fact_1486_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_89,M_13] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_89),M_13)),M_13) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_89),one_one_complex)),M_13) ).
fof(fact_1487_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_89,M_13] : hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_89),M_13)),M_13) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,A_89),one_on1645066479umeral)),M_13) ).
fof(fact_1488_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_89,M_13] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_89),M_13)),M_13) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_89),one_one_real)),M_13) ).
fof(fact_1489_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_89,M_13] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_89),M_13)),M_13) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_89),one_one_nat)),M_13) ).
fof(fact_1490_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A_89,M_13] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_89),M_13)),M_13) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_89),one_one_int)),M_13) ).
fof(fact_1491_power__add,axiom,
! [A_88,M_12,N_26] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_88),M_12)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_88),N_26)) ).
fof(fact_1492_power__add,axiom,
! [A_88,M_12,N_26] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_88),M_12)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_88),N_26)) ).
fof(fact_1493_power__add,axiom,
! [A_88,M_12,N_26] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_88),M_12)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_88),N_26)) ).
fof(fact_1494_power__add,axiom,
! [A_88,M_12,N_26] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_88),M_12)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_88),N_26)) ).
fof(fact_1495_power__add,axiom,
! [A_88,M_12,N_26] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_88),M_12)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_88),N_26)) ).
fof(fact_1496_power__add,axiom,
! [A_88,M_12,N_26] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_88),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_12),N_26)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_88),M_12)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_88),N_26)) ).
fof(fact_1497_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_53,P_7,Q_8] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_53),P_7)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_53),Q_8)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8)) ).
fof(fact_1498_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_53,P_7,Q_8] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_53),P_7)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_53),Q_8)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8)) ).
fof(fact_1499_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_53,P_7,Q_8] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_53),P_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_53),Q_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8)) ).
fof(fact_1500_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_53,P_7,Q_8] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_53),P_7)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_53),Q_8)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8)) ).
fof(fact_1501_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_53,P_7,Q_8] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_53),P_7)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_53),Q_8)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8)) ).
fof(fact_1502_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_53,P_7,Q_8] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_53),P_7)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_53),Q_8)) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,P_7),Q_8)) ).
fof(fact_1503_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_1504_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_1505_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y,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),Y))
=> 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),Y))) ) ) ).
fof(fact_1506_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ).
fof(fact_1507_zle__int,axiom,
! [Ma,Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_1508_zpower__zadd__distrib,axiom,
! [X,Y,Z_1] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),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),Y)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z_1)) ).
fof(fact_1509_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_52,N_25] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_52),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_25)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_52),N_25)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_52),N_25)) ).
fof(fact_1510_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_52,N_25] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_52),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_25)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_52),N_25)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_52),N_25)) ).
fof(fact_1511_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_52,N_25] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_52),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_25)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_52),N_25)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_52),N_25)) ).
fof(fact_1512_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_52,N_25] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_52),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_25)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_52),N_25)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_52),N_25)) ).
fof(fact_1513_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_52,N_25] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_52),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_25)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_52),N_25)),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_52),N_25)) ).
fof(fact_1514_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_52,N_25] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_52),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_25)) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_52),N_25)),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_52),N_25)) ).
fof(fact_1515_mult__left__le__imp__le,axiom,
! [C_33,A_87,B_60] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_33),A_87)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_33),B_60)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_33))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_87),B_60)) ) ) ).
fof(fact_1516_mult__left__le__imp__le,axiom,
! [C_33,A_87,B_60] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_33),A_87)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_33),B_60)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_33))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_87),B_60)) ) ) ).
fof(fact_1517_mult__left__le__imp__le,axiom,
! [C_33,A_87,B_60] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_33),A_87)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_33),B_60)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_33))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_87),B_60)) ) ) ).
fof(fact_1518_mult__left__le__imp__le,axiom,
! [C_33,A_87,B_60] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_33),A_87)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_33),B_60)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_33))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_87),B_60)) ) ) ).
fof(fact_1519_mult__left__le__imp__le,axiom,
! [C_33,A_87,B_60] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_33),A_87)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_33),B_60)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_33))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_87),B_60)) ) ) ).
fof(fact_1520_mult__left__le__imp__le,axiom,
! [C_33,A_87,B_60] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_33),A_87)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_33),B_60)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_33))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_87),B_60)) ) ) ).
fof(fact_1521_mult__right__le__imp__le,axiom,
! [A_86,C_32,B_59] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_86),C_32)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_59),C_32)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_32))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_86),B_59)) ) ) ).
fof(fact_1522_mult__right__le__imp__le,axiom,
! [A_86,C_32,B_59] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_86),C_32)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_59),C_32)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_32))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_86),B_59)) ) ) ).
fof(fact_1523_mult__right__le__imp__le,axiom,
! [A_86,C_32,B_59] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_86),C_32)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_59),C_32)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_32))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_86),B_59)) ) ) ).
fof(fact_1524_mult__right__le__imp__le,axiom,
! [A_86,C_32,B_59] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_86),C_32)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_59),C_32)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_32))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_86),B_59)) ) ) ).
fof(fact_1525_mult__right__le__imp__le,axiom,
! [A_86,C_32,B_59] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_86),C_32)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_59),C_32)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_32))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_86),B_59)) ) ) ).
fof(fact_1526_mult__right__le__imp__le,axiom,
! [A_86,C_32,B_59] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_86),C_32)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_59),C_32)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_32))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_86),B_59)) ) ) ).
fof(fact_1527_mult__less__imp__less__left,axiom,
! [C_31,A_85,B_58] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_31),A_85)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_31),B_58)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_31))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_85),B_58)) ) ) ).
fof(fact_1528_mult__less__imp__less__left,axiom,
! [C_31,A_85,B_58] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_31),A_85)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_31),B_58)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_31))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_85),B_58)) ) ) ).
fof(fact_1529_mult__less__imp__less__left,axiom,
! [C_31,A_85,B_58] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_31),A_85)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_31),B_58)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_31))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_85),B_58)) ) ) ).
fof(fact_1530_mult__less__imp__less__left,axiom,
! [C_31,A_85,B_58] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_31),A_85)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_31),B_58)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_31))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_85),B_58)) ) ) ).
fof(fact_1531_mult__less__imp__less__left,axiom,
! [C_31,A_85,B_58] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_31),A_85)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_31),B_58)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_31))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_85),B_58)) ) ) ).
fof(fact_1532_mult__less__imp__less__left,axiom,
! [C_31,A_85,B_58] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_31),A_85)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_31),B_58)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_31))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_85),B_58)) ) ) ).
fof(fact_1533_mult__left__less__imp__less,axiom,
! [C_30,A_84,B_57] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_30),A_84)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_30),B_57)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_30))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_84),B_57)) ) ) ).
fof(fact_1534_mult__left__less__imp__less,axiom,
! [C_30,A_84,B_57] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_30),A_84)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,C_30),B_57)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_30))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_84),B_57)) ) ) ).
fof(fact_1535_mult__left__less__imp__less,axiom,
! [C_30,A_84,B_57] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_30),A_84)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,C_30),B_57)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_30))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_84),B_57)) ) ) ).
fof(fact_1536_mult__left__less__imp__less,axiom,
! [C_30,A_84,B_57] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_30),A_84)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_30),B_57)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_30))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_84),B_57)) ) ) ).
fof(fact_1537_mult__left__less__imp__less,axiom,
! [C_30,A_84,B_57] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_30),A_84)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,C_30),B_57)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_30))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_84),B_57)) ) ) ).
fof(fact_1538_mult__left__less__imp__less,axiom,
! [C_30,A_84,B_57] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_30),A_84)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_30),B_57)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_30))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_84),B_57)) ) ) ).
fof(fact_1539_mult__less__imp__less__right,axiom,
! [A_83,C_29,B_56] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_83),C_29)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_56),C_29)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_29))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_83),B_56)) ) ) ).
fof(fact_1540_mult__less__imp__less__right,axiom,
! [A_83,C_29,B_56] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_83),C_29)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_56),C_29)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_29))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_83),B_56)) ) ) ).
fof(fact_1541_mult__less__imp__less__right,axiom,
! [A_83,C_29,B_56] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_83),C_29)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_56),C_29)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_29))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_83),B_56)) ) ) ).
fof(fact_1542_mult__less__imp__less__right,axiom,
! [A_83,C_29,B_56] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_83),C_29)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_56),C_29)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_29))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_83),B_56)) ) ) ).
fof(fact_1543_mult__less__imp__less__right,axiom,
! [A_83,C_29,B_56] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_83),C_29)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_56),C_29)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_83),B_56)) ) ) ).
fof(fact_1544_mult__less__imp__less__right,axiom,
! [A_83,C_29,B_56] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_83),C_29)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_56),C_29)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_29))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_83),B_56)) ) ) ).
fof(fact_1545_mult__right__less__imp__less,axiom,
! [A_82,C_28,B_55] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_82),C_28)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_55),C_28)))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_28))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_82),B_55)) ) ) ).
fof(fact_1546_mult__right__less__imp__less,axiom,
! [A_82,C_28,B_55] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_82),C_28)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_55),C_28)))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_28))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_82),B_55)) ) ) ).
fof(fact_1547_mult__right__less__imp__less,axiom,
! [A_82,C_28,B_55] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_82),C_28)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_55),C_28)))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_28))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_82),B_55)) ) ) ).
fof(fact_1548_mult__right__less__imp__less,axiom,
! [A_82,C_28,B_55] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_82),C_28)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_55),C_28)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_28))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_82),B_55)) ) ) ).
fof(fact_1549_mult__right__less__imp__less,axiom,
! [A_82,C_28,B_55] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_82),C_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_55),C_28)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_28))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_82),B_55)) ) ) ).
fof(fact_1550_mult__right__less__imp__less,axiom,
! [A_82,C_28,B_55] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_82),C_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_55),C_28)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_28))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_82),B_55)) ) ) ).
fof(fact_1551_mult__le__less__imp__less,axiom,
! [C_27,D_12,A_81,B_54] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_81),B_54))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_27),D_12))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_81))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_27))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_81),C_27)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_54),D_12))) ) ) ) ) ).
fof(fact_1552_mult__le__less__imp__less,axiom,
! [C_27,D_12,A_81,B_54] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_81),B_54))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_27),D_12))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_81))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_27))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_81),C_27)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_54),D_12))) ) ) ) ) ).
fof(fact_1553_mult__le__less__imp__less,axiom,
! [C_27,D_12,A_81,B_54] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,A_81),B_54))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_27),D_12))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_81))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_27))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_81),C_27)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_54),D_12))) ) ) ) ) ).
fof(fact_1554_mult__le__less__imp__less,axiom,
! [C_27,D_12,A_81,B_54] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_81),B_54))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_27),D_12))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_81))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_27))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_81),C_27)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_54),D_12))) ) ) ) ) ).
fof(fact_1555_mult__le__less__imp__less,axiom,
! [C_27,D_12,A_81,B_54] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_81),B_54))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_27),D_12))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_81))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_27))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_81),C_27)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_54),D_12))) ) ) ) ) ).
fof(fact_1556_mult__le__less__imp__less,axiom,
! [C_27,D_12,A_81,B_54] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_81),B_54))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_27),D_12))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_81))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_27))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_81),C_27)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_54),D_12))) ) ) ) ) ).
fof(fact_1557_mult__less__le__imp__less,axiom,
! [C_26,D_11,A_80,B_53] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_80),B_53))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_26),D_11))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_80))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),C_26))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_80),C_26)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_53),D_11))) ) ) ) ) ).
fof(fact_1558_mult__less__le__imp__less,axiom,
! [C_26,D_11,A_80,B_53] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_80),B_53))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_26),D_11))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_80))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),C_26))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_80),C_26)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_53),D_11))) ) ) ) ) ).
fof(fact_1559_mult__less__le__imp__less,axiom,
! [C_26,D_11,A_80,B_53] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_80),B_53))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,C_26),D_11))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_80))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),C_26))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_80),C_26)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_53),D_11))) ) ) ) ) ).
fof(fact_1560_mult__less__le__imp__less,axiom,
! [C_26,D_11,A_80,B_53] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_80),B_53))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_26),D_11))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_80))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),C_26))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_80),C_26)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_53),D_11))) ) ) ) ) ).
fof(fact_1561_mult__less__le__imp__less,axiom,
! [C_26,D_11,A_80,B_53] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_80),B_53))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_26),D_11))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_80))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),C_26))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_80),C_26)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_53),D_11))) ) ) ) ) ).
fof(fact_1562_mult__less__le__imp__less,axiom,
! [C_26,D_11,A_80,B_53] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_80),B_53))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_26),D_11))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_80))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_26))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_80),C_26)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_53),D_11))) ) ) ) ) ).
fof(fact_1563_mult__strict__mono_H,axiom,
! [C_25,D_10,A_79,B_52] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_79),B_52))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_25),D_10))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_79))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_25))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_79),C_25)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_52),D_10))) ) ) ) ) ).
fof(fact_1564_mult__strict__mono_H,axiom,
! [C_25,D_10,A_79,B_52] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_79),B_52))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_25),D_10))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),A_79))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_25))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_79),C_25)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_52),D_10))) ) ) ) ) ).
fof(fact_1565_mult__strict__mono_H,axiom,
! [C_25,D_10,A_79,B_52] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_79),B_52))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_25),D_10))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),A_79))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_25))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_79),C_25)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_52),D_10))) ) ) ) ) ).
fof(fact_1566_mult__strict__mono_H,axiom,
! [C_25,D_10,A_79,B_52] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_79),B_52))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_25),D_10))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_79))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_25))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_79),C_25)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_52),D_10))) ) ) ) ) ).
fof(fact_1567_mult__strict__mono_H,axiom,
! [C_25,D_10,A_79,B_52] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_79),B_52))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_25),D_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_79))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_79),C_25)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_52),D_10))) ) ) ) ) ).
fof(fact_1568_mult__strict__mono_H,axiom,
! [C_25,D_10,A_79,B_52] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_79),B_52))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_25),D_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_79))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_79),C_25)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_52),D_10))) ) ) ) ) ).
fof(fact_1569_mult__strict__mono,axiom,
! [C_24,D_9,A_78,B_51] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_78),B_51))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_24),D_9))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),B_51))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),C_24))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_78),C_24)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_51),D_9))) ) ) ) ) ).
fof(fact_1570_mult__strict__mono,axiom,
! [C_24,D_9,A_78,B_51] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_78),B_51))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_24),D_9))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),B_51))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,zero_z891286103de_int),C_24))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_78),C_24)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_51),D_9))) ) ) ) ) ).
fof(fact_1571_mult__strict__mono,axiom,
! [C_24,D_9,A_78,B_51] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_78),B_51))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,C_24),D_9))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),B_51))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le565307924umeral,zero_z126310315umeral),C_24))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_78),C_24)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,B_51),D_9))) ) ) ) ) ).
fof(fact_1572_mult__strict__mono,axiom,
! [C_24,D_9,A_78,B_51] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_78),B_51))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_24),D_9))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),B_51))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),C_24))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_78),C_24)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_51),D_9))) ) ) ) ) ).
fof(fact_1573_mult__strict__mono,axiom,
! [C_24,D_9,A_78,B_51] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_78),B_51))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_24),D_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_51))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_78),C_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_51),D_9))) ) ) ) ) ).
fof(fact_1574_mult__strict__mono,axiom,
! [C_24,D_9,A_78,B_51] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_78),B_51))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_24),D_9))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_51))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),C_24))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_78),C_24)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_51),D_9))) ) ) ) ) ).
fof(fact_1575_mult__le__cancel__left__neg,axiom,
! [A_1,B_1,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_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B_1)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_1),A_1)) ) ) ).
fof(fact_1576_mult__le__cancel__left__neg,axiom,
! [A_1,B_1,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_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_1),A_1)) ) ) ).
fof(fact_1577_mult__le__cancel__left__neg,axiom,
! [A_1,B_1,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_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1)) ) ) ).
fof(fact_1578_mult__le__cancel__left__pos,axiom,
! [A_1,B_1,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_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B_1)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),B_1)) ) ) ).
fof(fact_1579_mult__le__cancel__left__pos,axiom,
! [A_1,B_1,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_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) ) ) ).
fof(fact_1580_mult__le__cancel__left__pos,axiom,
! [A_1,B_1,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_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_1)) ) ) ).
fof(fact_1581_sum__squares__ge__zero,axiom,
! [X_51,Y_45] : 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_51),X_51)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_45),Y_45)))) ).
fof(fact_1582_sum__squares__ge__zero,axiom,
! [X_51,Y_45] : 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_51),X_51)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_45),Y_45)))) ).
fof(fact_1583_sum__squares__ge__zero,axiom,
! [X_51,Y_45] : 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_51),X_51)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_45),Y_45)))) ).
fof(fact_1584_sum__squares__le__zero__iff,axiom,
! [Xa,Ya] :
( 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,Xa),Xa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ya),Ya))),zero_zero_rat))
<=> ( Xa = zero_zero_rat
& Ya = zero_zero_rat ) ) ).
fof(fact_1585_sum__squares__le__zero__iff,axiom,
! [Xa,Ya] :
( 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,Xa),Xa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Ya))),zero_zero_real))
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ).
fof(fact_1586_sum__squares__le__zero__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( 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,Xa),Xa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ya),Ya))),zero_zero_int))
<=> ( Xa = zero_zero_int
& Ya = zero_zero_int ) ) ) ).
fof(fact_1587_not__sum__squares__lt__zero,axiom,
! [X_50,Y_44] : ~ 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_50),X_50)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_44),Y_44))),zero_zero_rat)) ).
fof(fact_1588_not__sum__squares__lt__zero,axiom,
! [X_50,Y_44] : ~ 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_50),X_50)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_44),Y_44))),zero_zero_real)) ).
fof(fact_1589_not__sum__squares__lt__zero,axiom,
! [X_50,Y_44] : ~ 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_50),X_50)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_44),Y_44))),zero_zero_int)) ).
fof(fact_1590_sum__squares__gt__zero__iff,axiom,
! [Xa,Ya] :
( 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,Xa),Xa)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Ya),Ya))))
<=> ( Xa != zero_zero_rat
| Ya != zero_zero_rat ) ) ).
fof(fact_1591_sum__squares__gt__zero__iff,axiom,
! [Xa,Ya] :
( 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,Xa),Xa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Ya))))
<=> ( Xa != zero_zero_real
| Ya != zero_zero_real ) ) ).
fof(fact_1592_sum__squares__gt__zero__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( 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,Xa),Xa)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ya),Ya))))
<=> ( Xa != zero_zero_int
| Ya != zero_zero_int ) ) ) ).
fof(fact_1593_mult__right__le__one__le,axiom,
! [Y_43,X_49] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_49))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_43))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_43),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_49),Y_43)),X_49)) ) ) ) ).
fof(fact_1594_mult__right__le__one__le,axiom,
! [Y_43,X_49] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_49))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_43))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_43),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_49),Y_43)),X_49)) ) ) ) ).
fof(fact_1595_mult__right__le__one__le,axiom,
! [Y_43,X_49] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_49))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_43))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_43),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_49),Y_43)),X_49)) ) ) ) ).
fof(fact_1596_mult__left__le__one__le,axiom,
! [Y_42,X_48] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_48))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),Y_42))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_42),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_42),X_48)),X_48)) ) ) ) ).
fof(fact_1597_mult__left__le__one__le,axiom,
! [Y_42,X_48] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_48))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y_42))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_42),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_42),X_48)),X_48)) ) ) ) ).
fof(fact_1598_mult__left__le__one__le,axiom,
! [Y_42,X_48] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_48))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y_42))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_42),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_42),X_48)),X_48)) ) ) ) ).
fof(fact_1599_power__gt1__lemma,axiom,
! [N_24,A_77] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_77))
=> 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_77),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_77),N_24)))) ) ).
fof(fact_1600_power__gt1__lemma,axiom,
! [N_24,A_77] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_77))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_77),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_77),N_24)))) ) ).
fof(fact_1601_power__gt1__lemma,axiom,
! [N_24,A_77] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_77))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_77),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_77),N_24)))) ) ).
fof(fact_1602_power__gt1__lemma,axiom,
! [N_24,A_77] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_77))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_77),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_77),N_24)))) ) ).
fof(fact_1603_power__gt1__lemma,axiom,
! [N_24,A_77] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_77))
=> 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_77),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_77),N_24)))) ) ).
fof(fact_1604_power__gt1__lemma,axiom,
! [N_24,A_77] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_77))
=> 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_77),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_77),N_24)))) ) ).
fof(fact_1605_power__less__power__Suc,axiom,
! [N_23,A_76] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),A_76))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_76),N_23)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_76),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_76),N_23)))) ) ).
fof(fact_1606_power__less__power__Suc,axiom,
! [N_23,A_76] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,one_on1684967323de_int),A_76))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_76),N_23)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_76),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_76),N_23)))) ) ).
fof(fact_1607_power__less__power__Suc,axiom,
! [N_23,A_76] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,one_on1645066479umeral),A_76))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_76),N_23)),hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_76),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_76),N_23)))) ) ).
fof(fact_1608_power__less__power__Suc,axiom,
! [N_23,A_76] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_76))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_76),N_23)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_76),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_76),N_23)))) ) ).
fof(fact_1609_power__less__power__Suc,axiom,
! [N_23,A_76] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),A_76))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_76),N_23)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_76),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_76),N_23)))) ) ).
fof(fact_1610_power__less__power__Suc,axiom,
! [N_23,A_76] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_76))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_76),N_23)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_76),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_76),N_23)))) ) ).
fof(fact_1611_mult__numeral__1__right,axiom,
! [A_75] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_75),hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))) = A_75 ).
fof(fact_1612_mult__numeral__1__right,axiom,
! [A_75] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_75),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls))) = A_75 ).
fof(fact_1613_mult__numeral__1__right,axiom,
! [A_75] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_75),hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))) = A_75 ).
fof(fact_1614_mult__numeral__1__right,axiom,
! [A_75] :
( is_int(A_75)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_75),hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))) = A_75 ) ).
fof(fact_1615_mult__numeral__1,axiom,
! [A_74] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit1,pls))),A_74) = A_74 ).
fof(fact_1616_mult__numeral__1,axiom,
! [A_74] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(bit1,pls))),A_74) = A_74 ).
fof(fact_1617_mult__numeral__1,axiom,
! [A_74] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit1,pls))),A_74) = A_74 ).
fof(fact_1618_mult__numeral__1,axiom,
! [A_74] :
( is_int(A_74)
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,pls))),A_74) = A_74 ) ).
fof(fact_1619_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_1620_pos__zmult__eq__1__iff,axiom,
! [Na,Ma] :
( ( is_int(Na)
& 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),Na) = one_one_int
<=> ( Ma = one_one_int
& Na = one_one_int ) ) ) ) ).
fof(fact_1621_le__nat__number__of,axiom,
! [Va,V] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_int_nat(number_number_of_nat,V)))
<=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),V))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Va),pls)) ) ) ).
fof(fact_1622_linorder__le__cases,axiom,
! [X_47,Y_41] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_47),Y_41))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_41),X_47)) ) ).
fof(fact_1623_linorder__le__cases,axiom,
! [X_47,Y_41] :
( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_47),Y_41))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_41),X_47)) ) ).
fof(fact_1624_linorder__le__cases,axiom,
! [X_47,Y_41] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_47),Y_41))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_41),X_47)) ) ).
fof(fact_1625_linorder__le__cases,axiom,
! [X_47,Y_41] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_47),Y_41))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_41),X_47)) ) ).
fof(fact_1626_linorder__le__cases,axiom,
! [X_47,Y_41] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_47),Y_41))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_41),X_47)) ) ).
fof(fact_1627_le__funE,axiom,
! [Xa,F,G] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,Xa)),hAPP_int_bool(G,Xa))) ) ).
fof(fact_1628_le__funE,axiom,
! [Xa,F,G] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,Xa)),hAPP_nat_bool(G,Xa))) ) ).
fof(fact_1629_xt1_I6_J,axiom,
! [Z_18,Y_40,X_46] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_40),X_46))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_18),Y_40))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_18),X_46)) ) ) ).
fof(fact_1630_xt1_I6_J,axiom,
! [Z_2,Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),Ya))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),Xa)) ) ) ).
fof(fact_1631_xt1_I6_J,axiom,
! [Z_2,Ya,Xa] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),Ya))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),Xa)) ) ) ).
fof(fact_1632_xt1_I6_J,axiom,
! [Z_18,Y_40,X_46] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_40),X_46))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Z_18),Y_40))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Z_18),X_46)) ) ) ).
fof(fact_1633_xt1_I6_J,axiom,
! [Z_18,Y_40,X_46] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_40),X_46))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_18),Y_40))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_18),X_46)) ) ) ).
fof(fact_1634_xt1_I6_J,axiom,
! [Z_18,Y_40,X_46] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_40),X_46))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_18),Y_40))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_18),X_46)) ) ) ).
fof(fact_1635_xt1_I6_J,axiom,
! [Z_18,Y_40,X_46] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_40),X_46))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_18),Y_40))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_18),X_46)) ) ) ).
fof(fact_1636_xt1_I5_J,axiom,
! [Y_39,X_45] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_39),X_45))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_45),Y_39))
=> X_45 = Y_39 ) ) ).
fof(fact_1637_xt1_I5_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
=> Xa = Ya ) ) ).
fof(fact_1638_xt1_I5_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
=> Xa = Ya ) ) ).
fof(fact_1639_xt1_I5_J,axiom,
! [Y_39,X_45] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_39),X_45))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_45),Y_39))
=> X_45 = Y_39 ) ) ).
fof(fact_1640_xt1_I5_J,axiom,
! [Y_39,X_45] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_39),X_45))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_45),Y_39))
=> X_45 = Y_39 ) ) ).
fof(fact_1641_xt1_I5_J,axiom,
! [Y_39,X_45] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_39),X_45))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_45),Y_39))
=> X_45 = Y_39 ) ) ).
fof(fact_1642_xt1_I5_J,axiom,
! [Y_39,X_45] :
( ( is_int(Y_39)
& is_int(X_45) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_39),X_45))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_45),Y_39))
=> X_45 = Y_39 ) ) ) ).
fof(fact_1643_order__trans,axiom,
! [Z_17,X_44,Y_38] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_44),Y_38))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_38),Z_17))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_44),Z_17)) ) ) ).
fof(fact_1644_order__trans,axiom,
! [Z_2,Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Z_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Z_2)) ) ) ).
fof(fact_1645_order__trans,axiom,
! [Z_2,Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Z_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Z_2)) ) ) ).
fof(fact_1646_order__trans,axiom,
! [Z_17,X_44,Y_38] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_44),Y_38))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_38),Z_17))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_44),Z_17)) ) ) ).
fof(fact_1647_order__trans,axiom,
! [Z_17,X_44,Y_38] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_44),Y_38))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_38),Z_17))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_44),Z_17)) ) ) ).
fof(fact_1648_order__trans,axiom,
! [Z_17,X_44,Y_38] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_44),Y_38))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_38),Z_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_44),Z_17)) ) ) ).
fof(fact_1649_order__trans,axiom,
! [Z_17,X_44,Y_38] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_44),Y_38))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_38),Z_17))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_44),Z_17)) ) ) ).
fof(fact_1650_order__antisym,axiom,
! [X_43,Y_37] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_43),Y_37))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_37),X_43))
=> X_43 = Y_37 ) ) ).
fof(fact_1651_order__antisym,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa))
=> Xa = Ya ) ) ).
fof(fact_1652_order__antisym,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Xa))
=> Xa = Ya ) ) ).
fof(fact_1653_order__antisym,axiom,
! [X_43,Y_37] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_43),Y_37))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_37),X_43))
=> X_43 = Y_37 ) ) ).
fof(fact_1654_order__antisym,axiom,
! [X_43,Y_37] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_43),Y_37))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_37),X_43))
=> X_43 = Y_37 ) ) ).
fof(fact_1655_order__antisym,axiom,
! [X_43,Y_37] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_43),Y_37))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_37),X_43))
=> X_43 = Y_37 ) ) ).
fof(fact_1656_order__antisym,axiom,
! [X_43,Y_37] :
( ( is_int(X_43)
& is_int(Y_37) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_43),Y_37))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_37),X_43))
=> X_43 = Y_37 ) ) ) ).
fof(fact_1657_xt1_I4_J,axiom,
! [C_23,B_50,A_73] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_50),A_73))
=> ( B_50 = C_23
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_23),A_73)) ) ) ).
fof(fact_1658_xt1_I4_J,axiom,
! [C,B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_1),A_1))
=> ( B_1 = C
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C),A_1)) ) ) ).
fof(fact_1659_xt1_I4_J,axiom,
! [C,B_1,A_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_1),A_1))
=> ( B_1 = C
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),A_1)) ) ) ).
fof(fact_1660_xt1_I4_J,axiom,
! [C_23,B_50,A_73] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_50),A_73))
=> ( B_50 = C_23
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_23),A_73)) ) ) ).
fof(fact_1661_xt1_I4_J,axiom,
! [C_23,B_50,A_73] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_50),A_73))
=> ( B_50 = C_23
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_23),A_73)) ) ) ).
fof(fact_1662_xt1_I4_J,axiom,
! [C_23,B_50,A_73] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_50),A_73))
=> ( B_50 = C_23
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_23),A_73)) ) ) ).
fof(fact_1663_xt1_I4_J,axiom,
! [C_23,B_50,A_73] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_50),A_73))
=> ( B_50 = C_23
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_23),A_73)) ) ) ).
fof(fact_1664_ord__le__eq__trans,axiom,
! [C_22,A_72,B_49] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_72),B_49))
=> ( B_49 = C_22
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_72),C_22)) ) ) ).
fof(fact_1665_ord__le__eq__trans,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),B_1))
=> ( B_1 = C
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),C)) ) ) ).
fof(fact_1666_ord__le__eq__trans,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1))
=> ( B_1 = C
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),C)) ) ) ).
fof(fact_1667_ord__le__eq__trans,axiom,
! [C_22,A_72,B_49] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_72),B_49))
=> ( B_49 = C_22
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_72),C_22)) ) ) ).
fof(fact_1668_ord__le__eq__trans,axiom,
! [C_22,A_72,B_49] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_72),B_49))
=> ( B_49 = C_22
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_72),C_22)) ) ) ).
fof(fact_1669_ord__le__eq__trans,axiom,
! [C_22,A_72,B_49] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_72),B_49))
=> ( B_49 = C_22
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_72),C_22)) ) ) ).
fof(fact_1670_ord__le__eq__trans,axiom,
! [C_22,A_72,B_49] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_72),B_49))
=> ( B_49 = C_22
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_72),C_22)) ) ) ).
fof(fact_1671_xt1_I3_J,axiom,
! [C_21,A_71,B_48] :
( A_71 = B_48
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_21),B_48))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C_21),A_71)) ) ) ).
fof(fact_1672_xt1_I3_J,axiom,
! [C,A_1,B_1] :
( A_1 = B_1
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C),B_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,C),A_1)) ) ) ).
fof(fact_1673_xt1_I3_J,axiom,
! [C,A_1,B_1] :
( A_1 = B_1
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),B_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),A_1)) ) ) ).
fof(fact_1674_xt1_I3_J,axiom,
! [C_21,A_71,B_48] :
( A_71 = B_48
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_21),B_48))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,C_21),A_71)) ) ) ).
fof(fact_1675_xt1_I3_J,axiom,
! [C_21,A_71,B_48] :
( A_71 = B_48
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_21),B_48))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C_21),A_71)) ) ) ).
fof(fact_1676_xt1_I3_J,axiom,
! [C_21,A_71,B_48] :
( A_71 = B_48
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_21),B_48))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_21),A_71)) ) ) ).
fof(fact_1677_xt1_I3_J,axiom,
! [C_21,A_71,B_48] :
( A_71 = B_48
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_21),B_48))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C_21),A_71)) ) ) ).
fof(fact_1678_ord__eq__le__trans,axiom,
! [C_20,A_70,B_47] :
( A_70 = B_47
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_47),C_20))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_70),C_20)) ) ) ).
fof(fact_1679_ord__eq__le__trans,axiom,
! [C,A_1,B_1] :
( A_1 = B_1
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_1),C))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),C)) ) ) ).
fof(fact_1680_ord__eq__le__trans,axiom,
! [C,A_1,B_1] :
( A_1 = B_1
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_1),C))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),C)) ) ) ).
fof(fact_1681_ord__eq__le__trans,axiom,
! [C_20,A_70,B_47] :
( A_70 = B_47
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_47),C_20))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_70),C_20)) ) ) ).
fof(fact_1682_ord__eq__le__trans,axiom,
! [C_20,A_70,B_47] :
( A_70 = B_47
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_47),C_20))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_70),C_20)) ) ) ).
fof(fact_1683_ord__eq__le__trans,axiom,
! [C_20,A_70,B_47] :
( A_70 = B_47
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_47),C_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_70),C_20)) ) ) ).
fof(fact_1684_ord__eq__le__trans,axiom,
! [C_20,A_70,B_47] :
( A_70 = B_47
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_47),C_20))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_70),C_20)) ) ) ).
fof(fact_1685_order__antisym__conv,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ya),Xa))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1686_order__antisym__conv,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1687_order__antisym__conv,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1688_order__antisym__conv,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Ya),Xa))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1689_order__antisym__conv,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1690_order__antisym__conv,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1691_order__antisym__conv,axiom,
! [Ya,Xa] :
( ( is_int(Ya)
& is_int(Xa) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya))
<=> Xa = Ya ) ) ) ).
fof(fact_1692_le__funD,axiom,
! [Xa,F,G] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,Xa)),hAPP_int_bool(G,Xa))) ) ).
fof(fact_1693_le__funD,axiom,
! [Xa,F,G] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,Xa)),hAPP_nat_bool(G,Xa))) ) ).
fof(fact_1694_order__eq__refl,axiom,
! [X_42,Y_36] :
( X_42 = Y_36
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_42),Y_36)) ) ).
fof(fact_1695_order__eq__refl,axiom,
! [Xa,Ya] :
( Xa = Ya
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)) ) ).
fof(fact_1696_order__eq__refl,axiom,
! [Xa,Ya] :
( Xa = Ya
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)) ) ).
fof(fact_1697_order__eq__refl,axiom,
! [X_42,Y_36] :
( X_42 = Y_36
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_42),Y_36)) ) ).
fof(fact_1698_order__eq__refl,axiom,
! [X_42,Y_36] :
( X_42 = Y_36
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_42),Y_36)) ) ).
fof(fact_1699_order__eq__refl,axiom,
! [X_42,Y_36] :
( X_42 = Y_36
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_42),Y_36)) ) ).
fof(fact_1700_order__eq__refl,axiom,
! [X_42,Y_36] :
( X_42 = Y_36
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_42),Y_36)) ) ).
fof(fact_1701_order__eq__iff,axiom,
! [Xa,Ya] :
( Xa = Ya
<=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya))
& hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ya),Xa)) ) ) ).
fof(fact_1702_order__eq__iff,axiom,
! [Xa,Ya] :
( Xa = Ya
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
& hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa)) ) ) ).
fof(fact_1703_order__eq__iff,axiom,
! [Xa,Ya] :
( Xa = Ya
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
& hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Xa)) ) ) ).
fof(fact_1704_order__eq__iff,axiom,
! [Xa,Ya] :
( Xa = Ya
<=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya))
& hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Ya),Xa)) ) ) ).
fof(fact_1705_order__eq__iff,axiom,
! [Xa,Ya] :
( Xa = Ya
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa)) ) ) ).
fof(fact_1706_order__eq__iff,axiom,
! [Xa,Ya] :
( Xa = Ya
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa)) ) ) ).
fof(fact_1707_order__eq__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( Xa = Ya
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),Xa)) ) ) ) ).
fof(fact_1708_linorder__linear,axiom,
! [X_41,Y_35] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_41),Y_35))
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_35),X_41)) ) ).
fof(fact_1709_linorder__linear,axiom,
! [X_41,Y_35] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_41),Y_35))
| hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_35),X_41)) ) ).
fof(fact_1710_linorder__linear,axiom,
! [X_41,Y_35] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_41),Y_35))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_35),X_41)) ) ).
fof(fact_1711_linorder__linear,axiom,
! [X_41,Y_35] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_41),Y_35))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_35),X_41)) ) ).
fof(fact_1712_linorder__linear,axiom,
! [X_41,Y_35] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_41),Y_35))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_35),X_41)) ) ).
fof(fact_1713_le__fun__def,axiom,
! [F,G] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
<=> ! [X_1] :
( is_int(X_1)
=> hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_int_bool(F,X_1)),hAPP_int_bool(G,X_1))) ) ) ).
fof(fact_1714_le__fun__def,axiom,
! [F,G] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
<=> ! [X_1] : hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(ord_less_eq_bool,hAPP_nat_bool(F,X_1)),hAPP_nat_bool(G,X_1))) ) ).
fof(fact_1715_linorder__cases,axiom,
! [X_40,Y_34] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_40),Y_34))
=> ( X_40 != Y_34
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_34),X_40)) ) ) ).
fof(fact_1716_linorder__cases,axiom,
! [X_40,Y_34] :
( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_40),Y_34))
=> ( X_40 != Y_34
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_34),X_40)) ) ) ).
fof(fact_1717_linorder__cases,axiom,
! [X_40,Y_34] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_40),Y_34))
=> ( X_40 != Y_34
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_34),X_40)) ) ) ).
fof(fact_1718_linorder__cases,axiom,
! [X_40,Y_34] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_40),Y_34))
=> ( X_40 != Y_34
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_34),X_40)) ) ) ).
fof(fact_1719_linorder__cases,axiom,
! [X_40,Y_34] :
( ( is_int(X_40)
& is_int(Y_34) )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_40),Y_34))
=> ( X_40 != Y_34
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_34),X_40)) ) ) ) ).
fof(fact_1720_order__less__asym,axiom,
! [X_39,Y_33] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_39),Y_33))
=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_33),X_39)) ) ).
fof(fact_1721_order__less__asym,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Xa)) ) ).
fof(fact_1722_order__less__asym,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Xa)) ) ).
fof(fact_1723_order__less__asym,axiom,
! [X_39,Y_33] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_39),Y_33))
=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_33),X_39)) ) ).
fof(fact_1724_order__less__asym,axiom,
! [X_39,Y_33] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_39),Y_33))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_33),X_39)) ) ).
fof(fact_1725_order__less__asym,axiom,
! [X_39,Y_33] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_39),Y_33))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_33),X_39)) ) ).
fof(fact_1726_order__less__asym,axiom,
! [X_39,Y_33] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_39),Y_33))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_33),X_39)) ) ).
fof(fact_1727_xt1_I10_J,axiom,
! [Z_16,Y_32,X_38] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_32),X_38))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_16),Y_32))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_16),X_38)) ) ) ).
fof(fact_1728_xt1_I10_J,axiom,
! [Z_2,Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Ya))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Xa)) ) ) ).
fof(fact_1729_xt1_I10_J,axiom,
! [Z_2,Ya,Xa] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Ya))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Xa)) ) ) ).
fof(fact_1730_xt1_I10_J,axiom,
! [Z_16,Y_32,X_38] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_32),X_38))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_16),Y_32))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_16),X_38)) ) ) ).
fof(fact_1731_xt1_I10_J,axiom,
! [Z_16,Y_32,X_38] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_32),X_38))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_16),Y_32))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_16),X_38)) ) ) ).
fof(fact_1732_xt1_I10_J,axiom,
! [Z_16,Y_32,X_38] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_32),X_38))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_16),Y_32))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_16),X_38)) ) ) ).
fof(fact_1733_xt1_I10_J,axiom,
! [Z_16,Y_32,X_38] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_32),X_38))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_16),Y_32))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_16),X_38)) ) ) ).
fof(fact_1734_order__less__trans,axiom,
! [Z_15,X_37,Y_31] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_37),Y_31))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_31),Z_15))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_37),Z_15)) ) ) ).
fof(fact_1735_order__less__trans,axiom,
! [Z_2,Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Z_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Z_2)) ) ) ).
fof(fact_1736_order__less__trans,axiom,
! [Z_2,Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Z_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Z_2)) ) ) ).
fof(fact_1737_order__less__trans,axiom,
! [Z_15,X_37,Y_31] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_37),Y_31))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_31),Z_15))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_37),Z_15)) ) ) ).
fof(fact_1738_order__less__trans,axiom,
! [Z_15,X_37,Y_31] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_37),Y_31))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_31),Z_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_37),Z_15)) ) ) ).
fof(fact_1739_order__less__trans,axiom,
! [Z_15,X_37,Y_31] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_37),Y_31))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_31),Z_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_37),Z_15)) ) ) ).
fof(fact_1740_order__less__trans,axiom,
! [Z_15,X_37,Y_31] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_37),Y_31))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_31),Z_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_37),Z_15)) ) ) ).
fof(fact_1741_xt1_I2_J,axiom,
! [C_19,B_46,A_69] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_46),A_69))
=> ( B_46 = C_19
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_19),A_69)) ) ) ).
fof(fact_1742_xt1_I2_J,axiom,
! [C,B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),A_1))
=> ( B_1 = C
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,C),A_1)) ) ) ).
fof(fact_1743_xt1_I2_J,axiom,
! [C,B_1,A_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1))
=> ( B_1 = C
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C),A_1)) ) ) ).
fof(fact_1744_xt1_I2_J,axiom,
! [C_19,B_46,A_69] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_46),A_69))
=> ( B_46 = C_19
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_19),A_69)) ) ) ).
fof(fact_1745_xt1_I2_J,axiom,
! [C_19,B_46,A_69] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_46),A_69))
=> ( B_46 = C_19
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_19),A_69)) ) ) ).
fof(fact_1746_xt1_I2_J,axiom,
! [C_19,B_46,A_69] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_46),A_69))
=> ( B_46 = C_19
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_19),A_69)) ) ) ).
fof(fact_1747_xt1_I2_J,axiom,
! [C_19,B_46,A_69] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_46),A_69))
=> ( B_46 = C_19
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_19),A_69)) ) ) ).
fof(fact_1748_ord__less__eq__trans,axiom,
! [C_18,A_68,B_45] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_68),B_45))
=> ( B_45 = C_18
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_68),C_18)) ) ) ).
fof(fact_1749_ord__less__eq__trans,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1))
=> ( B_1 = C
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),C)) ) ) ).
fof(fact_1750_ord__less__eq__trans,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1))
=> ( B_1 = C
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),C)) ) ) ).
fof(fact_1751_ord__less__eq__trans,axiom,
! [C_18,A_68,B_45] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_68),B_45))
=> ( B_45 = C_18
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_68),C_18)) ) ) ).
fof(fact_1752_ord__less__eq__trans,axiom,
! [C_18,A_68,B_45] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_68),B_45))
=> ( B_45 = C_18
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_68),C_18)) ) ) ).
fof(fact_1753_ord__less__eq__trans,axiom,
! [C_18,A_68,B_45] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_68),B_45))
=> ( B_45 = C_18
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_68),C_18)) ) ) ).
fof(fact_1754_ord__less__eq__trans,axiom,
! [C_18,A_68,B_45] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_68),B_45))
=> ( B_45 = C_18
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_68),C_18)) ) ) ).
fof(fact_1755_xt1_I1_J,axiom,
! [C_17,A_67,B_44] :
( A_67 = B_44
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_17),B_44))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C_17),A_67)) ) ) ).
fof(fact_1756_xt1_I1_J,axiom,
! [C,A_1,B_1] :
( A_1 = B_1
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,C),B_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,C),A_1)) ) ) ).
fof(fact_1757_xt1_I1_J,axiom,
! [C,A_1,B_1] :
( A_1 = B_1
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C),B_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,C),A_1)) ) ) ).
fof(fact_1758_xt1_I1_J,axiom,
! [C_17,A_67,B_44] :
( A_67 = B_44
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_17),B_44))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,C_17),A_67)) ) ) ).
fof(fact_1759_xt1_I1_J,axiom,
! [C_17,A_67,B_44] :
( A_67 = B_44
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_17),B_44))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C_17),A_67)) ) ) ).
fof(fact_1760_xt1_I1_J,axiom,
! [C_17,A_67,B_44] :
( A_67 = B_44
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_17),B_44))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_17),A_67)) ) ) ).
fof(fact_1761_xt1_I1_J,axiom,
! [C_17,A_67,B_44] :
( A_67 = B_44
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_17),B_44))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C_17),A_67)) ) ) ).
fof(fact_1762_ord__eq__less__trans,axiom,
! [C_16,A_66,B_43] :
( A_66 = B_43
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_43),C_16))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_66),C_16)) ) ) ).
fof(fact_1763_ord__eq__less__trans,axiom,
! [C,A_1,B_1] :
( A_1 = B_1
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),C))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),C)) ) ) ).
fof(fact_1764_ord__eq__less__trans,axiom,
! [C,A_1,B_1] :
( A_1 = B_1
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),C))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),C)) ) ) ).
fof(fact_1765_ord__eq__less__trans,axiom,
! [C_16,A_66,B_43] :
( A_66 = B_43
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_43),C_16))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_66),C_16)) ) ) ).
fof(fact_1766_ord__eq__less__trans,axiom,
! [C_16,A_66,B_43] :
( A_66 = B_43
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_43),C_16))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_66),C_16)) ) ) ).
fof(fact_1767_ord__eq__less__trans,axiom,
! [C_16,A_66,B_43] :
( A_66 = B_43
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_43),C_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_66),C_16)) ) ) ).
fof(fact_1768_ord__eq__less__trans,axiom,
! [C_16,A_66,B_43] :
( A_66 = B_43
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_43),C_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_66),C_16)) ) ) ).
fof(fact_1769_xt1_I9_J,axiom,
! [B_42,A_65] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_42),A_65))
=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_65),B_42)) ) ).
fof(fact_1770_xt1_I9_J,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),A_1))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1)) ) ).
fof(fact_1771_xt1_I9_J,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1)) ) ).
fof(fact_1772_xt1_I9_J,axiom,
! [B_42,A_65] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_42),A_65))
=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_65),B_42)) ) ).
fof(fact_1773_xt1_I9_J,axiom,
! [B_42,A_65] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_42),A_65))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_65),B_42)) ) ).
fof(fact_1774_xt1_I9_J,axiom,
! [B_42,A_65] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_42),A_65))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_65),B_42)) ) ).
fof(fact_1775_xt1_I9_J,axiom,
! [B_42,A_65] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_42),A_65))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_65),B_42)) ) ).
fof(fact_1776_order__less__asym_H,axiom,
! [A_64,B_41] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_64),B_41))
=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_41),A_64)) ) ).
fof(fact_1777_order__less__asym_H,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),A_1)) ) ).
fof(fact_1778_order__less__asym_H,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1)) ) ).
fof(fact_1779_order__less__asym_H,axiom,
! [A_64,B_41] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_64),B_41))
=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_41),A_64)) ) ).
fof(fact_1780_order__less__asym_H,axiom,
! [A_64,B_41] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_64),B_41))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_41),A_64)) ) ).
fof(fact_1781_order__less__asym_H,axiom,
! [A_64,B_41] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_64),B_41))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_41),A_64)) ) ).
fof(fact_1782_order__less__asym_H,axiom,
! [A_64,B_41] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_64),B_41))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_41),A_64)) ) ).
fof(fact_1783_order__less__imp__triv,axiom,
! [P_1,Xa,Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ya),Xa))
=> hBOOL(P_1) ) ) ).
fof(fact_1784_order__less__imp__triv,axiom,
! [P_1,Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Xa))
=> hBOOL(P_1) ) ) ).
fof(fact_1785_order__less__imp__triv,axiom,
! [P_1,Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Xa))
=> hBOOL(P_1) ) ) ).
fof(fact_1786_order__less__imp__triv,axiom,
! [P_1,Xa,Ya] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa))
=> hBOOL(P_1) ) ) ).
fof(fact_1787_order__less__imp__triv,axiom,
! [P_1,Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa))
=> hBOOL(P_1) ) ) ).
fof(fact_1788_order__less__imp__triv,axiom,
! [P_1,Xa,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ya),Xa))
=> hBOOL(P_1) ) ) ).
fof(fact_1789_order__less__imp__triv,axiom,
! [P_1,Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ya),Xa))
=> hBOOL(P_1) ) ) ).
fof(fact_1790_order__less__imp__not__eq2,axiom,
! [X_36,Y_30] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_36),Y_30))
=> Y_30 != X_36 ) ).
fof(fact_1791_order__less__imp__not__eq2,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
=> Ya != Xa ) ).
fof(fact_1792_order__less__imp__not__eq2,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
=> Ya != Xa ) ).
fof(fact_1793_order__less__imp__not__eq2,axiom,
! [X_36,Y_30] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_36),Y_30))
=> Y_30 != X_36 ) ).
fof(fact_1794_order__less__imp__not__eq2,axiom,
! [X_36,Y_30] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_36),Y_30))
=> Y_30 != X_36 ) ).
fof(fact_1795_order__less__imp__not__eq2,axiom,
! [X_36,Y_30] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_36),Y_30))
=> Y_30 != X_36 ) ).
fof(fact_1796_order__less__imp__not__eq2,axiom,
! [X_36,Y_30] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_36),Y_30))
=> Y_30 != X_36 ) ).
fof(fact_1797_order__less__imp__not__eq,axiom,
! [X_35,Y_29] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_35),Y_29))
=> X_35 != Y_29 ) ).
fof(fact_1798_order__less__imp__not__eq,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
=> Xa != Ya ) ).
fof(fact_1799_order__less__imp__not__eq,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
=> Xa != Ya ) ).
fof(fact_1800_order__less__imp__not__eq,axiom,
! [X_35,Y_29] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_35),Y_29))
=> X_35 != Y_29 ) ).
fof(fact_1801_order__less__imp__not__eq,axiom,
! [X_35,Y_29] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_35),Y_29))
=> X_35 != Y_29 ) ).
fof(fact_1802_order__less__imp__not__eq,axiom,
! [X_35,Y_29] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_35),Y_29))
=> X_35 != Y_29 ) ).
fof(fact_1803_order__less__imp__not__eq,axiom,
! [X_35,Y_29] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_35),Y_29))
=> X_35 != Y_29 ) ).
fof(fact_1804_order__less__imp__not__less,axiom,
! [X_34,Y_28] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_34),Y_28))
=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_28),X_34)) ) ).
fof(fact_1805_order__less__imp__not__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Xa)) ) ).
fof(fact_1806_order__less__imp__not__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Xa)) ) ).
fof(fact_1807_order__less__imp__not__less,axiom,
! [X_34,Y_28] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_34),Y_28))
=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_28),X_34)) ) ).
fof(fact_1808_order__less__imp__not__less,axiom,
! [X_34,Y_28] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_34),Y_28))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_28),X_34)) ) ).
fof(fact_1809_order__less__imp__not__less,axiom,
! [X_34,Y_28] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_34),Y_28))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_28),X_34)) ) ).
fof(fact_1810_order__less__imp__not__less,axiom,
! [X_34,Y_28] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_34),Y_28))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_28),X_34)) ) ).
fof(fact_1811_order__less__not__sym,axiom,
! [X_33,Y_27] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_33),Y_27))
=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_27),X_33)) ) ).
fof(fact_1812_order__less__not__sym,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Xa)) ) ).
fof(fact_1813_order__less__not__sym,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Xa)) ) ).
fof(fact_1814_order__less__not__sym,axiom,
! [X_33,Y_27] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_33),Y_27))
=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_27),X_33)) ) ).
fof(fact_1815_order__less__not__sym,axiom,
! [X_33,Y_27] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_33),Y_27))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_27),X_33)) ) ).
fof(fact_1816_order__less__not__sym,axiom,
! [X_33,Y_27] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_33),Y_27))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_27),X_33)) ) ).
fof(fact_1817_order__less__not__sym,axiom,
! [X_33,Y_27] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_33),Y_27))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_27),X_33)) ) ).
fof(fact_1818_less__imp__neq,axiom,
! [X_32,Y_26] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_32),Y_26))
=> X_32 != Y_26 ) ).
fof(fact_1819_less__imp__neq,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
=> Xa != Ya ) ).
fof(fact_1820_less__imp__neq,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
=> Xa != Ya ) ).
fof(fact_1821_less__imp__neq,axiom,
! [X_32,Y_26] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_32),Y_26))
=> X_32 != Y_26 ) ).
fof(fact_1822_less__imp__neq,axiom,
! [X_32,Y_26] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_32),Y_26))
=> X_32 != Y_26 ) ).
fof(fact_1823_less__imp__neq,axiom,
! [X_32,Y_26] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_32),Y_26))
=> X_32 != Y_26 ) ).
fof(fact_1824_less__imp__neq,axiom,
! [X_32,Y_26] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_32),Y_26))
=> X_32 != Y_26 ) ).
fof(fact_1825_linorder__neqE,axiom,
! [X_31,Y_25] :
( X_31 != Y_25
=> ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_31),Y_25))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_25),X_31)) ) ) ).
fof(fact_1826_linorder__neqE,axiom,
! [X_31,Y_25] :
( X_31 != Y_25
=> ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_31),Y_25))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_25),X_31)) ) ) ).
fof(fact_1827_linorder__neqE,axiom,
! [X_31,Y_25] :
( X_31 != Y_25
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_31),Y_25))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_25),X_31)) ) ) ).
fof(fact_1828_linorder__neqE,axiom,
! [X_31,Y_25] :
( X_31 != Y_25
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_31),Y_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_25),X_31)) ) ) ).
fof(fact_1829_linorder__neqE,axiom,
! [X_31,Y_25] :
( ( is_int(X_31)
& is_int(Y_25) )
=> ( X_31 != Y_25
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_31),Y_25))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_25),X_31)) ) ) ) ).
fof(fact_1830_linorder__antisym__conv3,axiom,
! [Ya,Xa] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ya),Xa))
=> ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1831_linorder__antisym__conv3,axiom,
! [Ya,Xa] :
( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa))
=> ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1832_linorder__antisym__conv3,axiom,
! [Ya,Xa] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa))
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1833_linorder__antisym__conv3,axiom,
! [Ya,Xa] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ya),Xa))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1834_linorder__antisym__conv3,axiom,
! [Ya,Xa] :
( ( is_int(Ya)
& is_int(Xa) )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ya),Xa))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
<=> Xa = Ya ) ) ) ).
fof(fact_1835_linorder__less__linear,axiom,
! [X_30,Y_24] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_30),Y_24))
| X_30 = Y_24
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_24),X_30)) ) ).
fof(fact_1836_linorder__less__linear,axiom,
! [X_30,Y_24] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_30),Y_24))
| X_30 = Y_24
| hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_24),X_30)) ) ).
fof(fact_1837_linorder__less__linear,axiom,
! [X_30,Y_24] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_30),Y_24))
| X_30 = Y_24
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_24),X_30)) ) ).
fof(fact_1838_linorder__less__linear,axiom,
! [X_30,Y_24] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_30),Y_24))
| X_30 = Y_24
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_24),X_30)) ) ).
fof(fact_1839_linorder__less__linear,axiom,
! [X_30,Y_24] :
( ( is_int(X_30)
& is_int(Y_24) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_30),Y_24))
| X_30 = Y_24
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_24),X_30)) ) ) ).
fof(fact_1840_not__less__iff__gr__or__eq,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))
<=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ya),Xa))
| Xa = Ya ) ) ).
fof(fact_1841_not__less__iff__gr__or__eq,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))
<=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa))
| Xa = Ya ) ) ).
fof(fact_1842_not__less__iff__gr__or__eq,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa))
| Xa = Ya ) ) ).
fof(fact_1843_not__less__iff__gr__or__eq,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ya),Xa))
| Xa = Ya ) ) ).
fof(fact_1844_not__less__iff__gr__or__eq,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ya),Xa))
| Xa = Ya ) ) ) ).
fof(fact_1845_linorder__neq__iff,axiom,
! [Xa,Ya] :
( Xa != Ya
<=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ya),Xa)) ) ) ).
fof(fact_1846_linorder__neq__iff,axiom,
! [Xa,Ya] :
( Xa != Ya
<=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))
| hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa)) ) ) ).
fof(fact_1847_linorder__neq__iff,axiom,
! [Xa,Ya] :
( Xa != Ya
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)) ) ) ).
fof(fact_1848_linorder__neq__iff,axiom,
! [Xa,Ya] :
( Xa != Ya
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ya),Xa)) ) ) ).
fof(fact_1849_linorder__neq__iff,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( Xa != Ya
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ya),Xa)) ) ) ) ).
fof(fact_1850_order__less__irrefl,axiom,
! [X_29] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_29),X_29)) ).
fof(fact_1851_order__less__irrefl,axiom,
! [Xa] : ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Xa)) ).
fof(fact_1852_order__less__irrefl,axiom,
! [Xa] : ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Xa)) ).
fof(fact_1853_order__less__irrefl,axiom,
! [X_29] : ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_29),X_29)) ).
fof(fact_1854_order__less__irrefl,axiom,
! [X_29] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_29),X_29)) ).
fof(fact_1855_order__less__irrefl,axiom,
! [X_29] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_29),X_29)) ).
fof(fact_1856_order__less__irrefl,axiom,
! [X_29] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_29),X_29)) ).
fof(fact_1857_convex__bound__le,axiom,
! [V_9,U_3,Y_23,X_28,A_63] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_28),A_63))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_23),A_63))
=> ( 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_9))
=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_3),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_3),X_28)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_9),Y_23))),A_63)) ) ) ) ) ) ).
fof(fact_1858_convex__bound__le,axiom,
! [V_9,U_3,Y_23,X_28,A_63] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_28),A_63))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_23),A_63))
=> ( 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_9))
=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_3),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_3),X_28)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_9),Y_23))),A_63)) ) ) ) ) ) ).
fof(fact_1859_convex__bound__le,axiom,
! [V_9,U_3,Y_23,X_28,A_63] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_28),A_63))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_23),A_63))
=> ( 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_9))
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_3),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_3),X_28)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_9),Y_23))),A_63)) ) ) ) ) ) ).
fof(fact_1860_power__Suc__less,axiom,
! [N_22,A_62] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_62))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_62),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_62),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_62),N_22))),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_62),N_22))) ) ) ).
fof(fact_1861_power__Suc__less,axiom,
! [N_22,A_62] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,zero_z891286103de_int),A_62))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_62),one_on1684967323de_int))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_62),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_62),N_22))),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_62),N_22))) ) ) ).
fof(fact_1862_power__Suc__less,axiom,
! [N_22,A_62] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,zero_z126310315umeral),A_62))
=> ( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,A_62),one_on1645066479umeral))
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(ord_le1304079648umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,A_62),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_62),N_22))),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,A_62),N_22))) ) ) ).
fof(fact_1863_power__Suc__less,axiom,
! [N_22,A_62] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_62))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_62),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_62),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_62),N_22))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_62),N_22))) ) ) ).
fof(fact_1864_power__Suc__less,axiom,
! [N_22,A_62] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_62))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_62),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_62),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_62),N_22))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_62),N_22))) ) ) ).
fof(fact_1865_power__Suc__less,axiom,
! [N_22,A_62] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_62))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_62),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_62),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_62),N_22))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_62),N_22))) ) ) ).
fof(fact_1866_double__number__of__Bit0,axiom,
! [W_5] : 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_5)) = hAPP_int_rat(number_number_of_rat,hAPP_int_int(bit0,W_5)) ).
fof(fact_1867_double__number__of__Bit0,axiom,
! [W_5] : 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_5)) = hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,W_5)) ).
fof(fact_1868_double__number__of__Bit0,axiom,
! [W_5] : 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_5)) = hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,W_5)) ).
fof(fact_1869_double__number__of__Bit0,axiom,
! [W_5] : 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_5)) = hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,W_5)) ).
fof(fact_1870_power3__eq__cube,axiom,
! [A_61] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_61),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_61),A_61)),A_61) ).
fof(fact_1871_power3__eq__cube,axiom,
! [A_61] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_61),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_61),A_61)),A_61) ).
fof(fact_1872_power3__eq__cube,axiom,
! [A_61] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_61),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_61),A_61)),A_61) ).
fof(fact_1873_power3__eq__cube,axiom,
! [A_61] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_61),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_61),A_61)),A_61) ).
fof(fact_1874_power3__eq__cube,axiom,
! [A_61] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_61),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_61),A_61)),A_61) ).
fof(fact_1875_power3__eq__cube,axiom,
! [A_61] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_61),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_61),A_61)),A_61) ).
fof(fact_1876_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_1877_convex__bound__lt,axiom,
! [V_8,U_2,Y_22,X_27,A_60] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_27),A_60))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_22),A_60))
=> ( 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_8))
=> ( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,U_2),V_8) = 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_2),X_27)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,V_8),Y_22))),A_60)) ) ) ) ) ) ).
fof(fact_1878_convex__bound__lt,axiom,
! [V_8,U_2,Y_22,X_27,A_60] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_27),A_60))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_22),A_60))
=> ( 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_8))
=> ( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,U_2),V_8) = 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_2),X_27)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,V_8),Y_22))),A_60)) ) ) ) ) ) ).
fof(fact_1879_convex__bound__lt,axiom,
! [V_8,U_2,Y_22,X_27,A_60] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_27),A_60))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_22),A_60))
=> ( 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_8))
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,U_2),V_8) = 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_2),X_27)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_8),Y_22))),A_60)) ) ) ) ) ) ).
fof(fact_1880_mult__2__right,axiom,
! [Z_14] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_14),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_14),Z_14) ).
fof(fact_1881_mult__2__right,axiom,
! [Z_14] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_14),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_14),Z_14) ).
fof(fact_1882_mult__2__right,axiom,
! [Z_14] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_14),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_14),Z_14) ).
fof(fact_1883_mult__2__right,axiom,
! [Z_14] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_14),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_14),Z_14) ).
fof(fact_1884_semiring__mult__2__right,axiom,
! [Z_13] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Z_13),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_13),Z_13) ).
fof(fact_1885_semiring__mult__2__right,axiom,
! [Z_13] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,Z_13),hAPP_int_complex(number528085621omplex,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,Z_13),Z_13) ).
fof(fact_1886_semiring__mult__2__right,axiom,
! [Z_13] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_13),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Z_13),Z_13) ).
fof(fact_1887_semiring__mult__2__right,axiom,
! [Z_13] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Z_13),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_13),Z_13) ).
fof(fact_1888_semiring__mult__2__right,axiom,
! [Z_13] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Z_13),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_13),Z_13) ).
fof(fact_1889_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_1890_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_1891_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_1892_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_1893_semiring__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_1894_semiring__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_1895_semiring__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_1896_semiring__mult__2,axiom,
! [Z_11] : 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_11) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Z_11),Z_11) ).
fof(fact_1897_semiring__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_1898_power2__eq__square,axiom,
! [A_59] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_59),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_59),A_59) ).
fof(fact_1899_power2__eq__square,axiom,
! [A_59] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_59),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_59),A_59) ).
fof(fact_1900_power2__eq__square,axiom,
! [A_59] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_59),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_59),A_59) ).
fof(fact_1901_power2__eq__square,axiom,
! [A_59] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_59),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_59),A_59) ).
fof(fact_1902_power2__eq__square,axiom,
! [A_59] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_59),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_59),A_59) ).
fof(fact_1903_power2__eq__square,axiom,
! [A_59] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,A_59),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_59),A_59) ).
fof(fact_1904_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_26] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_26),X_26) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_1905_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_26] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_26),X_26) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_1906_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_26] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X_26),X_26) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_1907_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_26] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_26),X_26) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_1908_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_26] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_26),X_26) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_1909_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_26] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,X_26),X_26) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_26),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_1910_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_1911_two__realpow__ge__one,axiom,
! [N] : 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))) ).
fof(fact_1912_power2__eq__square__number__of,axiom,
! [W_4] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_int_complex(number528085621omplex,W_4)),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_4)),hAPP_int_complex(number528085621omplex,W_4)) ).
fof(fact_1913_power2__eq__square__number__of,axiom,
! [W_4] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,W_4)),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_4)),hAPP_int_real(number267125858f_real,W_4)) ).
fof(fact_1914_power2__eq__square__number__of,axiom,
! [W_4] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,W_4)),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_4)),hAPP_int_nat(number_number_of_nat,W_4)) ).
fof(fact_1915_power2__eq__square__number__of,axiom,
! [W_4] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,W_4)),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_4)),hAPP_int_int(number_number_of_int,W_4)) ).
fof(fact_1916_power2__eq__square__number__of,axiom,
! [W_4] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_int_rat(number_number_of_rat,W_4)),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_4)),hAPP_int_rat(number_number_of_rat,W_4)) ).
fof(fact_1917_power2__eq__square__number__of,axiom,
! [W_4] : hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,hAPP_i1732201573de_int(number1226105091de_int,W_4)),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_4)),hAPP_i1732201573de_int(number1226105091de_int,W_4)) ).
fof(fact_1918_power2__eq__square__number__of,axiom,
! [W_4] : hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,hAPP_i769753017umeral(number1443263063umeral,W_4)),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_4)),hAPP_i769753017umeral(number1443263063umeral,W_4)) ).
fof(fact_1919_xt1_I8_J,axiom,
! [Z_10,Y_21,X_25] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_21),X_25))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_10),Y_21))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_10),X_25)) ) ) ).
fof(fact_1920_xt1_I8_J,axiom,
! [Z_2,Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Ya))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Xa)) ) ) ).
fof(fact_1921_xt1_I8_J,axiom,
! [Z_2,Ya,Xa] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Ya))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Xa)) ) ) ).
fof(fact_1922_xt1_I8_J,axiom,
! [Z_10,Y_21,X_25] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_21),X_25))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_10),Y_21))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_10),X_25)) ) ) ).
fof(fact_1923_xt1_I8_J,axiom,
! [Z_10,Y_21,X_25] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_21),X_25))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_10),Y_21))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_10),X_25)) ) ) ).
fof(fact_1924_xt1_I8_J,axiom,
! [Z_10,Y_21,X_25] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_21),X_25))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_10),Y_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_10),X_25)) ) ) ).
fof(fact_1925_xt1_I8_J,axiom,
! [Z_10,Y_21,X_25] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_21),X_25))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_10),Y_21))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_10),X_25)) ) ) ).
fof(fact_1926_order__le__less__trans,axiom,
! [Z_9,X_24,Y_20] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_24),Y_20))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_20),Z_9))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_24),Z_9)) ) ) ).
fof(fact_1927_order__le__less__trans,axiom,
! [Z_2,Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Z_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Z_2)) ) ) ).
fof(fact_1928_order__le__less__trans,axiom,
! [Z_2,Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Z_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Z_2)) ) ) ).
fof(fact_1929_order__le__less__trans,axiom,
! [Z_9,X_24,Y_20] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_24),Y_20))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_20),Z_9))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_24),Z_9)) ) ) ).
fof(fact_1930_order__le__less__trans,axiom,
! [Z_9,X_24,Y_20] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_24),Y_20))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_20),Z_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_24),Z_9)) ) ) ).
fof(fact_1931_order__le__less__trans,axiom,
! [Z_9,X_24,Y_20] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_24),Y_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_20),Z_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_24),Z_9)) ) ) ).
fof(fact_1932_order__le__less__trans,axiom,
! [Z_9,X_24,Y_20] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_24),Y_20))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_20),Z_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_24),Z_9)) ) ) ).
fof(fact_1933_xt1_I7_J,axiom,
! [Z_8,Y_19,X_23] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_19),X_23))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_8),Y_19))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_8),X_23)) ) ) ).
fof(fact_1934_xt1_I7_J,axiom,
! [Z_2,Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Z_2),Ya))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Z_2),Xa)) ) ) ).
fof(fact_1935_xt1_I7_J,axiom,
! [Z_2,Ya,Xa] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Ya),Xa))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Z_2),Ya))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Z_2),Xa)) ) ) ).
fof(fact_1936_xt1_I7_J,axiom,
! [Z_8,Y_19,X_23] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_19),X_23))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Z_8),Y_19))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Z_8),X_23)) ) ) ).
fof(fact_1937_xt1_I7_J,axiom,
! [Z_8,Y_19,X_23] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_19),X_23))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Z_8),Y_19))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_8),X_23)) ) ) ).
fof(fact_1938_xt1_I7_J,axiom,
! [Z_8,Y_19,X_23] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_19),X_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_8),Y_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_8),X_23)) ) ) ).
fof(fact_1939_xt1_I7_J,axiom,
! [Z_8,Y_19,X_23] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_19),X_23))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_8),Y_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z_8),X_23)) ) ) ).
fof(fact_1940_order__less__le__trans,axiom,
! [Z_7,X_22,Y_18] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_22),Y_18))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_18),Z_7))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_22),Z_7)) ) ) ).
fof(fact_1941_order__less__le__trans,axiom,
! [Z_2,Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Z_2))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Z_2)) ) ) ).
fof(fact_1942_order__less__le__trans,axiom,
! [Z_2,Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Z_2))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Z_2)) ) ) ).
fof(fact_1943_order__less__le__trans,axiom,
! [Z_7,X_22,Y_18] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_22),Y_18))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_18),Z_7))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_22),Z_7)) ) ) ).
fof(fact_1944_order__less__le__trans,axiom,
! [Z_7,X_22,Y_18] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_22),Y_18))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_18),Z_7))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_22),Z_7)) ) ) ).
fof(fact_1945_order__less__le__trans,axiom,
! [Z_7,X_22,Y_18] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_22),Y_18))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_18),Z_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_22),Z_7)) ) ) ).
fof(fact_1946_order__less__le__trans,axiom,
! [Z_7,X_22,Y_18] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_22),Y_18))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_18),Z_7))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_22),Z_7)) ) ) ).
fof(fact_1947_xt1_I11_J,axiom,
! [B_40,A_58] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_40),A_58))
=> ( A_58 != B_40
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_40),A_58)) ) ) ).
fof(fact_1948_xt1_I11_J,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_1),A_1))
=> ( A_1 != B_1
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),A_1)) ) ) ).
fof(fact_1949_xt1_I11_J,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_1),A_1))
=> ( A_1 != B_1
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1)) ) ) ).
fof(fact_1950_xt1_I11_J,axiom,
! [B_40,A_58] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_40),A_58))
=> ( A_58 != B_40
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_40),A_58)) ) ) ).
fof(fact_1951_xt1_I11_J,axiom,
! [B_40,A_58] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_40),A_58))
=> ( A_58 != B_40
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_40),A_58)) ) ) ).
fof(fact_1952_xt1_I11_J,axiom,
! [B_40,A_58] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_40),A_58))
=> ( A_58 != B_40
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_40),A_58)) ) ) ).
fof(fact_1953_xt1_I11_J,axiom,
! [B_40,A_58] :
( ( is_int(B_40)
& is_int(A_58) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_40),A_58))
=> ( A_58 != B_40
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_40),A_58)) ) ) ) ).
fof(fact_1954_order__le__neq__trans,axiom,
! [A_57,B_39] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_57),B_39))
=> ( A_57 != B_39
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_57),B_39)) ) ) ).
fof(fact_1955_order__le__neq__trans,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),B_1))
=> ( A_1 != B_1
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1)) ) ) ).
fof(fact_1956_order__le__neq__trans,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1))
=> ( A_1 != B_1
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1)) ) ) ).
fof(fact_1957_order__le__neq__trans,axiom,
! [A_57,B_39] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_57),B_39))
=> ( A_57 != B_39
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_57),B_39)) ) ) ).
fof(fact_1958_order__le__neq__trans,axiom,
! [A_57,B_39] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_57),B_39))
=> ( A_57 != B_39
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_57),B_39)) ) ) ).
fof(fact_1959_order__le__neq__trans,axiom,
! [A_57,B_39] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_57),B_39))
=> ( A_57 != B_39
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_57),B_39)) ) ) ).
fof(fact_1960_order__le__neq__trans,axiom,
! [A_57,B_39] :
( ( is_int(A_57)
& is_int(B_39) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_57),B_39))
=> ( A_57 != B_39
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_57),B_39)) ) ) ) ).
fof(fact_1961_order__le__imp__less__or__eq,axiom,
! [X_21,Y_17] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_21),Y_17))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_21),Y_17))
| X_21 = Y_17 ) ) ).
fof(fact_1962_order__le__imp__less__or__eq,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_1963_order__le__imp__less__or__eq,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_1964_order__le__imp__less__or__eq,axiom,
! [X_21,Y_17] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_21),Y_17))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_21),Y_17))
| X_21 = Y_17 ) ) ).
fof(fact_1965_order__le__imp__less__or__eq,axiom,
! [X_21,Y_17] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_21),Y_17))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_21),Y_17))
| X_21 = Y_17 ) ) ).
fof(fact_1966_order__le__imp__less__or__eq,axiom,
! [X_21,Y_17] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_21),Y_17))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_21),Y_17))
| X_21 = Y_17 ) ) ).
fof(fact_1967_order__le__imp__less__or__eq,axiom,
! [X_21,Y_17] :
( ( is_int(X_21)
& is_int(Y_17) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_21),Y_17))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_21),Y_17))
| X_21 = Y_17 ) ) ) ).
fof(fact_1968_linorder__antisym__conv2,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya))
=> ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1969_linorder__antisym__conv2,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya))
=> ( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1970_linorder__antisym__conv2,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
=> ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1971_linorder__antisym__conv2,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_1972_linorder__antisym__conv2,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
<=> Xa = Ya ) ) ) ).
fof(fact_1973_order__less__imp__le,axiom,
! [X_20,Y_16] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_20),Y_16))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_20),Y_16)) ) ).
fof(fact_1974_order__less__imp__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya)) ) ).
fof(fact_1975_order__less__imp__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya)) ) ).
fof(fact_1976_order__less__imp__le,axiom,
! [X_20,Y_16] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_20),Y_16))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_20),Y_16)) ) ).
fof(fact_1977_order__less__imp__le,axiom,
! [X_20,Y_16] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_20),Y_16))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_20),Y_16)) ) ).
fof(fact_1978_order__less__imp__le,axiom,
! [X_20,Y_16] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_20),Y_16))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_20),Y_16)) ) ).
fof(fact_1979_order__less__imp__le,axiom,
! [X_20,Y_16] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_20),Y_16))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_20),Y_16)) ) ).
fof(fact_1980_leD,axiom,
! [Y_15,X_19] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_15),X_19))
=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_19),Y_15)) ) ).
fof(fact_1981_leD,axiom,
! [Y_15,X_19] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_15),X_19))
=> ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_19),Y_15)) ) ).
fof(fact_1982_leD,axiom,
! [Y_15,X_19] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_15),X_19))
=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_19),Y_15)) ) ).
fof(fact_1983_leD,axiom,
! [Y_15,X_19] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_15),X_19))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_19),Y_15)) ) ).
fof(fact_1984_leD,axiom,
! [Y_15,X_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_15),X_19))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_19),Y_15)) ) ).
fof(fact_1985_xt1_I12_J,axiom,
! [A_56,B_38] :
( A_56 != B_38
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_38),A_56))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,B_38),A_56)) ) ) ).
fof(fact_1986_xt1_I12_J,axiom,
! [A_1,B_1] :
( A_1 != B_1
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,B_1),A_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,B_1),A_1)) ) ) ).
fof(fact_1987_xt1_I12_J,axiom,
! [A_1,B_1] :
( A_1 != B_1
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B_1),A_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,B_1),A_1)) ) ) ).
fof(fact_1988_xt1_I12_J,axiom,
! [A_56,B_38] :
( A_56 != B_38
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,B_38),A_56))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,B_38),A_56)) ) ) ).
fof(fact_1989_xt1_I12_J,axiom,
! [A_56,B_38] :
( A_56 != B_38
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_38),A_56))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_38),A_56)) ) ) ).
fof(fact_1990_xt1_I12_J,axiom,
! [A_56,B_38] :
( A_56 != B_38
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_38),A_56))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_38),A_56)) ) ) ).
fof(fact_1991_xt1_I12_J,axiom,
! [A_56,B_38] :
( ( is_int(A_56)
& is_int(B_38) )
=> ( A_56 != B_38
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_38),A_56))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_38),A_56)) ) ) ) ).
fof(fact_1992_order__neq__le__trans,axiom,
! [A_55,B_37] :
( A_55 != B_37
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_55),B_37))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_55),B_37)) ) ) ).
fof(fact_1993_order__neq__le__trans,axiom,
! [A_1,B_1] :
( A_1 != B_1
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A_1),B_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A_1),B_1)) ) ) ).
fof(fact_1994_order__neq__le__trans,axiom,
! [A_1,B_1] :
( A_1 != B_1
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A_1),B_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A_1),B_1)) ) ) ).
fof(fact_1995_order__neq__le__trans,axiom,
! [A_55,B_37] :
( A_55 != B_37
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,A_55),B_37))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,A_55),B_37)) ) ) ).
fof(fact_1996_order__neq__le__trans,axiom,
! [A_55,B_37] :
( A_55 != B_37
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_55),B_37))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_55),B_37)) ) ) ).
fof(fact_1997_order__neq__le__trans,axiom,
! [A_55,B_37] :
( A_55 != B_37
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_55),B_37))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_55),B_37)) ) ) ).
fof(fact_1998_order__neq__le__trans,axiom,
! [A_55,B_37] :
( ( is_int(A_55)
& is_int(B_37) )
=> ( A_55 != B_37
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_55),B_37))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_55),B_37)) ) ) ) ).
fof(fact_1999_linorder__antisym__conv1,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_2000_linorder__antisym__conv1,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_2001_linorder__antisym__conv1,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_2002_linorder__antisym__conv1,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_2003_linorder__antisym__conv1,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya))
<=> Xa = Ya ) ) ) ).
fof(fact_2004_not__leE,axiom,
! [Y_14,X_18] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_14),X_18))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_18),Y_14)) ) ).
fof(fact_2005_not__leE,axiom,
! [Y_14,X_18] :
( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_14),X_18))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_18),Y_14)) ) ).
fof(fact_2006_not__leE,axiom,
! [Y_14,X_18] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_14),X_18))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_18),Y_14)) ) ).
fof(fact_2007_not__leE,axiom,
! [Y_14,X_18] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_14),X_18))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_18),Y_14)) ) ).
fof(fact_2008_not__leE,axiom,
! [Y_14,X_18] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_14),X_18))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_18),Y_14)) ) ).
fof(fact_2009_leI,axiom,
! [X_17,Y_13] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_17),Y_13))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Y_13),X_17)) ) ).
fof(fact_2010_leI,axiom,
! [X_17,Y_13] :
( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_17),Y_13))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Y_13),X_17)) ) ).
fof(fact_2011_leI,axiom,
! [X_17,Y_13] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_17),Y_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_13),X_17)) ) ).
fof(fact_2012_leI,axiom,
! [X_17,Y_13] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_17),Y_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_13),X_17)) ) ).
fof(fact_2013_leI,axiom,
! [X_17,Y_13] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_17),Y_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y_13),X_17)) ) ).
fof(fact_2014_order__le__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya))
<=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_2015_order__le__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_2016_order__le__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_2017_order__le__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya))
<=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_2018_order__le__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_2019_order__le__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
| Xa = Ya ) ) ).
fof(fact_2020_order__le__less,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
| Xa = Ya ) ) ) ).
fof(fact_2021_less__le__not__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))
<=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya))
& ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ya),Xa)) ) ) ).
fof(fact_2022_less__le__not__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Ya),Xa)) ) ) ).
fof(fact_2023_less__le__not__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Ya),Xa)) ) ) ).
fof(fact_2024_less__le__not__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))
<=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya))
& ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Ya),Xa)) ) ) ).
fof(fact_2025_less__le__not__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
& ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa)) ) ) ).
fof(fact_2026_less__le__not__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa)) ) ) ).
fof(fact_2027_less__le__not__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya))
& ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),Xa)) ) ) ).
fof(fact_2028_order__less__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))
<=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_2029_order__less__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya))
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_2030_order__less__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_2031_order__less__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))
<=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_2032_order__less__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_2033_order__less__le,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_2034_order__less__le,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya))
& Xa != Ya ) ) ) ).
fof(fact_2035_linorder__le__less__linear,axiom,
! [X_16,Y_12] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_16),Y_12))
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Y_12),X_16)) ) ).
fof(fact_2036_linorder__le__less__linear,axiom,
! [X_16,Y_12] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_16),Y_12))
| hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Y_12),X_16)) ) ).
fof(fact_2037_linorder__le__less__linear,axiom,
! [X_16,Y_12] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_16),Y_12))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_12),X_16)) ) ).
fof(fact_2038_linorder__le__less__linear,axiom,
! [X_16,Y_12] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_16),Y_12))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_12),X_16)) ) ).
fof(fact_2039_linorder__le__less__linear,axiom,
! [X_16,Y_12] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_16),Y_12))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y_12),X_16)) ) ).
fof(fact_2040_linorder__not__le,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Ya),Xa)) ) ).
fof(fact_2041_linorder__not__le,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Xa),Ya))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Ya),Xa)) ) ).
fof(fact_2042_linorder__not__le,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa)) ) ).
fof(fact_2043_linorder__not__le,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ya),Xa)) ) ).
fof(fact_2044_linorder__not__le,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ya),Xa)) ) ).
fof(fact_2045_linorder__not__less,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Ya),Xa)) ) ).
fof(fact_2046_linorder__not__less,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Xa),Ya))
<=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Ya),Xa)) ) ).
fof(fact_2047_linorder__not__less,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa)) ) ).
fof(fact_2048_linorder__not__less,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa)) ) ).
fof(fact_2049_linorder__not__less,axiom,
! [Xa,Ya] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),Xa)) ) ).
fof(fact_2050_conj__le__cong,axiom,
! [P_2,P_1,Xa] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
& hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
& hBOOL(P_2) ) ) ) ).
fof(fact_2051_imp__le__cong,axiom,
! [P_2,P_1,Xa] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hBOOL(P_1) )
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hBOOL(P_2) ) ) ) ).
fof(fact_2052_power2__sum,axiom,
! [X_15,Y_11] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_15),Y_11)),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_15),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_11),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_15)),Y_11)) ).
fof(fact_2053_power2__sum,axiom,
! [X_15,Y_11] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_15),Y_11)),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_15),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_11),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_15)),Y_11)) ).
fof(fact_2054_power2__sum,axiom,
! [X_15,Y_11] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_15),Y_11)),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_15),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_11),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_15)),Y_11)) ).
fof(fact_2055_power2__sum,axiom,
! [X_15,Y_11] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_15),Y_11)),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_15),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_11),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_15)),Y_11)) ).
fof(fact_2056_power2__sum,axiom,
! [X_15,Y_11] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_15),Y_11)),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_15),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_11),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_15)),Y_11)) ).
fof(fact_2057_self__quotient__aux2,axiom,
! [R_2,Q_1,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_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),one_one_int)) ) ) ) ).
fof(fact_2058_self__quotient__aux1,axiom,
! [R_2,Q_1,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_2),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Q_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),A_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),Q_1)) ) ) ) ).
fof(fact_2059_ex__least__nat__less,axiom,
! [Na,P_1] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,Na))
=> ? [K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),Na))
& ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K_1))
=> ~ 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_1),one_one_nat))) ) ) ) ).
fof(fact_2060_incr__mult__lemma,axiom,
! [K,P_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D_1))) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X_1] :
( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_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,K),D_1)))) ) ) ) ) ).
fof(fact_2061_smaller_I2_J,axiom,
~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),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(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)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))))) ) ).
fof(fact_2062_q__pos__lemma,axiom,
! [B_5,Q_5,R_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,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_5)),R_4)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_4),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_5)) ) ) ) ).
fof(fact_2063_q__neg__lemma,axiom,
! [B_5,Q_5,R_4] :
( 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_5)),R_4)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_4))
=> ( 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_5),zero_zero_int)) ) ) ) ).
fof(fact_2064_unique__quotient__lemma,axiom,
! [B_2,Q_5,R_4,Q_1,R_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,B_2),Q_5)),R_4)),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_1)),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_4))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_4),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_5),Q_1)) ) ) ) ) ).
fof(fact_2065_zdiv__mono2__lemma,axiom,
! [B_2,Q_1,R_2,B_5,Q_5,R_4] :
( 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_1)),R_2) = 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_5)),R_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,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_5),Q_5)),R_4)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_4),B_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_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_1),Q_5)) ) ) ) ) ) ) ).
fof(fact_2066_unique__quotient__lemma__neg,axiom,
! [B_2,Q_5,R_4,Q_1,R_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,B_2),Q_5)),R_4)),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_1)),R_2)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_4))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Q_1),Q_5)) ) ) ) ) ).
fof(fact_2067_nQ1,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)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,zero_zero_nat))))) ).
fof(fact_2068_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_2069_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_2070_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_2071_IH,axiom,
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),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(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)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))))) ) ).
fof(fact_2072__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_,axiom,
~ ! [X_1,Y_1] :
( ( is_int(X_1)
& is_int(Y_1) )
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_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_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))))) != 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)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))) ) ).
fof(fact_2073_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_2074_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_2075__096t_A_061_A1_A_061_061_062_AEX_Ax_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A,axiom,
( t = one_one_int
=> ? [X_1,Y_1] :
( is_int(X_1)
& is_int(Y_1)
& hAPP_int_int(hAPP_int_fun_int_int(plus_plus_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_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))))) = 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_2076_less__fun__def,axiom,
! [F,G] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,F),G))
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,F),G))
& ~ hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,G),F)) ) ) ).
fof(fact_2077_less__fun__def,axiom,
! [F,G] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,F),G))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,F),G))
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,G),F)) ) ) ).
fof(fact_2078_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_2079_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_2080_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_2081_real__le__refl,axiom,
! [W] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,W),W)) ).
fof(fact_2082_nat__mult__commute,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M) ).
fof(fact_2083_nat__mult__assoc,axiom,
! [M,N,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)),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),K_2)) ).
fof(fact_2084_mult__0,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).
fof(fact_2085_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_2086_mult__is__0,axiom,
! [Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na) = zero_zero_nat
<=> ( Ma = zero_zero_nat
| Na = zero_zero_nat ) ) ).
fof(fact_2087_mult__cancel1,axiom,
! [K,Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)
<=> ( Ma = Na
| K = zero_zero_nat ) ) ).
fof(fact_2088_mult__cancel2,axiom,
! [Ma,K,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K)
<=> ( Ma = Na
| K = zero_zero_nat ) ) ).
fof(fact_2089_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_14,P_6,Q_7] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_14),P_6)),Q_7) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_6),Q_7)) ).
fof(fact_2090_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_14,P_6,Q_7] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_14),P_6)),Q_7) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_6),Q_7)) ).
fof(fact_2091_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_14,P_6,Q_7] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_14),P_6)),Q_7) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_6),Q_7)) ).
fof(fact_2092_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_14,P_6,Q_7] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_14),P_6)),Q_7) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_14),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_6),Q_7)) ).
fof(fact_2093_power__mult,axiom,
! [A_54,M_11,N_21] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_54),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_11),N_21)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_54),M_11)),N_21) ).
fof(fact_2094_power__mult,axiom,
! [A_54,M_11,N_21] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_54),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_11),N_21)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_54),M_11)),N_21) ).
fof(fact_2095_power__mult,axiom,
! [A_54,M_11,N_21] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_54),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_11),N_21)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_54),M_11)),N_21) ).
fof(fact_2096_power__mult,axiom,
! [A_54,M_11,N_21] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_54),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_11),N_21)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_54),M_11)),N_21) ).
fof(fact_2097_real__mult__left__cancel,axiom,
! [A_1,B_1,C] :
( C != zero_zero_real
=> ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A_1) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1)
<=> A_1 = B_1 ) ) ).
fof(fact_2098_real__mult__right__cancel,axiom,
! [A_1,B_1,C] :
( C != zero_zero_real
=> ( hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),C) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),C)
<=> A_1 = B_1 ) ) ).
fof(fact_2099_add__mult__distrib,axiom,
! [M,N,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)),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),K_2)) ).
fof(fact_2100_add__mult__distrib2,axiom,
! [K_2,M,N] : 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)) = 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)) ).
fof(fact_2101_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_2102_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_2103_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_2104_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_2105_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_2106_nat__mult__eq__one,axiom,
! [Na,Ma] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),Ma) = one_one_nat
<=> ( Na = one_one_nat
& Ma = one_one_nat ) ) ).
fof(fact_2107_nat__mult__eq__1__iff,axiom,
! [Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na) = one_one_nat
<=> ( Ma = one_one_nat
& Na = one_one_nat ) ) ).
fof(fact_2108_nat__mult__1__right,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).
fof(fact_2109_nat__1__eq__mult__iff,axiom,
! [Ma,Na] :
( one_one_nat = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na)
<=> ( Ma = one_one_nat
& Na = one_one_nat ) ) ).
fof(fact_2110_nat__mult__1,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).
fof(fact_2111_real__mult__1,axiom,
! [Z_1] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,one_one_real),Z_1) = Z_1 ).
fof(fact_2112_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_2113_zpower__zpower,axiom,
! [X,Y,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),Y)),Z_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z_1)) ).
fof(fact_2114_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_2115_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_2116_mult__less__cancel2,axiom,
! [Ma,K,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ) ).
fof(fact_2117_mult__less__cancel1,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ) ).
fof(fact_2118_nat__0__less__mult__iff,axiom,
! [Ma,Na] :
( 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),Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ma))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ) ).
fof(fact_2119_not__real__square__gt__zero,axiom,
! [Xa] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Xa)))
<=> Xa = zero_zero_real ) ).
fof(fact_2120_real__mult__less__mono2,axiom,
! [X,Y,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),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_1),Y))) ) ) ).
fof(fact_2121_real__mult__order,axiom,
! [Y,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,zero_zero_real),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y))) ) ) ).
fof(fact_2122_real__mult__less__iff1,axiom,
! [Xa,Ya,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,Xa),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Z_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) ) ) ).
fof(fact_2123_real__mult__le__cancel__iff2,axiom,
! [Xa,Ya,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),Xa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Z_2),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ) ).
fof(fact_2124_real__mult__le__cancel__iff1,axiom,
! [Xa,Ya,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,Xa),Z_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Z_2)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ) ).
fof(fact_2125_mult__eq__self__implies__10,axiom,
! [M,N] :
( M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)
=> ( N = one_one_nat
| M = zero_zero_nat ) ) ).
fof(fact_2126_real__two__squares__add__zero__iff,axiom,
! [Xa,Ya] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Xa),Xa)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Ya),Ya)) = zero_zero_real
<=> ( Xa = zero_zero_real
& Ya = zero_zero_real ) ) ).
fof(fact_2127_int__mult,axiom,
! [M,N] : hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) ).
fof(fact_2128_zmult__int,axiom,
! [M,N] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) ).
fof(fact_2129_Nat__Transfer_Otransfer__int__nat__functions_I2_J,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)) ).
fof(fact_2130_is__mult__sum2sq,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,X))
=> ( hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,Y))
=> hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).
fof(fact_2131_mult__le__cancel2,axiom,
! [Ma,K,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ) ).
fof(fact_2132_mult__le__cancel1,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ) ).
fof(fact_2133_power__even__eq,axiom,
! [A_53,N_20] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_53),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_20)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_53),N_20)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_2134_power__even__eq,axiom,
! [A_53,N_20] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_53),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_20)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_53),N_20)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_2135_power__even__eq,axiom,
! [A_53,N_20] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_53),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_20)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_53),N_20)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_2136_power__even__eq,axiom,
! [A_53,N_20] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_53),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_20)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_53),N_20)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_2137_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_2138_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_2139_nat__number__of__mult__left,axiom,
! [V_2,K_2,V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_2)),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_1),V_2))),K_2) ) ) ).
fof(fact_2140_mult__nat__number__of,axiom,
! [V_2,V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,V_1),V_2)) ) ) ).
fof(fact_2141_four__x__squared,axiom,
! [X] : 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),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)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_2142_even__power__le__0__imp__0,axiom,
! [A_52,K_7] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_52),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_7))),zero_zero_rat))
=> A_52 = zero_zero_rat ) ).
fof(fact_2143_even__power__le__0__imp__0,axiom,
! [A_52,K_7] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_52),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_7))),zero_zero_real))
=> A_52 = zero_zero_real ) ).
fof(fact_2144_even__power__le__0__imp__0,axiom,
! [A_52,K_7] :
( is_int(A_52)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_52),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_7))),zero_zero_int))
=> A_52 = zero_zero_int ) ) ).
fof(fact_2145_zero__le__even__power_H,axiom,
! [A_51,N_19] : 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_51),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_19)))) ).
fof(fact_2146_zero__le__even__power_H,axiom,
! [A_51,N_19] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_51),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_19)))) ).
fof(fact_2147_zero__le__even__power_H,axiom,
! [A_51,N_19] : 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_51),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_19)))) ).
fof(fact_2148_zdiv__mono2__neg__lemma,axiom,
! [B_2,Q_1,R_2,B_5,Q_5,R_4] :
( 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_1)),R_2) = 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_5)),R_4)
=> ( 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_5)),R_4)),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_4))
=> ( 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_5),Q_1)) ) ) ) ) ) ) ).
fof(fact_2149_xy,axiom,
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(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)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))) ).
fof(fact_2150_real__sum__squared__expand,axiom,
! [X,Y] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),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),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),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)),Y)) ).
fof(fact_2151__096_B_Bthesis_O_A_I_B_Bt_O_As_____A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A,axiom,
~ ! [T] :
( 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_2152_nat__mult__le__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ) ).
fof(fact_2153_mono__nat__linear__lb,axiom,
! [Ma,K,F] :
( ! [M_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_1)),hAPP_nat_nat(F,N_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(F,Ma)),K)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K)))) ) ).
fof(fact_2154_ex__least__nat__le,axiom,
! [Na,P_1] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,Na))
=> ? [K_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),Na))
& ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),K_1))
=> ~ hBOOL(hAPP_nat_bool(P_1,I_2)) )
& hBOOL(hAPP_nat_bool(P_1,K_1)) ) ) ) ).
fof(fact_2155__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_2156_mult__left__cancel,axiom,
! [N,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) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),M)
=> N = M ) ) ).
fof(fact_2157_nat__mult__less__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ) ).
fof(fact_2158_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_2159_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_2160_Bit1__Min,axiom,
hAPP_int_int(bit1,min) = min ).
fof(fact_2161_rel__simps_I43_J,axiom,
! [L] :
( is_int(L)
=> ( min = hAPP_int_int(bit1,L)
<=> min = L ) ) ).
fof(fact_2162_rel__simps_I47_J,axiom,
! [K] :
( is_int(K)
=> ( hAPP_int_int(bit1,K) = min
<=> K = min ) ) ).
fof(fact_2163_rel__simps_I40_J,axiom,
min != pls ).
fof(fact_2164_rel__simps_I37_J,axiom,
pls != min ).
fof(fact_2165_rel__simps_I42_J,axiom,
! [L_2] : min != hAPP_int_int(bit0,L_2) ).
fof(fact_2166_rel__simps_I45_J,axiom,
! [K_2] : hAPP_int_int(bit0,K_2) != min ).
fof(fact_2167_rel__simps_I7_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),min)) ).
fof(fact_2168_rel__simps_I24_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),min)) ).
fof(fact_2169_rel__simps_I9_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit1,K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) ) ).
fof(fact_2170_rel__simps_I13_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit1,K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),min)) ) ).
fof(fact_2171_rel__simps_I6_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),pls)) ).
fof(fact_2172_rel__simps_I3_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),min)) ).
fof(fact_2173_rel__simps_I8_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),hAPP_int_int(bit0,K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) ) ).
fof(fact_2174_rel__simps_I30_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit1,K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) ) ).
fof(fact_2175_rel__simps_I26_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit1,K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),K)) ) ).
fof(fact_2176_bin__less__0__simps_I2_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),zero_zero_int)) ).
fof(fact_2177_rel__simps_I20_J,axiom,
~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,pls),min)) ).
fof(fact_2178_rel__simps_I23_J,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),pls)) ).
fof(fact_2179_rel__simps_I28_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(bit0,K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) ) ).
fof(fact_2180_eq__number__of__Pls__Min,axiom,
hAPP_int_int(number_number_of_int,pls) != hAPP_int_int(number_number_of_int,min) ).
fof(fact_2181_succ__Min,axiom,
hAPP_int_int(succ,min) = pls ).
fof(fact_2182_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_2183_rel__simps_I11_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(bit0,K)),min))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),min)) ) ).
fof(fact_2184_rel__simps_I25_J,axiom,
! [K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,min),hAPP_int_int(bit0,K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,min),K)) ) ).
fof(fact_2185_pos__zmult__eq__1__iff__lemma,axiom,
! [M,N] :
( is_int(M)
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N) = one_one_int
=> ( M = one_one_int
| M = hAPP_int_int(number_number_of_int,min) ) ) ) ).
fof(fact_2186_zmult__eq__1__iff,axiom,
! [Ma,Na] :
( ( is_int(Ma)
& is_int(Na) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),Na) = one_one_int
<=> ( ( Ma = one_one_int
& Na = one_one_int )
| ( Ma = hAPP_int_int(number_number_of_int,min)
& Na = hAPP_int_int(number_number_of_int,min) ) ) ) ) ).
fof(fact_2187_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_2188_nat__mult__eq__cancel__disj,axiom,
! [K,Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)
<=> ( K = zero_zero_nat
| Ma = Na ) ) ).
fof(fact_2189_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_2190_power__m1__even,axiom,
! [N_18] : 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_18)) = one_one_rat ).
fof(fact_2191_power__m1__even,axiom,
! [N_18] : 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_18)) = one_one_complex ).
fof(fact_2192_power__m1__even,axiom,
! [N_18] : 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_18)) = one_one_real ).
fof(fact_2193_power__m1__even,axiom,
! [N_18] : 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_18)) = one_one_int ).
fof(fact_2194_nat__mult__eq__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)
<=> Ma = Na ) ) ).
fof(fact_2195__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_2196__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_2197_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_2198__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_2199__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_2200_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_2201_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_2202_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_2203__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_2204_dvd__0__right,axiom,
! [A_50] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_50),zero_zero_nat)) ).
fof(fact_2205_dvd__0__right,axiom,
! [A_50] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_50),zero_zero_int)) ).
fof(fact_2206_dvd__0__right,axiom,
! [A_50] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_50),zero_zero_rat)) ).
fof(fact_2207_dvd__0__right,axiom,
! [A_50] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_50),zero_z891286103de_int)) ).
fof(fact_2208_dvd__0__right,axiom,
! [A_50] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_50),zero_zero_complex)) ).
fof(fact_2209_dvd__0__right,axiom,
! [A_50] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,A_50),zero_z126310315umeral)) ).
fof(fact_2210_dvd__0__right,axiom,
! [A_50] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_50),zero_zero_real)) ).
fof(fact_2211__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_2212__096_091s_____A_094_A2_A_061_As1_A_094_A2_093_A_Imod_A4_A_K_Am_A_L_A1_,axiom,
hBOOL(hAPP_int_bool(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_2213__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_2214__096EX_B_As_O_A0_A_060_061_As_A_G_As_A_060_A4_A_K_Am_A_L_A1_A_G_A_091s,axiom,
? [X_1] :
( 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(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_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(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int)))
& ! [Y_1] :
( is_int(Y_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_int,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_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_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(bit0,hAPP_int_int(bit1,pls))))),m)),one_one_int))) )
=> Y_1 = X_1 ) ) ) ).
fof(fact_2215_dvd__diff,axiom,
! [Z_6,X_13,Y_10] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_13),Y_10))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_13),Z_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_13),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_10),Z_6))) ) ) ).
fof(fact_2216_dvd__diff,axiom,
! [Z_6,X_13,Y_10] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_13),Y_10))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_13),Z_6))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_13),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_10),Z_6))) ) ) ).
fof(fact_2217_dvd__diff,axiom,
! [Z_6,X_13,Y_10] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_13),Y_10))
=> ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_13),Z_6))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_13),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_10),Z_6))) ) ) ).
fof(fact_2218_dvd__diff,axiom,
! [Z_6,X_13,Y_10] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_13),Y_10))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_13),Z_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_13),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_10),Z_6))) ) ) ).
fof(fact_2219_dvd__trans,axiom,
! [C_15,A_49,B_36] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_49),B_36))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_36),C_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_49),C_15)) ) ) ).
fof(fact_2220_dvd__trans,axiom,
! [C_15,A_49,B_36] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_49),B_36))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_36),C_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_49),C_15)) ) ) ).
fof(fact_2221_dvd__refl,axiom,
! [A_48] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_48),A_48)) ).
fof(fact_2222_dvd__refl,axiom,
! [A_48] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_48),A_48)) ).
fof(fact_2223_zdvd__zdiffD,axiom,
! [K_2,M,N] :
( 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)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).
fof(fact_2224_diff__eq__diff__eq,axiom,
! [A_1,B_1,C,D_1] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D_1)
=> ( A_1 = B_1
<=> C = D_1 ) ) ).
fof(fact_2225_diff__eq__diff__eq,axiom,
! [A_1,B_1,C,D_1] :
( hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_1),B_1) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C),D_1)
=> ( A_1 = B_1
<=> C = D_1 ) ) ).
fof(fact_2226_diff__eq__diff__eq,axiom,
! [A_1,B_1,C,D_1] :
( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D_1)
=> ( A_1 = B_1
<=> C = D_1 ) ) ).
fof(fact_2227_diff__eq__diff__eq,axiom,
! [A_1,B_1,C,D_1] :
( ( is_int(A_1)
& is_int(B_1)
& is_int(C)
& is_int(D_1) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D_1)
=> ( A_1 = B_1
<=> C = D_1 ) ) ) ).
fof(fact_2228_number__of__diff,axiom,
! [V_7,W_3] : hAPP_int_rat(number_number_of_rat,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),W_3)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,V_7)),hAPP_int_rat(number_number_of_rat,W_3)) ).
fof(fact_2229_number__of__diff,axiom,
! [V_7,W_3] : hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),W_3)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,V_7)),hAPP_int_complex(number528085621omplex,W_3)) ).
fof(fact_2230_number__of__diff,axiom,
! [V_7,W_3] : hAPP_int_real(number267125858f_real,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),W_3)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,V_7)),hAPP_int_real(number267125858f_real,W_3)) ).
fof(fact_2231_number__of__diff,axiom,
! [V_7,W_3] : hAPP_int_int(number_number_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,V_7),W_3)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_7)),hAPP_int_int(number_number_of_int,W_3)) ).
fof(fact_2232_inv__not__p__minus__1__aux,axiom,
! [A_1,P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),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_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int)),P_5)) ) ).
fof(fact_2233_inf__period_I4_J,axiom,
! [Ta,D_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),D))
=> ! [X_1,K_1] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_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,K_1),D))),Ta))) ) ) ).
fof(fact_2234_inf__period_I4_J,axiom,
! [Ta,D_1,D] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D_1),D))
=> ! [X_1,K_1] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),Ta)))
<=> ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K_1),D))),Ta))) ) ) ).
fof(fact_2235_inf__period_I4_J,axiom,
! [Ta,D_1,D] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D_1),D))
=> ! [X_1,K_1] :
( ~ hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D_1),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Ta)))
<=> ~ hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D_1),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K_1),D))),Ta))) ) ) ).
fof(fact_2236_inf__period_I4_J,axiom,
! [Ta,D_1,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),D))
=> ! [X_1,K_1] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Ta)))
<=> ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K_1),D))),Ta))) ) ) ).
fof(fact_2237_inf__period_I3_J,axiom,
! [Ta,D_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),D))
=> ! [X_1,K_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_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,K_1),D))),Ta))) ) ) ).
fof(fact_2238_inf__period_I3_J,axiom,
! [Ta,D_1,D] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D_1),D))
=> ! [X_1,K_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),Ta)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,D_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_1),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,K_1),D))),Ta))) ) ) ).
fof(fact_2239_inf__period_I3_J,axiom,
! [Ta,D_1,D] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D_1),D))
=> ! [X_1,K_1] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D_1),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),Ta)))
<=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,D_1),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_1),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,K_1),D))),Ta))) ) ) ).
fof(fact_2240_inf__period_I3_J,axiom,
! [Ta,D_1,D] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),D))
=> ! [X_1,K_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),Ta)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,D_1),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,K_1),D))),Ta))) ) ) ).
fof(fact_2241_of__nat__diff,axiom,
! [N_17,M_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_17),M_10))
=> hAPP_nat_rat(semiri151668891at_rat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_10),N_17)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_nat_rat(semiri151668891at_rat,M_10)),hAPP_nat_rat(semiri151668891at_rat,N_17)) ) ).
fof(fact_2242_of__nat__diff,axiom,
! [N_17,M_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_17),M_10))
=> hAPP_nat_complex(semiri2020571505omplex,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_10),N_17)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_nat_complex(semiri2020571505omplex,M_10)),hAPP_nat_complex(semiri2020571505omplex,N_17)) ) ).
fof(fact_2243_of__nat__diff,axiom,
! [N_17,M_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_17),M_10))
=> hAPP_nat_real(semiri132038758t_real,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_10),N_17)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(semiri132038758t_real,M_10)),hAPP_nat_real(semiri132038758t_real,N_17)) ) ).
fof(fact_2244_of__nat__diff,axiom,
! [N_17,M_10] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_17),M_10))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_10),N_17)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M_10)),hAPP_nat_int(semiri1621563631at_int,N_17)) ) ).
fof(fact_2245_dvd__0__left,axiom,
! [A_47] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_47))
=> A_47 = zero_zero_nat ) ).
fof(fact_2246_dvd__0__left,axiom,
! [A_47] :
( is_int(A_47)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,zero_zero_int),A_47))
=> A_47 = zero_zero_int ) ) ).
fof(fact_2247_dvd__0__left,axiom,
! [A_47] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,zero_zero_rat),A_47))
=> A_47 = zero_zero_rat ) ).
fof(fact_2248_dvd__0__left,axiom,
! [A_47] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,zero_z891286103de_int),A_47))
=> A_47 = zero_z891286103de_int ) ).
fof(fact_2249_dvd__0__left,axiom,
! [A_47] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,zero_zero_complex),A_47))
=> A_47 = zero_zero_complex ) ).
fof(fact_2250_dvd__0__left,axiom,
! [A_47] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,zero_z126310315umeral),A_47))
=> A_47 = zero_z126310315umeral ) ).
fof(fact_2251_dvd__0__left,axiom,
! [A_47] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,zero_zero_real),A_47))
=> A_47 = zero_zero_real ) ).
fof(fact_2252_dvd__mult__right,axiom,
! [A_46,B_35,C_14] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_46),B_35)),C_14))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_35),C_14)) ) ).
fof(fact_2253_dvd__mult__right,axiom,
! [A_46,B_35,C_14] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_46),B_35)),C_14))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_35),C_14)) ) ).
fof(fact_2254_dvd__mult__right,axiom,
! [A_46,B_35,C_14] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_46),B_35)),C_14))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_35),C_14)) ) ).
fof(fact_2255_dvd__mult__right,axiom,
! [A_46,B_35,C_14] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_46),B_35)),C_14))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_35),C_14)) ) ).
fof(fact_2256_dvd__mult__right,axiom,
! [A_46,B_35,C_14] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_46),B_35)),C_14))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_35),C_14)) ) ).
fof(fact_2257_dvd__mult__right,axiom,
! [A_46,B_35,C_14] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_46),B_35)),C_14))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_35),C_14)) ) ).
fof(fact_2258_dvd__mult__left,axiom,
! [A_45,B_34,C_13] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_45),B_34)),C_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_45),C_13)) ) ).
fof(fact_2259_dvd__mult__left,axiom,
! [A_45,B_34,C_13] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_45),B_34)),C_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_45),C_13)) ) ).
fof(fact_2260_dvd__mult__left,axiom,
! [A_45,B_34,C_13] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_45),B_34)),C_13))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_45),C_13)) ) ).
fof(fact_2261_dvd__mult__left,axiom,
! [A_45,B_34,C_13] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_45),B_34)),C_13))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_45),C_13)) ) ).
fof(fact_2262_dvd__mult__left,axiom,
! [A_45,B_34,C_13] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_45),B_34)),C_13))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_45),C_13)) ) ).
fof(fact_2263_dvd__mult__left,axiom,
! [A_45,B_34,C_13] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_45),B_34)),C_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_45),C_13)) ) ).
fof(fact_2264_dvdI,axiom,
! [A_44,B_33,K_6] :
( A_44 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_33),K_6)
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_33),A_44)) ) ).
fof(fact_2265_dvdI,axiom,
! [A_44,B_33,K_6] :
( A_44 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_33),K_6)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,B_33),A_44)) ) ).
fof(fact_2266_dvdI,axiom,
! [A_44,B_33,K_6] :
( A_44 = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_33),K_6)
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,B_33),A_44)) ) ).
fof(fact_2267_dvdI,axiom,
! [A_44,B_33,K_6] :
( A_44 = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_33),K_6)
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,B_33),A_44)) ) ).
fof(fact_2268_dvdI,axiom,
! [A_44,B_33,K_6] :
( A_44 = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_33),K_6)
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,B_33),A_44)) ) ).
fof(fact_2269_dvdI,axiom,
! [A_44,B_33,K_6] :
( A_44 = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_33),K_6)
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,B_33),A_44)) ) ).
fof(fact_2270_mult__dvd__mono,axiom,
! [C_12,D_8,A_43,B_32] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_43),B_32))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,C_12),D_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_43),C_12)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_32),D_8))) ) ) ).
fof(fact_2271_mult__dvd__mono,axiom,
! [C_12,D_8,A_43,B_32] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_43),B_32))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,C_12),D_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_43),C_12)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_32),D_8))) ) ) ).
fof(fact_2272_mult__dvd__mono,axiom,
! [C_12,D_8,A_43,B_32] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_43),B_32))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,C_12),D_8))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_43),C_12)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_32),D_8))) ) ) ).
fof(fact_2273_mult__dvd__mono,axiom,
! [C_12,D_8,A_43,B_32] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_43),B_32))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,C_12),D_8))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_43),C_12)),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_32),D_8))) ) ) ).
fof(fact_2274_mult__dvd__mono,axiom,
! [C_12,D_8,A_43,B_32] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_43),B_32))
=> ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,C_12),D_8))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_43),C_12)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_32),D_8))) ) ) ).
fof(fact_2275_mult__dvd__mono,axiom,
! [C_12,D_8,A_43,B_32] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_43),B_32))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,C_12),D_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_43),C_12)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_32),D_8))) ) ) ).
fof(fact_2276_dvd__mult,axiom,
! [B_31,A_42,C_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_42),C_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_42),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_31),C_11))) ) ).
fof(fact_2277_dvd__mult,axiom,
! [B_31,A_42,C_11] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_42),C_11))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_42),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_31),C_11))) ) ).
fof(fact_2278_dvd__mult,axiom,
! [B_31,A_42,C_11] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_42),C_11))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_42),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_31),C_11))) ) ).
fof(fact_2279_dvd__mult,axiom,
! [B_31,A_42,C_11] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_42),C_11))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_42),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_31),C_11))) ) ).
fof(fact_2280_dvd__mult,axiom,
! [B_31,A_42,C_11] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_42),C_11))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_42),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_31),C_11))) ) ).
fof(fact_2281_dvd__mult,axiom,
! [B_31,A_42,C_11] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_42),C_11))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_42),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_31),C_11))) ) ).
fof(fact_2282_dvd__mult2,axiom,
! [C_10,A_41,B_30] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_41),B_30))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_41),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_30),C_10))) ) ).
fof(fact_2283_dvd__mult2,axiom,
! [C_10,A_41,B_30] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_41),B_30))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_41),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_30),C_10))) ) ).
fof(fact_2284_dvd__mult2,axiom,
! [C_10,A_41,B_30] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_41),B_30))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_41),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_30),C_10))) ) ).
fof(fact_2285_dvd__mult2,axiom,
! [C_10,A_41,B_30] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_41),B_30))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_41),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_30),C_10))) ) ).
fof(fact_2286_dvd__mult2,axiom,
! [C_10,A_41,B_30] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_41),B_30))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_41),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_30),C_10))) ) ).
fof(fact_2287_dvd__mult2,axiom,
! [C_10,A_41,B_30] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_41),B_30))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_41),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_30),C_10))) ) ).
fof(fact_2288_dvd__triv__right,axiom,
! [A_40,B_29] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_29),A_40))) ).
fof(fact_2289_dvd__triv__right,axiom,
! [A_40,B_29] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_40),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_29),A_40))) ).
fof(fact_2290_dvd__triv__right,axiom,
! [A_40,B_29] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_40),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_29),A_40))) ).
fof(fact_2291_dvd__triv__right,axiom,
! [A_40,B_29] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_40),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,B_29),A_40))) ).
fof(fact_2292_dvd__triv__right,axiom,
! [A_40,B_29] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_40),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_29),A_40))) ).
fof(fact_2293_dvd__triv__right,axiom,
! [A_40,B_29] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_40),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_29),A_40))) ).
fof(fact_2294_dvd__triv__left,axiom,
! [A_39,B_28] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_39),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_39),B_28))) ).
fof(fact_2295_dvd__triv__left,axiom,
! [A_39,B_28] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_39),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_39),B_28))) ).
fof(fact_2296_dvd__triv__left,axiom,
! [A_39,B_28] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_39),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_39),B_28))) ).
fof(fact_2297_dvd__triv__left,axiom,
! [A_39,B_28] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_39),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,A_39),B_28))) ).
fof(fact_2298_dvd__triv__left,axiom,
! [A_39,B_28] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_39),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_39),B_28))) ).
fof(fact_2299_dvd__triv__left,axiom,
! [A_39,B_28] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_39),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_39),B_28))) ).
fof(fact_2300_dvd__add,axiom,
! [C_9,A_38,B_27] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_38),B_27))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_38),C_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_38),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_27),C_9))) ) ) ).
fof(fact_2301_dvd__add,axiom,
! [C_9,A_38,B_27] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_38),B_27))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_38),C_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_38),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_27),C_9))) ) ) ).
fof(fact_2302_dvd__add,axiom,
! [C_9,A_38,B_27] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_38),B_27))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_38),C_9))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_38),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_27),C_9))) ) ) ).
fof(fact_2303_dvd__add,axiom,
! [C_9,A_38,B_27] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_38),B_27))
=> ( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_38),C_9))
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,A_38),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,B_27),C_9))) ) ) ).
fof(fact_2304_dvd__add,axiom,
! [C_9,A_38,B_27] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_38),B_27))
=> ( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_38),C_9))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_38),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_27),C_9))) ) ) ).
fof(fact_2305_dvd__add,axiom,
! [C_9,A_38,B_27] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_38),B_27))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_38),C_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_38),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_27),C_9))) ) ) ).
fof(fact_2306_one__dvd,axiom,
! [A_37] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_37)) ).
fof(fact_2307_one__dvd,axiom,
! [A_37] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,one_one_int),A_37)) ).
fof(fact_2308_one__dvd,axiom,
! [A_37] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,one_one_rat),A_37)) ).
fof(fact_2309_one__dvd,axiom,
! [A_37] : hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,one_on1684967323de_int),A_37)) ).
fof(fact_2310_one__dvd,axiom,
! [A_37] : hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,one_one_complex),A_37)) ).
fof(fact_2311_one__dvd,axiom,
! [A_37] : hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,one_on1645066479umeral),A_37)) ).
fof(fact_2312_one__dvd,axiom,
! [A_37] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,one_one_real),A_37)) ).
fof(fact_2313_dvd__power__same,axiom,
! [N_16,X_12,Y_9] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_12),Y_9))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_12),N_16)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_9),N_16))) ) ).
fof(fact_2314_dvd__power__same,axiom,
! [N_16,X_12,Y_9] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_12),Y_9))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_12),N_16)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_9),N_16))) ) ).
fof(fact_2315_dvd__power__same,axiom,
! [N_16,X_12,Y_9] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_12),Y_9))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_12),N_16)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_9),N_16))) ) ).
fof(fact_2316_dvd__power__same,axiom,
! [N_16,X_12,Y_9] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_12),Y_9))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_12),N_16)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_9),N_16))) ) ).
fof(fact_2317_right__minus__eq,axiom,
! [A_1,B_1] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1) = zero_zero_rat
<=> A_1 = B_1 ) ).
fof(fact_2318_right__minus__eq,axiom,
! [A_1,B_1] :
( hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_1),B_1) = zero_zero_complex
<=> A_1 = B_1 ) ).
fof(fact_2319_right__minus__eq,axiom,
! [A_1,B_1] :
( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1) = zero_zero_real
<=> A_1 = B_1 ) ).
fof(fact_2320_right__minus__eq,axiom,
! [A_1,B_1] :
( ( is_int(A_1)
& is_int(B_1) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1) = zero_zero_int
<=> A_1 = B_1 ) ) ).
fof(fact_2321_eq__iff__diff__eq__0,axiom,
! [A_1,B_1] :
( A_1 = B_1
<=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1) = zero_zero_rat ) ).
fof(fact_2322_eq__iff__diff__eq__0,axiom,
! [A_1,B_1] :
( A_1 = B_1
<=> hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_1),B_1) = zero_zero_complex ) ).
fof(fact_2323_eq__iff__diff__eq__0,axiom,
! [A_1,B_1] :
( A_1 = B_1
<=> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1) = zero_zero_real ) ).
fof(fact_2324_eq__iff__diff__eq__0,axiom,
! [A_1,B_1] :
( ( is_int(A_1)
& is_int(B_1) )
=> ( A_1 = B_1
<=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1) = zero_zero_int ) ) ).
fof(fact_2325_diff__self,axiom,
! [A_36] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_36),A_36) = zero_zero_rat ).
fof(fact_2326_diff__self,axiom,
! [A_36] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_36),A_36) = zero_zero_complex ).
fof(fact_2327_diff__self,axiom,
! [A_36] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_36),A_36) = zero_zero_real ).
fof(fact_2328_diff__self,axiom,
! [A_36] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_36),A_36) = zero_zero_int ).
fof(fact_2329_diff__0__right,axiom,
! [A_35] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_35),zero_zero_rat) = A_35 ).
fof(fact_2330_diff__0__right,axiom,
! [A_35] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_35),zero_zero_complex) = A_35 ).
fof(fact_2331_diff__0__right,axiom,
! [A_35] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_35),zero_zero_real) = A_35 ).
fof(fact_2332_diff__0__right,axiom,
! [A_35] :
( is_int(A_35)
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_35),zero_zero_int) = A_35 ) ).
fof(fact_2333_diff__eq__diff__less__eq,axiom,
! [A_1,B_1,C,D_1] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D_1)
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),B_1))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,C),D_1)) ) ) ).
fof(fact_2334_diff__eq__diff__less__eq,axiom,
! [A_1,B_1,C,D_1] :
( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D_1)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,C),D_1)) ) ) ).
fof(fact_2335_diff__eq__diff__less__eq,axiom,
! [A_1,B_1,C,D_1] :
( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,C),D_1)) ) ) ).
fof(fact_2336_diff__eq__diff__less,axiom,
! [A_1,B_1,C,D_1] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C),D_1)
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),B_1))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,C),D_1)) ) ) ).
fof(fact_2337_diff__eq__diff__less,axiom,
! [A_1,B_1,C,D_1] :
( hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C),D_1)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,C),D_1)) ) ) ).
fof(fact_2338_diff__eq__diff__less,axiom,
! [A_1,B_1,C,D_1] :
( hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C),D_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,C),D_1)) ) ) ).
fof(fact_2339_add__diff__cancel,axiom,
! [A_34,B_26] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_34),B_26)),B_26) = A_34 ).
fof(fact_2340_add__diff__cancel,axiom,
! [A_34,B_26] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_34),B_26)),B_26) = A_34 ).
fof(fact_2341_add__diff__cancel,axiom,
! [A_34,B_26] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_34),B_26)),B_26) = A_34 ).
fof(fact_2342_add__diff__cancel,axiom,
! [A_34,B_26] :
( is_int(A_34)
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_34),B_26)),B_26) = A_34 ) ).
fof(fact_2343_diff__add__cancel,axiom,
! [A_33,B_25] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_33),B_25)),B_25) = A_33 ).
fof(fact_2344_diff__add__cancel,axiom,
! [A_33,B_25] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_33),B_25)),B_25) = A_33 ).
fof(fact_2345_diff__add__cancel,axiom,
! [A_33,B_25] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_33),B_25)),B_25) = A_33 ).
fof(fact_2346_diff__add__cancel,axiom,
! [A_33,B_25] :
( is_int(A_33)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_33),B_25)),B_25) = A_33 ) ).
fof(fact_2347_add__diff__add,axiom,
! [A_32,C_8,B_24,D_7] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_32),C_8)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,B_24),D_7)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_32),B_24)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,C_8),D_7)) ).
fof(fact_2348_add__diff__add,axiom,
! [A_32,C_8,B_24,D_7] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,A_32),C_8)),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,B_24),D_7)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_32),B_24)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,C_8),D_7)) ).
fof(fact_2349_add__diff__add,axiom,
! [A_32,C_8,B_24,D_7] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_32),C_8)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,B_24),D_7)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_32),B_24)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,C_8),D_7)) ).
fof(fact_2350_add__diff__add,axiom,
! [A_32,C_8,B_24,D_7] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_32),C_8)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,B_24),D_7)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_32),B_24)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,C_8),D_7)) ).
fof(fact_2351_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_2352_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_2353_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_2354_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_2355_dvd__mult__cancel__right,axiom,
! [A_1,C,B_1] :
( 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_1),C)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),C)))
<=> ( C = zero_zero_int
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),B_1)) ) ) ) ).
fof(fact_2356_dvd__mult__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),C)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_1),C)))
<=> ( C = zero_zero_rat
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_1),B_1)) ) ) ).
fof(fact_2357_dvd__mult__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),C)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_1),C)))
<=> ( C = zero_zero_complex
| hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_1),B_1)) ) ) ).
fof(fact_2358_dvd__mult__cancel__right,axiom,
! [A_1,C,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),C)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),C)))
<=> ( C = zero_zero_real
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_1),B_1)) ) ) ).
fof(fact_2359_dvd__mult__cancel__left,axiom,
! [C,A_1,B_1] :
( 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_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)))
<=> ( C = zero_zero_int
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),B_1)) ) ) ) ).
fof(fact_2360_dvd__mult__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),A_1)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C),B_1)))
<=> ( C = zero_zero_rat
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,A_1),B_1)) ) ) ).
fof(fact_2361_dvd__mult__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),A_1)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,C),B_1)))
<=> ( C = zero_zero_complex
| hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,A_1),B_1)) ) ) ).
fof(fact_2362_dvd__mult__cancel__left,axiom,
! [C,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),A_1)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C),B_1)))
<=> ( C = zero_zero_real
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,A_1),B_1)) ) ) ).
fof(fact_2363_power__le__dvd,axiom,
! [M_9,A_31,N_15,B_23] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_31),N_15)),B_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_31),M_9)),B_23)) ) ) ).
fof(fact_2364_power__le__dvd,axiom,
! [M_9,A_31,N_15,B_23] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_31),N_15)),B_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_15))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_31),M_9)),B_23)) ) ) ).
fof(fact_2365_power__le__dvd,axiom,
! [M_9,A_31,N_15,B_23] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_31),N_15)),B_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_15))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_31),M_9)),B_23)) ) ) ).
fof(fact_2366_power__le__dvd,axiom,
! [M_9,A_31,N_15,B_23] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_31),N_15)),B_23))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_9),N_15))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_31),M_9)),B_23)) ) ) ).
fof(fact_2367_dvd__power__le,axiom,
! [N_14,M_8,X_11,Y_8] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_11),Y_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_11),N_14)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y_8),M_8))) ) ) ).
fof(fact_2368_dvd__power__le,axiom,
! [N_14,M_8,X_11,Y_8] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_11),Y_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_8))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_11),N_14)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y_8),M_8))) ) ) ).
fof(fact_2369_dvd__power__le,axiom,
! [N_14,M_8,X_11,Y_8] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_11),Y_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_8))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_11),N_14)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,Y_8),M_8))) ) ) ).
fof(fact_2370_dvd__power__le,axiom,
! [N_14,M_8,X_11,Y_8] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_11),Y_8))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_14),M_8))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_11),N_14)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y_8),M_8))) ) ) ).
fof(fact_2371_le__imp__power__dvd,axiom,
! [A_30,M_7,N_13] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_7),N_13))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_30),M_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_30),N_13))) ) ).
fof(fact_2372_le__imp__power__dvd,axiom,
! [A_30,M_7,N_13] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_7),N_13))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_30),M_7)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_30),N_13))) ) ).
fof(fact_2373_le__imp__power__dvd,axiom,
! [A_30,M_7,N_13] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_7),N_13))
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_30),M_7)),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,A_30),N_13))) ) ).
fof(fact_2374_le__imp__power__dvd,axiom,
! [A_30,M_7,N_13] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_7),N_13))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_30),M_7)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_30),N_13))) ) ).
fof(fact_2375_le__iff__diff__le__0,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_1),B_1))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1)),zero_zero_rat)) ) ).
fof(fact_2376_le__iff__diff__le__0,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1)),zero_zero_real)) ) ).
fof(fact_2377_le__iff__diff__le__0,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1)),zero_zero_int)) ) ).
fof(fact_2378_less__iff__diff__less__0,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,A_1),B_1))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_1),B_1)),zero_zero_rat)) ) ).
fof(fact_2379_less__iff__diff__less__0,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1)),zero_zero_real)) ) ).
fof(fact_2380_less__iff__diff__less__0,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1)),zero_zero_int)) ) ).
fof(fact_2381_eq__add__iff1,axiom,
! [A_1,E_2,C,B_1,D_1] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),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_1),E_2)),D_1)
<=> 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_1),B_1)),E_2)),C) = D_1 ) ).
fof(fact_2382_eq__add__iff1,axiom,
! [A_1,E_2,C,B_1,D_1] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_1),E_2)),D_1)
<=> 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_1),B_1)),E_2)),C) = D_1 ) ).
fof(fact_2383_eq__add__iff1,axiom,
! [A_1,E_2,C,B_1,D_1] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),E_2)),D_1)
<=> 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_1),B_1)),E_2)),C) = D_1 ) ).
fof(fact_2384_eq__add__iff1,axiom,
! [A_1,E_2,C,B_1,D_1] :
( is_int(D_1)
=> ( hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),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_1),E_2)),D_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_1),B_1)),E_2)),C) = D_1 ) ) ).
fof(fact_2385_eq__add__iff2,axiom,
! [A_1,E_2,C,B_1,D_1] :
( hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_1),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_1),E_2)),D_1)
<=> 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_1),A_1)),E_2)),D_1) ) ).
fof(fact_2386_eq__add__iff2,axiom,
! [A_1,E_2,C,B_1,D_1] :
( hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_1),E_2)),C) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_1),E_2)),D_1)
<=> 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_1),A_1)),E_2)),D_1) ) ).
fof(fact_2387_eq__add__iff2,axiom,
! [A_1,E_2,C,B_1,D_1] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_1),E_2)),C) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),E_2)),D_1)
<=> 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_1),A_1)),E_2)),D_1) ) ).
fof(fact_2388_eq__add__iff2,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),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_1),E_2)),D_1)
<=> 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_1),A_1)),E_2)),D_1) ) ) ).
fof(fact_2389_mult__diff__mult,axiom,
! [X_10,Y_7,A_29,B_22] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_10),Y_7)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_29),B_22)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_10),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Y_7),B_22))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_10),A_29)),B_22)) ).
fof(fact_2390_mult__diff__mult,axiom,
! [X_10,Y_7,A_29,B_22] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_10),Y_7)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_29),B_22)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_10),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,Y_7),B_22))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_10),A_29)),B_22)) ).
fof(fact_2391_mult__diff__mult,axiom,
! [X_10,Y_7,A_29,B_22] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_10),Y_7)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_29),B_22)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_10),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Y_7),B_22))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_10),A_29)),B_22)) ).
fof(fact_2392_mult__diff__mult,axiom,
! [X_10,Y_7,A_29,B_22] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_10),Y_7)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_29),B_22)) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_10),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Y_7),B_22))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_10),A_29)),B_22)) ).
fof(fact_2393_right__diff__distrib__number__of,axiom,
! [V_6,B_21,C_7] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_6)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_21),C_7)) = 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_6)),B_21)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_int_rat(number_number_of_rat,V_6)),C_7)) ).
fof(fact_2394_right__diff__distrib__number__of,axiom,
! [V_6,B_21,C_7] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_6)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,B_21),C_7)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_6)),B_21)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_int_complex(number528085621omplex,V_6)),C_7)) ).
fof(fact_2395_right__diff__distrib__number__of,axiom,
! [V_6,B_21,C_7] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_6)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_21),C_7)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_6)),B_21)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,V_6)),C_7)) ).
fof(fact_2396_right__diff__distrib__number__of,axiom,
! [V_6,B_21,C_7] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_6)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_21),C_7)) = 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_6)),B_21)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(number_number_of_int,V_6)),C_7)) ).
fof(fact_2397_left__diff__distrib__number__of,axiom,
! [A_28,B_20,V_5] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_28),B_20)),hAPP_int_rat(number_number_of_rat,V_5)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_28),hAPP_int_rat(number_number_of_rat,V_5))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,B_20),hAPP_int_rat(number_number_of_rat,V_5))) ).
fof(fact_2398_left__diff__distrib__number__of,axiom,
! [A_28,B_20,V_5] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,A_28),B_20)),hAPP_int_complex(number528085621omplex,V_5)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,A_28),hAPP_int_complex(number528085621omplex,V_5))),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,B_20),hAPP_int_complex(number528085621omplex,V_5))) ).
fof(fact_2399_left__diff__distrib__number__of,axiom,
! [A_28,B_20,V_5] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_28),B_20)),hAPP_int_real(number267125858f_real,V_5)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_28),hAPP_int_real(number267125858f_real,V_5))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_20),hAPP_int_real(number267125858f_real,V_5))) ).
fof(fact_2400_left__diff__distrib__number__of,axiom,
! [A_28,B_20,V_5] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_28),B_20)),hAPP_int_int(number_number_of_int,V_5)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_28),hAPP_int_int(number_number_of_int,V_5))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_20),hAPP_int_int(number_number_of_int,V_5))) ).
fof(fact_2401_zdvd__not__zless,axiom,
! [N,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))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M)) ) ) ).
fof(fact_2402_zdvd__mult__cancel,axiom,
! [K_2,M,N] :
( 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)))
=> ( K_2 != zero_zero_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N)) ) ) ) ).
fof(fact_2403_zdvd__mono,axiom,
! [Ma,Ta,K] :
( is_int(K)
=> ( K != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),Ta))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),Ma)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),Ta))) ) ) ) ).
fof(fact_2404_zdvd__antisym__nonneg,axiom,
! [N,M] :
( ( is_int(N)
& 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))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))
=> M = N ) ) ) ) ) ).
fof(fact_2405_zcong__zpower__zmult,axiom,
! [Z_1,X,Y,P_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),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),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Y),Z_1))),one_one_int),P_3)) ) ).
fof(fact_2406_zdvd__reduce,axiom,
! [K,Na,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Na),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),Ma))))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Na)) ) ).
fof(fact_2407_zdvd__period,axiom,
! [C,Xa,Ta,A_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),D_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),Ta)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),D_1))),Ta))) ) ) ).
fof(fact_2408_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_2409_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_2410_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_2411_less__bin__lemma,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),L))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K),L)),zero_zero_int)) ) ).
fof(fact_2412_zprime__zdvd__power,axiom,
! [A_3,N,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)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),A_3)) ) ) ).
fof(fact_2413_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_2414_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X_1] : hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,L),X_1)))
<=> ? [X_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,L),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),zero_zero_nat)))
& hBOOL(hAPP_nat_bool(P_1,X_1)) ) ) ).
fof(fact_2415_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,L),X_1))) )
<=> ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,L),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),zero_zero_int)))
& hBOOL(hAPP_int_bool(P_1,X_1)) ) ) ).
fof(fact_2416_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X_1] : hBOOL(hAPP_rat_bool(P_1,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,L),X_1)))
<=> ? [X_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,L),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X_1),zero_zero_rat)))
& hBOOL(hAPP_rat_bool(P_1,X_1)) ) ) ).
fof(fact_2417_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X_1] : hBOOL(hAPP_Q2096512830t_bool(P_1,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,L),X_1)))
<=> ? [X_1] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,L),hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,X_1),zero_z891286103de_int)))
& hBOOL(hAPP_Q2096512830t_bool(P_1,X_1)) ) ) ).
fof(fact_2418_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X_1] : hBOOL(hAPP_complex_bool(P_1,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,L),X_1)))
<=> ? [X_1] :
( hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,L),hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_1),zero_zero_complex)))
& hBOOL(hAPP_complex_bool(P_1,X_1)) ) ) ).
fof(fact_2419_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X_1] : hBOOL(hAPP_real_bool(P_1,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,L),X_1)))
<=> ? [X_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,L),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_1),zero_zero_real)))
& hBOOL(hAPP_real_bool(P_1,X_1)) ) ) ).
fof(fact_2420_unity__coeff__ex,axiom,
! [P_1,L] :
( ? [X_1] : hBOOL(hAPP_C24501650l_bool(P_1,hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,L),X_1)))
<=> ? [X_1] :
( hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,L),hAPP_C498520661umeral(hAPP_C1594335432umeral(plus_p1627245867umeral,X_1),zero_z126310315umeral)))
& hBOOL(hAPP_C24501650l_bool(P_1,X_1)) ) ) ).
fof(fact_2421_le__add__iff2,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),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_1),E_2)),D_1)))
<=> 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_1),A_1)),E_2)),D_1))) ) ).
fof(fact_2422_le__add__iff2,axiom,
! [A_1,E_2,C,B_1,D_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,A_1),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),E_2)),D_1)))
<=> 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_1),A_1)),E_2)),D_1))) ) ).
fof(fact_2423_le__add__iff2,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),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_1),E_2)),D_1)))
<=> 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_1),A_1)),E_2)),D_1))) ) ).
fof(fact_2424_le__add__iff1,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),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_1),E_2)),D_1)))
<=> 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_1),B_1)),E_2)),C)),D_1)) ) ).
fof(fact_2425_le__add__iff1,axiom,
! [A_1,E_2,C,B_1,D_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,A_1),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),E_2)),D_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_real_real(hAPP_r1250527377l_real(minus_minus_real,A_1),B_1)),E_2)),C)),D_1)) ) ).
fof(fact_2426_le__add__iff1,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),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_1),E_2)),D_1)))
<=> 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_1),B_1)),E_2)),C)),D_1)) ) ).
fof(fact_2427_less__add__iff2,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),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_1),E_2)),D_1)))
<=> 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_1),A_1)),E_2)),D_1))) ) ).
fof(fact_2428_less__add__iff2,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),E_2)),D_1)))
<=> 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_1),A_1)),E_2)),D_1))) ) ).
fof(fact_2429_less__add__iff2,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),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_1),E_2)),D_1)))
<=> 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_1),A_1)),E_2)),D_1))) ) ).
fof(fact_2430_less__add__iff1,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),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_1),E_2)),D_1)))
<=> 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_1),B_1)),E_2)),C)),D_1)) ) ).
fof(fact_2431_less__add__iff1,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),E_2)),C)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B_1),E_2)),D_1)))
<=> 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_1),B_1)),E_2)),C)),D_1)) ) ).
fof(fact_2432_less__add__iff1,axiom,
! [A_1,E_2,C,B_1,D_1] :
( 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_1),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_1),E_2)),D_1)))
<=> 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_1),B_1)),E_2)),C)),D_1)) ) ).
fof(fact_2433_real__squared__diff__one__factored,axiom,
! [X_9] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X_9),X_9)),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_9),one_one_rat)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_9),one_one_rat)) ).
fof(fact_2434_real__squared__diff__one__factored,axiom,
! [X_9] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X_9),X_9)),one_one_complex) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X_9),one_one_complex)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_9),one_one_complex)) ).
fof(fact_2435_real__squared__diff__one__factored,axiom,
! [X_9] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X_9),X_9)),one_one_real) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X_9),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_9),one_one_real)) ).
fof(fact_2436_real__squared__diff__one__factored,axiom,
! [X_9] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X_9),X_9)),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_9),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_9),one_one_int)) ).
fof(fact_2437_add__number__of__diff1,axiom,
! [V_4,W_2,C_6] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(number_number_of_rat,V_4)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_int_rat(number_number_of_rat,W_2)),C_6)) = 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_4),W_2))),C_6) ).
fof(fact_2438_add__number__of__diff1,axiom,
! [V_4,W_2,C_6] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_int_complex(number528085621omplex,V_4)),hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,W_2)),C_6)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_int_complex(number528085621omplex,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,V_4),W_2))),C_6) ).
fof(fact_2439_add__number__of__diff1,axiom,
! [V_4,W_2,C_6] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(number267125858f_real,V_4)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(number267125858f_real,W_2)),C_6)) = 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_4),W_2))),C_6) ).
fof(fact_2440_add__number__of__diff1,axiom,
! [V_4,W_2,C_6] : hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(number_number_of_int,V_4)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,W_2)),C_6)) = 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_4),W_2))),C_6) ).
fof(fact_2441_zdvd__imp__le,axiom,
! [Z_1,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Z_1),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Z_1),N)) ) ) ).
fof(fact_2442_zle__diff1__eq,axiom,
! [Wa,Z_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),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,Wa),Z_2)) ) ).
fof(fact_2443_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_2444_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_2445_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_2446_zprime__power__zdvd__cancel__left,axiom,
! [N,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)),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)),B_2)) ) ) ) ).
fof(fact_2447_zprime__power__zdvd__cancel__right,axiom,
! [N,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)),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)),A_3)) ) ) ) ).
fof(fact_2448_dvd__power,axiom,
! [X_8,N_12] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12))
| X_8 = one_one_nat )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X_8),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_8),N_12))) ) ).
fof(fact_2449_dvd__power,axiom,
! [X_8,N_12] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12))
| X_8 = one_one_int )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X_8),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_8),N_12))) ) ).
fof(fact_2450_dvd__power,axiom,
! [X_8,N_12] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12))
| X_8 = one_one_rat )
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,X_8),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_8),N_12))) ) ).
fof(fact_2451_dvd__power,axiom,
! [X_8,N_12] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12))
| X_8 = one_on1684967323de_int )
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(dvd_dv1760642554de_int,X_8),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_8),N_12))) ) ).
fof(fact_2452_dvd__power,axiom,
! [X_8,N_12] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12))
| X_8 = one_one_complex )
=> hBOOL(hAPP_complex_bool(hAPP_c1403687985x_bool(dvd_dvd_complex,X_8),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_8),N_12))) ) ).
fof(fact_2453_dvd__power,axiom,
! [X_8,N_12] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12))
| X_8 = one_on1645066479umeral )
=> hBOOL(hAPP_C24501650l_bool(hAPP_C10902773l_bool(dvd_dv174992974umeral,X_8),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,X_8),N_12))) ) ).
fof(fact_2454_dvd__power,axiom,
! [X_8,N_12] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_12))
| X_8 = one_one_real )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,X_8),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_8),N_12))) ) ).
fof(fact_2455_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_2456_power2__diff,axiom,
! [X_7,Y_6] : hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X_7),Y_6)),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_7),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_6),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_7)),Y_6)) ).
fof(fact_2457_power2__diff,axiom,
! [X_7,Y_6] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X_7),Y_6)),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_7),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_6),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_7)),Y_6)) ).
fof(fact_2458_power2__diff,axiom,
! [X_7,Y_6] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X_7),Y_6)),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_7),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_6),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_7)),Y_6)) ).
fof(fact_2459_power2__diff,axiom,
! [X_7,Y_6] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_7),Y_6)),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_7),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_6),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_7)),Y_6)) ).
fof(fact_2460_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_2461_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_2462_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_2463_Euler_Oaux____1,axiom,
! [Y,X,P_3] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X),P_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y)) ) ) ).
fof(fact_2464_zcong__neg__1__impl__ne__1,axiom,
! [X,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),hAPP_int_int(number_number_of_int,min)),P_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),one_one_int),P_3)) ) ) ).
fof(fact_2465_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_2466_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_2467_QuadRes__def,axiom,
! [Ma,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,Ma),Xa))
<=> ? [Y_1] :
( is_int(Y_1)
& 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))))),Xa),Ma)) ) ) ).
fof(fact_2468_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_2469_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_2470_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_2471_tn,axiom,
tn = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,t)),one_one_nat) ).
fof(fact_2472_dvd__diff__nat,axiom,
! [N,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))
=> 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))) ) ) ).
fof(fact_2473_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_2474_dvd__diffD,axiom,
! [K_2,M,N] :
( 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)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) ) ) ) ).
fof(fact_2475_dvd__diffD1,axiom,
! [K_2,M,N] :
( 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)))
=> ( 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),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)) ) ) ) ).
fof(fact_2476_diffs0__imp__equal,axiom,
! [M,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat
=> M = N ) ) ).
fof(fact_2477_diff__self__eq__0,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).
fof(fact_2478_minus__nat_Odiff__0,axiom,
! [M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).
fof(fact_2479_diff__0__eq__0,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).
fof(fact_2480_less__imp__diff__less,axiom,
! [N,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)),K_2)) ) ).
fof(fact_2481_diff__less__mono2,axiom,
! [L_2,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( 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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M))) ) ) ).
fof(fact_2482_diff__add__inverse2,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).
fof(fact_2483_diff__add__inverse,axiom,
! [N,M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).
fof(fact_2484_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_2485_Nat_Odiff__cancel,axiom,
! [K_2,M,N] : 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)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).
fof(fact_2486_diff__cancel2,axiom,
! [M,K_2,N] : 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),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).
fof(fact_2487_le__diff__iff,axiom,
! [Na,K,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ) ) ).
fof(fact_2488_Nat_Odiff__diff__eq,axiom,
! [N,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))
=> 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),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).
fof(fact_2489_eq__diff__iff,axiom,
! [Na,K,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K)
<=> Ma = Na ) ) ) ).
fof(fact_2490_diff__diff__cancel,axiom,
! [I,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ).
fof(fact_2491_diff__le__mono,axiom,
! [L_2,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,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),L_2))) ) ).
fof(fact_2492_diff__le__mono2,axiom,
! [L_2,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L_2),M))) ) ).
fof(fact_2493_Nat_Odiff__le__self,axiom,
! [M,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).
fof(fact_2494_diff__mult__distrib,axiom,
! [M,N,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)),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),K_2)) ).
fof(fact_2495_diff__mult__distrib2,axiom,
! [K_2,M,N] : 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)) = 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)) ).
fof(fact_2496_dvd__reduce,axiom,
! [K,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Na),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),Na)) ) ).
fof(fact_2497_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_2498_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_2499_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_2500_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_2501_divides__exp,axiom,
! [N,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),N))) ) ).
fof(fact_2502_real__le__eq__diff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Xa),Ya)),zero_zero_real)) ) ).
fof(fact_2503_zero__less__diff,axiom,
! [Na,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,Na),Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_2504_diff__less,axiom,
! [M,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,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)),M)) ) ) ).
fof(fact_2505_diff__add__0,axiom,
! [N,M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).
fof(fact_2506_nat__dvd__not__less,axiom,
! [N,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))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M)) ) ) ).
fof(fact_2507_diff__is__0__eq,axiom,
! [Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),Na) = zero_zero_nat
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_2508_diff__is__0__eq_H,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ).
fof(fact_2509_less__diff__conv,axiom,
! [I_1,J,K] :
( 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)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K)),J)) ) ).
fof(fact_2510_add__diff__inverse,axiom,
! [M,N] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ).
fof(fact_2511_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_2512_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_2513_less__diff__iff,axiom,
! [Na,K,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),K)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ) ) ).
fof(fact_2514_nat__mult__dvd__cancel__disj,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
<=> ( K = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) ) ) ).
fof(fact_2515_nat__mult__dvd__cancel__disj_H,axiom,
! [Ma,K,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),K)))
<=> ( K = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) ) ) ).
fof(fact_2516_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_2517_le__diff__conv,axiom,
! [J,K,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)),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))) ) ).
fof(fact_2518_le__add__diff,axiom,
! [M,K_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N))
=> 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),M)),K_2))) ) ).
fof(fact_2519_le__add__diff__inverse,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ).
fof(fact_2520_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_2521_le__diff__conv2,axiom,
! [I_1,K,J] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)))
<=> 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)),J)) ) ) ).
fof(fact_2522_le__add__diff__inverse2,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),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)),N) = M ) ).
fof(fact_2523_le__imp__diff__is__add,axiom,
! [K,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
<=> J = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),I_1) ) ) ).
fof(fact_2524_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_2525_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_2526_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_2527_divides__rev,axiom,
! [A_3,N,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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N)))
=> ( N != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),B_2)) ) ) ).
fof(fact_2528_divides__exp2,axiom,
! [X,Y,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,X),N)),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
fof(fact_2529_zdvd__int,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya))) ) ).
fof(fact_2530_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) ) ).
fof(fact_2531_nat__diff__split__asm,axiom,
! [P_1,A_1,B_1] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),B_1)))
<=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1))
& ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
| ? [D_2] :
( A_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),D_2)
& ~ hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).
fof(fact_2532_nat__diff__split,axiom,
! [P_1,A_1,B_1] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_1),B_1)))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
& ! [D_2] :
( A_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_1),D_2)
=> hBOOL(hAPP_nat_bool(P_1,D_2)) ) ) ) ).
fof(fact_2533_dvd__imp__le,axiom,
! [K_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N)) ) ) ).
fof(fact_2534_nat__mult__dvd__cancel1,axiom,
! [Ma,Na,K] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ma),Na)) ) ) ).
fof(fact_2535_dvd__mult__cancel,axiom,
! [K_2,M,N] :
( 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)))
=> ( 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)) ) ) ).
fof(fact_2536_nat__eq__add__iff2,axiom,
! [U,Ma,Na,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)),Na)
<=> 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)),Na) ) ) ).
fof(fact_2537_nat__diff__add__eq2,axiom,
! [U_1,M,N,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)) = 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)) ) ).
fof(fact_2538_nat__le__add__iff2,axiom,
! [U,Ma,Na,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)),Na)))
<=> 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)),Na))) ) ) ).
fof(fact_2539_nat__eq__add__iff1,axiom,
! [U,Ma,Na,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)),Na)
<=> 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) = Na ) ) ).
fof(fact_2540_nat__diff__add__eq1,axiom,
! [U_1,M,N,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)) = 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) ) ).
fof(fact_2541_nat__le__add__iff1,axiom,
! [U,Ma,Na,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)),Na)))
<=> 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)),Na)) ) ) ).
fof(fact_2542_zdiff__int,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,M)),hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ).
fof(fact_2543_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_2544_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_2545_zcong__refl,axiom,
! [K_2,M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(K_2),K_2),M)) ).
fof(fact_2546_zcong__sym,axiom,
! [A_1,B_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(B_1),A_1),Ma)) ) ).
fof(fact_2547_divides__div__not,axiom,
! [X,Q_1,N,R_2] :
( X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_1),N)),R_2)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),R_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,R_2),N))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),X)) ) ) ) ).
fof(fact_2548_mult__eq__if,axiom,
! [N,M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
& ( M != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),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)) ) ) ).
fof(fact_2549_nat__less__add__iff1,axiom,
! [U,Ma,Na,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)),Na)))
<=> 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)),Na)) ) ) ).
fof(fact_2550_nat__less__add__iff2,axiom,
! [U,Ma,Na,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)),Na)))
<=> 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)),Na))) ) ) ).
fof(fact_2551_dvd__mult__cancel1,axiom,
! [Na,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),Na)),Ma))
<=> Na = one_one_nat ) ) ).
fof(fact_2552_dvd__mult__cancel2,axiom,
! [Na,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,Na),Ma)),Ma))
<=> Na = one_one_nat ) ) ).
fof(fact_2553_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_2554_power__dvd__imp__le,axiom,
! [I,M,N] :
( 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)))
=> ( 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)) ) ) ).
fof(fact_2555_realpow__minus__mult,axiom,
! [X_6,N_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11))
=> hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X_6),N_11) ) ).
fof(fact_2556_realpow__minus__mult,axiom,
! [X_6,N_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11))
=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_6),N_11) ) ).
fof(fact_2557_realpow__minus__mult,axiom,
! [X_6,N_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X_6),N_11) ) ).
fof(fact_2558_realpow__minus__mult,axiom,
! [X_6,N_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X_6),N_11) ) ).
fof(fact_2559_realpow__minus__mult,axiom,
! [X_6,N_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11))
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,X_6),N_11) ) ).
fof(fact_2560_realpow__minus__mult,axiom,
! [X_6,N_11] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_11))
=> hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_11),one_one_nat))),X_6) = hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,X_6),N_11) ) ).
fof(fact_2561_zdiff__int__split,axiom,
! [P_1,Xa,Ya] :
( hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Xa),Ya))))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ya),Xa))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_nat_int(semiri1621563631at_int,Xa)),hAPP_nat_int(semiri1621563631at_int,Ya)))) )
& ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
=> hBOOL(hAPP_int_bool(P_1,zero_zero_int)) ) ) ) ).
fof(fact_2562_order__le__neq__implies__less,axiom,
! [X_5,Y_5] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,X_5),Y_5))
=> ( X_5 != Y_5
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,X_5),Y_5)) ) ) ).
fof(fact_2563_order__le__neq__implies__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,Xa),Ya))
=> ( Xa != Ya
=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,Xa),Ya)) ) ) ).
fof(fact_2564_order__le__neq__implies__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,Xa),Ya))
=> ( Xa != Ya
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,Xa),Ya)) ) ) ).
fof(fact_2565_order__le__neq__implies__less,axiom,
! [X_5,Y_5] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,X_5),Y_5))
=> ( X_5 != Y_5
=> hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,X_5),Y_5)) ) ) ).
fof(fact_2566_order__le__neq__implies__less,axiom,
! [X_5,Y_5] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_5),Y_5))
=> ( X_5 != Y_5
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_5),Y_5)) ) ) ).
fof(fact_2567_order__le__neq__implies__less,axiom,
! [X_5,Y_5] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_5),Y_5))
=> ( X_5 != Y_5
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_5),Y_5)) ) ) ).
fof(fact_2568_order__le__neq__implies__less,axiom,
! [X_5,Y_5] :
( ( is_int(X_5)
& is_int(Y_5) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_5),Y_5))
=> ( X_5 != Y_5
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_5),Y_5)) ) ) ) ).
fof(fact_2569_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_2570_divides__cases,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))
=> ( M = zero_zero_nat
| M = N
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,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)),M)) ) ) ).
fof(fact_2571_diff__square,axiom,
! [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),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),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),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y)) ).
fof(fact_2572_zcong__id,axiom,
! [M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(M),zero_zero_int),M)) ).
fof(fact_2573_IntPrimes_Ozcong__zero,axiom,
! [A_1,B_1] :
( ( is_int(A_1)
& is_int(B_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),zero_zero_int))
<=> A_1 = B_1 ) ) ).
fof(fact_2574_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_2575_zcong__1,axiom,
! [A_3,B_2] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),B_2),one_one_int)) ).
fof(fact_2576_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_2577_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_2578_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_2579_zcong__zmult__prop2,axiom,
! [C,D_1,A_1,B_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_1),A_1)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_1),B_1)),Ma)) ) ) ).
fof(fact_2580_zcong__zmult__prop1,axiom,
! [C,D_1,A_1,B_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),D_1)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(C),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1)),Ma)) ) ) ).
fof(fact_2581_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_2582_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_2583_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_2584_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_2585_zcong__zpower,axiom,
! [Z_1,X,Y,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Z_1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),Z_1)),M)) ) ).
fof(fact_2586_zcong__less__eq,axiom,
! [M,Y,X] :
( ( is_int(Y)
& is_int(X) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),M))
=> X = Y ) ) ) ) ) ) ) ).
fof(fact_2587_zcong__not__zero,axiom,
! [M,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),M))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),M)) ) ) ).
fof(fact_2588_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_2589_xzgcda__linear__aux1,axiom,
! [A_3,R_2,B_2,M,C_1,D_3,N] : 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_2),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_2),D_3))),N)) = 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))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_2),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)))) ).
fof(fact_2590_zdvd__bounds,axiom,
! [N,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),zero_zero_int))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),M)) ) ) ).
fof(fact_2591_zcong__iff__lin,axiom,
! [A_1,B_1,Ma] :
( is_int(B_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma))
<=> ? [K_1] :
( is_int(K_1)
& B_1 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),K_1)) ) ) ) ).
fof(fact_2592_zcong__eq__zdvd__prop,axiom,
! [Xa,P_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),P_5))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_5),Xa)) ) ).
fof(fact_2593_zcong__zero__equiv__div,axiom,
! [A_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),zero_zero_int),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),A_1)) ) ).
fof(fact_2594_zcong__def,axiom,
! [A_1,B_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),B_1))) ) ).
fof(fact_2595_zprime__zdvd__zmult__better,axiom,
! [M,N,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)))
=> ( 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)) ) ) ) ).
fof(fact_2596_Euler_Oaux1,axiom,
! [A_3,X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),A_3))
=> ( X != 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),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_3),one_one_int))) ) ) ) ) ).
fof(fact_2597_Int2_Ozcong__zero,axiom,
! [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(X),zero_zero_int),M))
=> X = zero_zero_int ) ) ) ) ).
fof(fact_2598_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_2599_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_2600_zpower__zdvd__prop1,axiom,
! [P_3,Y,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Y),N))) ) ) ).
fof(fact_2601_neg__one__power,axiom,
! [N] :
( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N) = one_one_int
| hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),N) = hAPP_int_int(number_number_of_int,min) ) ).
fof(fact_2602_zcong__zmult__prop3,axiom,
! [Y,X,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y),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),Y)),zero_zero_int),P_3)) ) ) ) ).
fof(fact_2603_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_2604_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_2605_zprime__zdvd__zmult,axiom,
! [N,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)))
=> ( 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)) ) ) ) ) ).
fof(fact_2606_zpower__zdvd__prop2,axiom,
! [Y,N,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),N)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),Y)) ) ) ) ).
fof(fact_2607_realpow__num__eq__if,axiom,
! [M_6,N_10] :
( ( N_10 = zero_zero_nat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_6),N_10) = one_one_rat )
& ( N_10 != zero_zero_nat
=> hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_6),N_10) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,M_6),hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) ) ) ).
fof(fact_2608_realpow__num__eq__if,axiom,
! [M_6,N_10] :
( ( N_10 = zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_6),N_10) = one_on1684967323de_int )
& ( N_10 != zero_zero_nat
=> hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_6),N_10) = hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,M_6),hAPP_n522471361de_int(hAPP_Q125967240de_int(power_881366806de_int,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) ) ) ).
fof(fact_2609_realpow__num__eq__if,axiom,
! [M_6,N_10] :
( ( N_10 = zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_6),N_10) = one_one_complex )
& ( N_10 != zero_zero_nat
=> hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_6),N_10) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,M_6),hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) ) ) ).
fof(fact_2610_realpow__num__eq__if,axiom,
! [M_6,N_10] :
( ( N_10 = zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_6),N_10) = one_on1645066479umeral )
& ( N_10 != zero_zero_nat
=> hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_6),N_10) = hAPP_C498520661umeral(hAPP_C1594335432umeral(times_1655362735umeral,M_6),hAPP_n1108039445umeral(hAPP_C471253896umeral(power_2100829034umeral,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) ) ) ).
fof(fact_2611_realpow__num__eq__if,axiom,
! [M_6,N_10] :
( ( N_10 = zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_6),N_10) = one_one_real )
& ( N_10 != zero_zero_nat
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_6),N_10) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,M_6),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) ) ) ).
fof(fact_2612_realpow__num__eq__if,axiom,
! [M_6,N_10] :
( ( N_10 = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_10) = one_one_nat )
& ( N_10 != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),N_10) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) ) ) ).
fof(fact_2613_realpow__num__eq__if,axiom,
! [M_6,N_10] :
( ( N_10 = zero_zero_nat
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_6),N_10) = one_one_int )
& ( N_10 != zero_zero_nat
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_6),N_10) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M_6),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_10),one_one_nat))) ) ) ).
fof(fact_2614__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An,axiom,
~ ! [R_1,V_3] :
( ( is_int(R_1)
& is_int(V_3) )
=> ~ ( V_3 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,x),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,R_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))))
& 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,V_3))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))) ) ) ).
fof(fact_2615__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An,axiom,
~ ! [S_2,W_1] :
( ( is_int(S_2)
& is_int(W_1) )
=> ~ ( W_1 = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,y),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,S_2),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))))
& 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,W_1))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))) ) ) ).
fof(fact_2616_decr__mult__lemma,axiom,
! [K,P_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D_1))) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),K))
=> ! [X_1] :
( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),D_1)))) ) ) ) ) ).
fof(fact_2617_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_2618_zcong__zless__unique,axiom,
! [A_3,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),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(A_3),X_1),M))
& ! [Y_1] :
( is_int(Y_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_int,Y_1),M))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_3),Y_1),M)) )
=> Y_1 = X_1 ) ) ) ) ).
fof(fact_2619__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_2620_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_2621_dvd_Oorder__refl,axiom,
! [X] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) ).
fof(fact_2622__096_B_Bthesis_O_A_I_B_Btn_O_A_091_124_Atn_A_061_Anat_At_A_N_A1_059_A0,axiom,
hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,t)),one_one_nat))) ).
fof(fact_2623_dvd_Oless__asym,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
fof(fact_2624_dvd_Oless__trans,axiom,
! [Z_1,X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X)) ) ) ) ).
fof(fact_2625_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_2626_dvd_Oless__le__trans,axiom,
! [Z_1,X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X)) ) ) ) ).
fof(fact_2627_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_2628_dvd_Oless__imp__triv,axiom,
! [P_1,Xa,Ya] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya)) )
=> hBOOL(P_1) ) ) ).
fof(fact_2629_dvd_Oless__imp__not__eq2,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> Y != X ) ).
fof(fact_2630_dvd_Oless__imp__not__eq,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> X != Y ) ).
fof(fact_2631_dvd_Oless__imp__not__less,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
fof(fact_2632_dvd_Oless__imp__le,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ).
fof(fact_2633_dvd_Oless__not__sym,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ) ).
fof(fact_2634_dvd_Oless__imp__neq,axiom,
! [X,Y] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
=> X != Y ) ).
fof(fact_2635_dvd_Ole__less__trans,axiom,
! [Z_1,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),Y)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Z_1),X)) ) ) ) ).
fof(fact_2636_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_2637_dvd_Oorder__trans,axiom,
! [Z_1,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1)) ) ) ).
fof(fact_2638_dvd_Oantisym,axiom,
! [X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
=> X = Y ) ) ).
fof(fact_2639_dvd__antisym,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),M))
=> M = N ) ) ).
fof(fact_2640_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_2641_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_2642_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_2643_dvd_Ole__imp__less__or__eq,axiom,
! [X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X)) )
| X = Y ) ) ).
fof(fact_2644_dvd_Oantisym__conv,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
<=> Xa = Ya ) ) ).
fof(fact_2645_dvd_Oeq__refl,axiom,
! [X,Y] :
( X = Y
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y)) ) ).
fof(fact_2646_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_2647_dvd_Oless__le__not__le,axiom,
! [Xa,Ya] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) ) ) ).
fof(fact_2648_dvd_Oless__le,axiom,
! [Xa,Ya] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& Xa != Ya ) ) ).
fof(fact_2649_dvd_Ole__less,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
| Xa = Ya ) ) ).
fof(fact_2650_dvd_Oeq__iff,axiom,
! [Xa,Ya] :
( Xa = Ya
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) ) ) ).
fof(fact_2651_dvd_Oless__irrefl,axiom,
! [X] :
~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),X)) ) ).
fof(fact_2652_divides__antisym,axiom,
! [Xa,Ya] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Ya),Xa)) )
<=> Xa = Ya ) ).
fof(fact_2653_nat__if__cong,axiom,
! [Xa,Ya,P_1] :
( ( hBOOL(P_1)
=> hAPP_int_nat(nat,Xa) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),Xa),Ya)) )
& ( ~ hBOOL(P_1)
=> hAPP_int_nat(nat,Ya) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,P_1),Xa),Ya)) ) ) ).
fof(fact_2654_abs__idempotent,axiom,
! [A_27] : hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(abs_abs_rat,A_27)) = hAPP_rat_rat(abs_abs_rat,A_27) ).
fof(fact_2655_abs__idempotent,axiom,
! [A_27] : hAPP_real_real(abs_abs_real,hAPP_real_real(abs_abs_real,A_27)) = hAPP_real_real(abs_abs_real,A_27) ).
fof(fact_2656_abs__idempotent,axiom,
! [A_27] : hAPP_int_int(abs_abs_int,hAPP_int_int(abs_abs_int,A_27)) = hAPP_int_int(abs_abs_int,A_27) ).
fof(fact_2657_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_2658_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_2659_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_2660_abs__eq__0,axiom,
! [A_1] :
( hAPP_rat_rat(abs_abs_rat,A_1) = zero_zero_rat
<=> A_1 = zero_zero_rat ) ).
fof(fact_2661_abs__eq__0,axiom,
! [A_1] :
( hAPP_real_real(abs_abs_real,A_1) = zero_zero_real
<=> A_1 = zero_zero_real ) ).
fof(fact_2662_abs__eq__0,axiom,
! [A_1] :
( is_int(A_1)
=> ( hAPP_int_int(abs_abs_int,A_1) = zero_zero_int
<=> A_1 = zero_zero_int ) ) ).
fof(fact_2663_abs__zero,axiom,
hAPP_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).
fof(fact_2664_abs__zero,axiom,
hAPP_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).
fof(fact_2665_abs__zero,axiom,
hAPP_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).
fof(fact_2666_abs__ge__self,axiom,
! [A_26] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_26),hAPP_rat_rat(abs_abs_rat,A_26))) ).
fof(fact_2667_abs__ge__self,axiom,
! [A_26] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_26),hAPP_real_real(abs_abs_real,A_26))) ).
fof(fact_2668_abs__ge__self,axiom,
! [A_26] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_26),hAPP_int_int(abs_abs_int,A_26))) ).
fof(fact_2669_abs__le__D1,axiom,
! [A_25,B_19] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,A_25)),B_19))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_25),B_19)) ) ).
fof(fact_2670_abs__le__D1,axiom,
! [A_25,B_19] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,A_25)),B_19))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_25),B_19)) ) ).
fof(fact_2671_abs__le__D1,axiom,
! [A_25,B_19] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,A_25)),B_19))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_25),B_19)) ) ).
fof(fact_2672_abs__mult__self,axiom,
! [A_24] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_24)),hAPP_rat_rat(abs_abs_rat,A_24)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_24),A_24) ).
fof(fact_2673_abs__mult__self,axiom,
! [A_24] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_24)),hAPP_real_real(abs_abs_real,A_24)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_24),A_24) ).
fof(fact_2674_abs__mult__self,axiom,
! [A_24] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_24)),hAPP_int_int(abs_abs_int,A_24)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_24),A_24) ).
fof(fact_2675_abs__mult,axiom,
! [A_23,B_18] : hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_23),B_18)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_23)),hAPP_rat_rat(abs_abs_rat,B_18)) ).
fof(fact_2676_abs__mult,axiom,
! [A_23,B_18] : hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_23),B_18)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_23)),hAPP_real_real(abs_abs_real,B_18)) ).
fof(fact_2677_abs__mult,axiom,
! [A_23,B_18] : hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_23),B_18)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_23)),hAPP_int_int(abs_abs_int,B_18)) ).
fof(fact_2678_abs__add__abs,axiom,
! [A_22,B_17] : hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_22)),hAPP_rat_rat(abs_abs_rat,B_17))) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_22)),hAPP_rat_rat(abs_abs_rat,B_17)) ).
fof(fact_2679_abs__add__abs,axiom,
! [A_22,B_17] : hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_22)),hAPP_real_real(abs_abs_real,B_17))) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_22)),hAPP_real_real(abs_abs_real,B_17)) ).
fof(fact_2680_abs__add__abs,axiom,
! [A_22,B_17] : hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_22)),hAPP_int_int(abs_abs_int,B_17))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_22)),hAPP_int_int(abs_abs_int,B_17)) ).
fof(fact_2681_abs__one,axiom,
hAPP_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).
fof(fact_2682_abs__one,axiom,
hAPP_real_real(abs_abs_real,one_one_real) = one_one_real ).
fof(fact_2683_abs__one,axiom,
hAPP_int_int(abs_abs_int,one_one_int) = one_one_int ).
fof(fact_2684_abs__minus__commute,axiom,
! [A_21,B_16] : hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_21),B_16)) = hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_16),A_21)) ).
fof(fact_2685_abs__minus__commute,axiom,
! [A_21,B_16] : hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_21),B_16)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_16),A_21)) ).
fof(fact_2686_abs__minus__commute,axiom,
! [A_21,B_16] : hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_21),B_16)) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_16),A_21)) ).
fof(fact_2687_power__abs,axiom,
! [A_20,N_9] : hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,A_20),N_9)) = hAPP_nat_rat(hAPP_rat_fun_nat_rat(power_power_rat,hAPP_rat_rat(abs_abs_rat,A_20)),N_9) ).
fof(fact_2688_power__abs,axiom,
! [A_20,N_9] : hAPP_real_real(abs_abs_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,A_20),N_9)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,A_20)),N_9) ).
fof(fact_2689_power__abs,axiom,
! [A_20,N_9] : hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_20),N_9)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(abs_abs_int,A_20)),N_9) ).
fof(fact_2690_dvd__if__abs__eq,axiom,
! [L_3,K_5] :
( hAPP_int_int(abs_abs_int,L_3) = hAPP_int_int(abs_abs_int,K_5)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,L_3),K_5)) ) ).
fof(fact_2691_dvd__if__abs__eq,axiom,
! [L_3,K_5] :
( hAPP_rat_rat(abs_abs_rat,L_3) = hAPP_rat_rat(abs_abs_rat,K_5)
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,L_3),K_5)) ) ).
fof(fact_2692_dvd__if__abs__eq,axiom,
! [L_3,K_5] :
( hAPP_real_real(abs_abs_real,L_3) = hAPP_real_real(abs_abs_real,K_5)
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,L_3),K_5)) ) ).
fof(fact_2693_abs__dvd__iff,axiom,
! [Ma,K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(abs_abs_int,Ma)),K))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),K)) ) ).
fof(fact_2694_abs__dvd__iff,axiom,
! [Ma,K] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,hAPP_rat_rat(abs_abs_rat,Ma)),K))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,Ma),K)) ) ).
fof(fact_2695_abs__dvd__iff,axiom,
! [Ma,K] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,hAPP_real_real(abs_abs_real,Ma)),K))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,Ma),K)) ) ).
fof(fact_2696_dvd__abs__iff,axiom,
! [Ma,K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),hAPP_int_int(abs_abs_int,K)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Ma),K)) ) ).
fof(fact_2697_dvd__abs__iff,axiom,
! [Ma,K] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,Ma),hAPP_rat_rat(abs_abs_rat,K)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(dvd_dvd_rat,Ma),K)) ) ).
fof(fact_2698_dvd__abs__iff,axiom,
! [Ma,K] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,Ma),hAPP_real_real(abs_abs_real,K)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(dvd_dvd_real,Ma),K)) ) ).
fof(fact_2699_abs__of__nat,axiom,
! [N_8] : hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(semiri151668891at_rat,N_8)) = hAPP_nat_rat(semiri151668891at_rat,N_8) ).
fof(fact_2700_abs__of__nat,axiom,
! [N_8] : hAPP_real_real(abs_abs_real,hAPP_nat_real(semiri132038758t_real,N_8)) = hAPP_nat_real(semiri132038758t_real,N_8) ).
fof(fact_2701_abs__of__nat,axiom,
! [N_8] : hAPP_int_int(abs_abs_int,hAPP_nat_int(semiri1621563631at_int,N_8)) = hAPP_nat_int(semiri1621563631at_int,N_8) ).
fof(fact_2702_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_2703_nat__int,axiom,
! [N] : hAPP_int_nat(nat,hAPP_nat_int(semiri1621563631at_int,N)) = N ).
fof(fact_2704_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_2705_abs__ge__zero,axiom,
! [A_19] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_rat_rat(abs_abs_rat,A_19))) ).
fof(fact_2706_abs__ge__zero,axiom,
! [A_19] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(abs_abs_real,A_19))) ).
fof(fact_2707_abs__ge__zero,axiom,
! [A_19] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(abs_abs_int,A_19))) ).
fof(fact_2708_abs__le__zero__iff,axiom,
! [A_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,A_1)),zero_zero_rat))
<=> A_1 = zero_zero_rat ) ).
fof(fact_2709_abs__le__zero__iff,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,A_1)),zero_zero_real))
<=> A_1 = zero_zero_real ) ).
fof(fact_2710_abs__le__zero__iff,axiom,
! [A_1] :
( is_int(A_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(abs_abs_int,A_1)),zero_zero_int))
<=> A_1 = zero_zero_int ) ) ).
fof(fact_2711_abs__of__nonneg,axiom,
! [A_18] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_18))
=> hAPP_rat_rat(abs_abs_rat,A_18) = A_18 ) ).
fof(fact_2712_abs__of__nonneg,axiom,
! [A_18] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_18))
=> hAPP_real_real(abs_abs_real,A_18) = A_18 ) ).
fof(fact_2713_abs__of__nonneg,axiom,
! [A_18] :
( is_int(A_18)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_18))
=> hAPP_int_int(abs_abs_int,A_18) = A_18 ) ) ).
fof(fact_2714_abs__of__pos,axiom,
! [A_17] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_17))
=> hAPP_rat_rat(abs_abs_rat,A_17) = A_17 ) ).
fof(fact_2715_abs__of__pos,axiom,
! [A_17] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),A_17))
=> hAPP_real_real(abs_abs_real,A_17) = A_17 ) ).
fof(fact_2716_abs__of__pos,axiom,
! [A_17] :
( is_int(A_17)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_17))
=> hAPP_int_int(abs_abs_int,A_17) = A_17 ) ) ).
fof(fact_2717_zero__less__abs__iff,axiom,
! [A_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_rat_rat(abs_abs_rat,A_1)))
<=> A_1 != zero_zero_rat ) ).
fof(fact_2718_zero__less__abs__iff,axiom,
! [A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(abs_abs_real,A_1)))
<=> A_1 != zero_zero_real ) ).
fof(fact_2719_zero__less__abs__iff,axiom,
! [A_1] :
( is_int(A_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(abs_abs_int,A_1)))
<=> A_1 != zero_zero_int ) ) ).
fof(fact_2720_abs__not__less__zero,axiom,
! [A_16] : ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,A_16)),zero_zero_rat)) ).
fof(fact_2721_abs__not__less__zero,axiom,
! [A_16] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,A_16)),zero_zero_real)) ).
fof(fact_2722_abs__not__less__zero,axiom,
! [A_16] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,A_16)),zero_zero_int)) ).
fof(fact_2723_abs__mult__less,axiom,
! [B_15,D_6,A_15,C_5] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,A_15)),C_5))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,B_15)),D_6))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_15)),hAPP_rat_rat(abs_abs_rat,B_15))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,C_5),D_6))) ) ) ).
fof(fact_2724_abs__mult__less,axiom,
! [B_15,D_6,A_15,C_5] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,A_15)),C_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,B_15)),D_6))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_15)),hAPP_real_real(abs_abs_real,B_15))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_5),D_6))) ) ) ).
fof(fact_2725_abs__mult__less,axiom,
! [B_15,D_6,A_15,C_5] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,A_15)),C_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(abs_abs_int,B_15)),D_6))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_15)),hAPP_int_int(abs_abs_int,B_15))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C_5),D_6))) ) ) ).
fof(fact_2726_abs__triangle__ineq,axiom,
! [A_14,B_14] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_14),B_14))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_14)),hAPP_rat_rat(abs_abs_rat,B_14)))) ).
fof(fact_2727_abs__triangle__ineq,axiom,
! [A_14,B_14] : 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,A_14),B_14))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_14)),hAPP_real_real(abs_abs_real,B_14)))) ).
fof(fact_2728_abs__triangle__ineq,axiom,
! [A_14,B_14] : 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(plus_plus_int,A_14),B_14))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_14)),hAPP_int_int(abs_abs_int,B_14)))) ).
fof(fact_2729_abs__triangle__ineq2__sym,axiom,
! [A_13,B_13] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(abs_abs_rat,A_13)),hAPP_rat_rat(abs_abs_rat,B_13))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_13),A_13)))) ).
fof(fact_2730_abs__triangle__ineq2__sym,axiom,
! [A_13,B_13] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(abs_abs_real,A_13)),hAPP_real_real(abs_abs_real,B_13))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_13),A_13)))) ).
fof(fact_2731_abs__triangle__ineq2__sym,axiom,
! [A_13,B_13] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,A_13)),hAPP_int_int(abs_abs_int,B_13))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_13),A_13)))) ).
fof(fact_2732_abs__triangle__ineq2,axiom,
! [A_12,B_12] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(abs_abs_rat,A_12)),hAPP_rat_rat(abs_abs_rat,B_12))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_12),B_12)))) ).
fof(fact_2733_abs__triangle__ineq2,axiom,
! [A_12,B_12] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(abs_abs_real,A_12)),hAPP_real_real(abs_abs_real,B_12))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_12),B_12)))) ).
fof(fact_2734_abs__triangle__ineq2,axiom,
! [A_12,B_12] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(abs_abs_int,A_12)),hAPP_int_int(abs_abs_int,B_12))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_12),B_12)))) ).
fof(fact_2735_abs__triangle__ineq3,axiom,
! [A_11,B_11] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(abs_abs_rat,A_11)),hAPP_rat_rat(abs_abs_rat,B_11)))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_11),B_11)))) ).
fof(fact_2736_abs__triangle__ineq3,axiom,
! [A_11,B_11] : 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(abs_abs_real,A_11)),hAPP_real_real(abs_abs_real,B_11)))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_11),B_11)))) ).
fof(fact_2737_abs__triangle__ineq3,axiom,
! [A_11,B_11] : 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_int_int(abs_abs_int,A_11)),hAPP_int_int(abs_abs_int,B_11)))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_11),B_11)))) ).
fof(fact_2738_nat__0,axiom,
hAPP_int_nat(nat,zero_zero_int) = zero_zero_nat ).
fof(fact_2739_transfer__nat__int__numerals_I1_J,axiom,
zero_zero_nat = hAPP_int_nat(nat,zero_zero_int) ).
fof(fact_2740_abs__zmult__eq__1,axiom,
! [M,N] :
( hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,M),N)) = one_one_int
=> hAPP_int_int(abs_abs_int,M) = one_one_int ) ).
fof(fact_2741_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_2742_transfer__nat__int__relations_I1_J,axiom,
! [Ya,Xa] :
( ( is_int(Ya)
& is_int(Xa) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hAPP_int_nat(nat,Xa) = hAPP_int_nat(nat,Ya)
<=> Xa = Ya ) ) ) ) ).
fof(fact_2743_all__nat,axiom,
! [P_1] :
( ! [X1] : hBOOL(hAPP_nat_bool(P_1,X1))
<=> ! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,X_1))) ) ) ) ).
fof(fact_2744_ex__nat,axiom,
! [P_1] :
( ? [X1] : hBOOL(hAPP_nat_bool(P_1,X1))
<=> ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1))
& hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,X_1))) ) ) ).
fof(fact_2745_nat__number__of__def,axiom,
! [V_1] : hAPP_int_nat(number_number_of_nat,V_1) = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,V_1)) ).
fof(fact_2746_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_2747_transfer__nat__int__numerals_I2_J,axiom,
one_one_nat = hAPP_int_nat(nat,one_one_int) ).
fof(fact_2748_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_2749_zdvd1__eq,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Xa),one_one_int))
<=> hAPP_int_int(abs_abs_int,Xa) = one_one_int ) ).
fof(fact_2750_mult__sum2sq,axiom,
! [A_3,B_2,P_3,Q_1] : 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_1))) = 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_1))),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_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),P_3)))) ).
fof(fact_2751_abs__mult__pos,axiom,
! [Y_4,X_4] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),X_4))
=> hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,Y_4)),X_4) = hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Y_4),X_4)) ) ).
fof(fact_2752_abs__mult__pos,axiom,
! [Y_4,X_4] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X_4))
=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,Y_4)),X_4) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y_4),X_4)) ) ).
fof(fact_2753_abs__mult__pos,axiom,
! [Y_4,X_4] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_4))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,Y_4)),X_4) = hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y_4),X_4)) ) ).
fof(fact_2754_abs__eq__mult,axiom,
! [B_10,A_10] :
( ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),A_10))
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,A_10),zero_zero_rat)) )
& ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),B_10))
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,B_10),zero_zero_rat)) ) )
=> hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,A_10),B_10)) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,hAPP_rat_rat(abs_abs_rat,A_10)),hAPP_rat_rat(abs_abs_rat,B_10)) ) ).
fof(fact_2755_abs__eq__mult,axiom,
! [B_10,A_10] :
( ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),A_10))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_10),zero_zero_real)) )
& ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),B_10))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,B_10),zero_zero_real)) ) )
=> hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_10),B_10)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(abs_abs_real,A_10)),hAPP_real_real(abs_abs_real,B_10)) ) ).
fof(fact_2756_abs__eq__mult,axiom,
! [B_10,A_10] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_10))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_10),zero_zero_int)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),B_10))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_10),zero_zero_int)) ) )
=> hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_10),B_10)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_10)),hAPP_int_int(abs_abs_int,B_10)) ) ).
fof(fact_2757_zero__le__power__abs,axiom,
! [A_9,N_7] : 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,hAPP_rat_rat(abs_abs_rat,A_9)),N_7))) ).
fof(fact_2758_zero__le__power__abs,axiom,
! [A_9,N_7] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,A_9)),N_7))) ).
fof(fact_2759_zero__le__power__abs,axiom,
! [A_9,N_7] : 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,A_9)),N_7))) ).
fof(fact_2760_abs__triangle__ineq4,axiom,
! [A_8,B_9] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_8),B_9))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,A_8)),hAPP_rat_rat(abs_abs_rat,B_9)))) ).
fof(fact_2761_abs__triangle__ineq4,axiom,
! [A_8,B_9] : 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,A_8),B_9))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,A_8)),hAPP_real_real(abs_abs_real,B_9)))) ).
fof(fact_2762_abs__triangle__ineq4,axiom,
! [A_8,B_9] : 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,A_8),B_9))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(abs_abs_int,A_8)),hAPP_int_int(abs_abs_int,B_9)))) ).
fof(fact_2763_abs__diff__triangle__ineq,axiom,
! [A_7,B_8,C_4,D_5] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,A_7),B_8)),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,C_4),D_5)))),hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,A_7),C_4))),hAPP_rat_rat(abs_abs_rat,hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,B_8),D_5))))) ).
fof(fact_2764_abs__diff__triangle__ineq,axiom,
! [A_7,B_8,C_4,D_5] : 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,A_7),B_8)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,C_4),D_5)))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_7),C_4))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_8),D_5))))) ).
fof(fact_2765_abs__diff__triangle__ineq,axiom,
! [A_7,B_8,C_4,D_5] : 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_int_int(hAPP_int_fun_int_int(plus_plus_int,A_7),B_8)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,C_4),D_5)))),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,A_7),C_4))),hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_8),D_5))))) ).
fof(fact_2766_abs__minus__one,axiom,
hAPP_rat_rat(abs_abs_rat,hAPP_int_rat(number_number_of_rat,min)) = one_one_rat ).
fof(fact_2767_abs__minus__one,axiom,
hAPP_real_real(abs_abs_real,hAPP_int_real(number267125858f_real,min)) = one_one_real ).
fof(fact_2768_abs__minus__one,axiom,
hAPP_int_int(abs_abs_int,hAPP_int_int(number_number_of_int,min)) = one_one_int ).
fof(fact_2769_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_2770_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_2771_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_2772_zless__nat__conj,axiom,
! [Wa,Z_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,Wa)),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,Wa),Z_2)) ) ) ).
fof(fact_2773_nat__mono__iff,axiom,
! [Wa,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,Wa)),hAPP_int_nat(nat,Z_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_2)) ) ) ).
fof(fact_2774_transfer__nat__int__relations_I3_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,Xa)),hAPP_int_nat(nat,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ) ) ).
fof(fact_2775_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_2776_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_2777_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_2778_zero__le__zpower__abs,axiom,
! [X,N] : 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)),N))) ).
fof(fact_2779_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_2780_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_2781_transfer__nat__int__relations_I4_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_int_nat(nat,Xa)),hAPP_int_nat(nat,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Xa),Ya)) ) ) ) ).
fof(fact_2782_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_2783_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_2784_nat__less__eq__zless,axiom,
! [Z_2,Wa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,Wa)),hAPP_int_nat(nat,Z_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_2)) ) ) ).
fof(fact_2785_transfer__nat__int__relations_I2_J,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,Xa)),hAPP_int_nat(nat,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ) ) ).
fof(fact_2786_abs__power3__distrib,axiom,
! [X] : hAPP_int_int(abs_abs_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),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)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,hAPP_int_int(bit1,pls)))) ).
fof(fact_2787_zero__less__zpower__abs__iff,axiom,
! [Xa,Na] :
( is_int(Xa)
=> ( 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,Xa)),Na)))
<=> ( Xa != zero_zero_int
| Na = zero_zero_nat ) ) ) ).
fof(fact_2788_split__nat,axiom,
! [P_1,I_1] :
( is_int(I_1)
=> ( hBOOL(hAPP_nat_bool(P_1,hAPP_int_nat(nat,I_1)))
<=> ( ! [N_1] :
( I_1 = hAPP_nat_int(semiri1621563631at_int,N_1)
=> hBOOL(hAPP_nat_bool(P_1,N_1)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),zero_zero_int))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ) ) ).
fof(fact_2789_nat__eq__iff,axiom,
! [Wa,Ma] :
( is_int(Wa)
=> ( hAPP_int_nat(nat,Wa) = Ma
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> Wa = hAPP_nat_int(semiri1621563631at_int,Ma) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> Ma = zero_zero_nat ) ) ) ) ).
fof(fact_2790_nat__eq__iff2,axiom,
! [Ma,Wa] :
( is_int(Wa)
=> ( Ma = hAPP_int_nat(nat,Wa)
<=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> Wa = hAPP_nat_int(semiri1621563631at_int,Ma) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> Ma = zero_zero_nat ) ) ) ) ).
fof(fact_2791_nat__le__eq__zle,axiom,
! [Z_2,Wa] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Wa))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Z_2)) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(nat,Wa)),hAPP_int_nat(nat,Z_2)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Wa),Z_2)) ) ) ).
fof(fact_2792_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
! [Y,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),Y))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)) ) ) ).
fof(fact_2793_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_2794_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_2795_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
! [Y,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),Y))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)) ) ) ).
fof(fact_2796_zdvd__mult__cancel1,axiom,
! [Na,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),Na)),Ma))
<=> hAPP_int_int(abs_abs_int,Na) = one_one_int ) ) ) ).
fof(fact_2797_int__eq__iff__number__of,axiom,
! [Ma,Va] :
( hAPP_nat_int(semiri1621563631at_int,Ma) = hAPP_int_int(number_number_of_int,Va)
<=> ( Ma = hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,Va))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(number_number_of_int,Va))) ) ) ).
fof(fact_2798_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_2799_nat__power__eq,axiom,
! [N,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)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat,Z_1)),N) ) ).
fof(fact_2800_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
! [N,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(nat,X)),N) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N)) ) ).
fof(fact_2801_Little__Fermat,axiom,
! [X,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),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_2802_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_2803_power2__eq__iff__abs__eq,axiom,
! [A_1,B_1] :
( hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),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_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))
<=> hAPP_int_int(abs_abs_int,A_1) = hAPP_int_int(abs_abs_int,B_1) ) ).
fof(fact_2804_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_2805_nat__less__iff,axiom,
! [Ma,Wa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Wa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(nat,Wa)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),hAPP_nat_int(semiri1621563631at_int,Ma))) ) ) ).
fof(fact_2806_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_2807_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_2808_decr__lemma,axiom,
! [X,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),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),Z_1))),one_one_int)),D_3))),Z_1)) ) ).
fof(fact_2809_incr__lemma,axiom,
! [Z_1,X,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),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),Z_1))),one_one_int)),D_3)))) ) ).
fof(fact_2810_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_2811_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_2812_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_2813_best__division__abs,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ? [N_1] :
( is_int(N_1)
& 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),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),X))))),X)) ) ) ).
fof(fact_2814_nat0__intermed__int__val,axiom,
! [K,F,Na] :
( is_int(K)
=> ( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),Na))
=> 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))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),Na))
& hAPP_nat_int(F,I_2) = K ) ) ) ) ) ).
fof(fact_2815_int__val__lemma,axiom,
! [K,F,Na] :
( is_int(K)
=> ( ! [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_2),Na))
=> 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))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),Na))
& hAPP_nat_int(F,I_2) = K ) ) ) ) ) ).
fof(fact_2816_is__sum2sq__def,axiom,
! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_int_bool(twoSqu1770640450sum2sq,Xa))
<=> ? [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)) = Xa ) ) ) ).
fof(fact_2817_nat__intermed__int__val,axiom,
! [K,F,Na,Ma] :
( is_int(K)
=> ( ! [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),Na)) )
=> 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),Na))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(F,Ma)),K))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),hAPP_nat_int(F,Na)))
=> ? [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),Na))
& hAPP_nat_int(F,I_2) = K ) ) ) ) ) ) ).
fof(fact_2818_nat__aux__def,axiom,
! [I,N] : hAPP_nat_nat(nat_aux(I),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,I)),N) ).
fof(fact_2819_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_2820_Euler__part1,axiom,
! [X,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),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),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_2821_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_2822_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_2823_pos__imp__zdiv__neg__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_1),B_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)) ) ) ).
fof(fact_2824_neg__imp__zdiv__neg__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),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_1),B_1)),zero_zero_int))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1)) ) ) ).
fof(fact_2825_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_2826_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,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),Y))
=> 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),Y))) ) ) ).
fof(fact_2827_zdiv__number__of__Bit0,axiom,
! [V_1,W] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_1))),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_1)),hAPP_int_int(number_number_of_int,W)) ).
fof(fact_2828_zdvd__mult__div__cancel,axiom,
! [N,M] :
( is_int(M)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),M))
=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N)) = M ) ) ).
fof(fact_2829_abs__div,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y),X))
=> hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_int(abs_abs_int,Y)) ) ).
fof(fact_2830_int__div__less__self,axiom,
! [K_2,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),X))
=> ( 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),K_2)),X)) ) ) ).
fof(fact_2831_div__prop1,axiom,
! [X,Y,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),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),Z_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Z_1)),Y)) ) ) ).
fof(fact_2832_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_2833_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_2834_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_2835_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_2836_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_2837_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_2838_neg__imp__zdiv__nonneg__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),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_1),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int)) ) ) ).
fof(fact_2839_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_2840_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_2841_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_2842_nonneg1__imp__zdiv__pos__iff,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_1),B_1)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) ) ) ) ).
fof(fact_2843_pos__imp__zdiv__pos__iff,axiom,
! [I_1,K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
=> ( 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)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),I_1)) ) ) ).
fof(fact_2844_pos__imp__zdiv__nonneg__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_1),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1)) ) ) ).
fof(fact_2845_zdiv__eq__0__iff,axiom,
! [I_1,K] :
( is_int(K)
=> ( hAPP_int_int(hAPP_int_fun_int_int(div_div_int,I_1),K) = zero_zero_int
<=> ( K = 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)) )
| ( 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),I_1)) ) ) ) ) ).
fof(fact_2846_abs__add__one__not__less__self,axiom,
! [X] : ~ 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)),one_one_real)),X)) ).
fof(fact_2847_zdiv__leq__prop,axiom,
! [X,Y] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))),X)) ) ).
fof(fact_2848_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_2849_abs__add__one__gt__zero,axiom,
! [X] : 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)))) ).
fof(fact_2850_div__prop2,axiom,
! [X,Y,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),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Y),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),Z_1)),Y)) ) ) ).
fof(fact_2851_divmod__int__rel__div__eq,axiom,
! [A_6,B_7,Y,R_5] :
( ( is_int(B_7)
& is_int(Y) )
=> ( A_6 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_7),Y)),R_5)
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_7))
=> ( 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_7)) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_7),R_5))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_5),zero_zero_int)) ) ) )
=> ( B_7 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_6),B_7) = Y ) ) ) ) ).
fof(fact_2852_split__zdiv,axiom,
! [P_1,Na,K] :
( ( is_int(Na)
& is_int(K) )
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Na),K)))
<=> ( ( K = 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))
=> ! [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))
& Na = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(P_1,I_2)) ) ) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),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),J_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
& Na = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(P_1,I_2)) ) ) ) ) ) ) ).
fof(fact_2853_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_2854_zdiv__number__of__Bit1,axiom,
! [V_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_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_1))),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_1)),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_1))),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_1)),one_one_int)),hAPP_int_int(number_number_of_int,W)) ) ) ).
fof(fact_2855_less__one__imp__sqr__less__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_real)) ) ).
fof(fact_2856_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_2857_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_2858_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_2859_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_2860_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_2861_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_2862_Euler__part3,axiom,
! [X,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),zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_3),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),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_2863_zfact_Osimps,axiom,
! [N] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int))
=> hAPP_int_int(zfact,N) = one_one_int )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),zero_zero_int))
=> hAPP_int_int(zfact,N) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(zfact,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int))) ) ) ).
fof(fact_2864_rv,axiom,
( v = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,x),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))))
& 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,v))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))) ) ).
fof(fact_2865_sw,axiom,
( w = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,y),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))))
& 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,w))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))) ) ).
fof(fact_2866_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_2867_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_2868_int__le__induct,axiom,
! [P_1,I_1,K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_1),K))
=> ( hBOOL(hAPP_int_bool(P_1,K))
=> ( ! [I_2] :
( is_int(I_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,I_2),K))
=> ( 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_2869_m1,axiom,
hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),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(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),m1) ).
fof(fact_2870__0961_A_L_Aint_An_Advd_Av_A_094_A2_A_L_Aw_A_094_A2_096,axiom,
hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_2871__096_Ix_A_094_A2_A_L_Ay_A_094_A2_J_A_K_A_Iv_A_094_A2_A_L_Aw_A_094_A2_J,axiom,
hAPP_int_int(hAPP_int_fun_int_int(times_times_int,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_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),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_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,x),v)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,y),w))),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(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,x),w)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,y),v))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_2872__096_I_Ir_A_K_A_I1_A_L_Aint_An_J_A_L_Av_J_A_K_Av_A_L_A_Is_A_K_A_I1_A_L,axiom,
hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_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(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),v)),v)),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,sa),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),w)),w))),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(hAPP_int_fun_int_int(minus_minus_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,r),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),v)),w)),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,sa),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),w)),v))),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_nat_int(hAPP_int_fun_nat_int(power_power_int,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(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),v))),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),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),w)))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),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,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(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),w))),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),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),v)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_2873__096_I_I1_A_L_Aint_An_J_A_K_A_Ir_A_K_Av_J_A_L_A_I1_A_L_Aint_An_J_A_K_A,axiom,
hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,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(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),v))),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),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),w)))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),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,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(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),w))),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),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),v)))),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_nat_int(hAPP_int_fun_nat_int(power_power_int,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(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),v))),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),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),w)))),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),hAPP_nat_int(semiri1621563631at_int,n))),m1))),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(hAPP_int_fun_int_int(minus_minus_int,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),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),w))),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),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),v)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_2874__096_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_K_A_Iv_A_094_A2_A_L_A,axiom,
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(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,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),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(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_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_2875__096_Ix_A_K_Av_A_L_Ay_A_K_Aw_J_A_094_A2_A_L_A_Ix_A_K_Aw_A_N_Ay_A_K_Av_,axiom,
hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,x),v)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,y),w))),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(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,x),w)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,y),v))),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_nat_int(hAPP_int_fun_nat_int(power_power_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(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),v)),v)),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,sa),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),w)),w))),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(hAPP_int_fun_int_int(minus_minus_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,r),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),v)),w)),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,sa),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),w)),v))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_2876__096_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_K_A_Iv_A_094_A2_A_L_A,axiom,
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(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,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),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_nat_int(hAPP_int_fun_nat_int(power_power_int,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(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),v))),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),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),w)))),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),hAPP_nat_int(semiri1621563631at_int,n))),m1))),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(hAPP_int_fun_int_int(minus_minus_int,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),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),w))),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),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),v)))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_2877__096_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_K_A_Iv_A_094_A2_A_L_A,axiom,
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(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,one_one_int),hAPP_nat_int(semiri1621563631at_int,n)))),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),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_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),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(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,r),v)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),w))),m1)),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_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),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(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),w)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),v))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_2878__096_I1_A_L_Aint_An_J_A_094_A2_A_K_A_I_I4_A_K_Am_A_L_A1_J_A_K_Am1_J_A_,axiom,
hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),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(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)),m1)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),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_nat_int(hAPP_int_fun_nat_int(power_power_int,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,r),v)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),w))),m1)),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(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),w)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),v))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_2879_calculation,axiom,
hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),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(minus_minus_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)),m1)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,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,r),v)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),w))),m1)),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(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,r),w)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,sa),v))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = zero_zero_int ).
fof(fact_2880__096_B_Bthesis_O_A_I_B_Bm1_O_Av_A_094_A2_A_L_Aw_A_094_A2_A_061_A_I1_A_,axiom,
~ ! [M1] :
( is_int(M1)
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,v),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,w),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(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),M1) ) ).
fof(fact_2881_div__le__mono,axiom,
! [K_2,M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,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),K_2))) ) ).
fof(fact_2882_div__le__dividend,axiom,
! [M,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),M)) ).
fof(fact_2883_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_2884_div__less,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = zero_zero_nat ) ).
fof(fact_2885_nat__mult__div__cancel__disj,axiom,
! [M,N,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)) = 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)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) ) ) ).
fof(fact_2886_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_2887_Divides_Otransfer__int__nat__functions_I1_J,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y)) ).
fof(fact_2888_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_2889_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_2890_div__le__mono2,axiom,
! [K_2,N,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))
=> 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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,K_2),M))) ) ) ).
fof(fact_2891_div__mult__self1__is__m,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ).
fof(fact_2892_div__mult__self__is__m,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ).
fof(fact_2893_nat__mult__div__cancel1,axiom,
! [M,N,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)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) ) ).
fof(fact_2894_div__less__dividend,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),M)) ) ) ).
fof(fact_2895_split__div,axiom,
! [P_1,Na,K] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Na),K)))
<=> ( ( K = zero_zero_nat
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
& ( K != zero_zero_nat
=> ! [I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K))
=> ( Na = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),I_2)),J_1)
=> hBOOL(hAPP_nat_bool(P_1,I_2)) ) ) ) ) ) ).
fof(fact_2896_Divides_Otransfer__nat__int__functions_I1_J,axiom,
! [Y,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),Y))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) ) ) ).
fof(fact_2897_nat__div__distrib,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) ) ).
fof(fact_2898_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_2899_div__2__gt__zero,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N))
=> 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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_2900_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_2901_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_2902_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_2903_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_2904_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_2905_MultInv__prop1,axiom,
! [X,Y,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),Y),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),hAPP_int_int(multInv(P_3),Y)),P_3)) ) ) ).
fof(fact_2906_Suc__n__div__2__gt__zero,axiom,
! [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,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),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_2907_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_2908_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_2909_MultInv__def,axiom,
! [P_3,X] : hAPP_int_int(multInv(P_3),X) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),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_2910_MultInv__prop5,axiom,
! [Y,X,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),zero_zero_int),P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Y),zero_zero_int),P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),hAPP_int_int(multInv(P_3),Y)),P_3))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),P_3)) ) ) ) ) ) ).
fof(fact_2911_MultInv__prop4,axiom,
! [X,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),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))),X),P_3)) ) ) ) ).
fof(fact_2912_MultInv__prop3,axiom,
! [X,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),zero_zero_int),P_3))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(multInv(P_3),X)),zero_zero_int),P_3)) ) ) ) ).
fof(fact_2913_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_2914_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_2915_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_2916_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_2917_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_2918_Int2_Oaux____2,axiom,
! [X,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),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),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X)))),X),P_3)) ) ) ) ).
fof(fact_2919_Int2_Oaux____1,axiom,
! [X,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),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))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),hAPP_int_int(multInv(P_3),X))),hAPP_int_int(multInv(P_3),hAPP_int_int(multInv(P_3),X)))),P_3)) ) ) ) ).
fof(fact_2920_MultInv__prop2a,axiom,
! [X,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),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)),X)),one_one_int),P_3)) ) ) ) ).
fof(fact_2921_MultInv__prop2,axiom,
! [X,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),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),hAPP_int_int(multInv(P_3),X))),one_one_int),P_3)) ) ) ) ).
fof(fact_2922_sin__bound__lemma,axiom,
! [U_1,V_1,X,Y] :
( X = Y
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,U_1)),V_1))
=> 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),U_1)),Y))),V_1)) ) ) ).
fof(fact_2923_wset__mem__inv__mem,axiom,
! [B_1,A_1,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_1),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_1),hAPP_i1948725293t_bool(wset(A_1),P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(inv(P_5),B_1)),hAPP_i1948725293t_bool(wset(A_1),P_5))) ) ) ) ) ).
fof(fact_2924_wset__inv__mem__mem,axiom,
! [B_1,A_1,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_1),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_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),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_1)),hAPP_i1948725293t_bool(wset(A_1),P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(A_1),P_5))) ) ) ) ) ) ) ).
fof(fact_2925_rabs__ratiotest__lemma,axiom,
! [X,Y,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)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,C_1),hAPP_real_real(abs_abs_real,Y))))
=> X = zero_zero_real ) ) ).
fof(fact_2926_int__less__induct,axiom,
! [P_1,I_1,K] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_1),K))
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,K),one_one_int)))
=> ( ! [I_2] :
( is_int(I_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,I_2),K))
=> ( 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_2927_d22set__induct__old,axiom,
! [Xa,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,Xa)) ) ).
fof(fact_2928_wset__mem__mem,axiom,
! [P_5,A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),hAPP_i1948725293t_bool(wset(A_1),P_5))) ) ).
fof(fact_2929_wset__subset,axiom,
! [B_1,P_5,A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),one_one_int)),P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(A_1),P_5))) ) ) ).
fof(fact_2930_wset__g__1,axiom,
! [B_1,A_1,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),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_1),hAPP_i1948725293t_bool(wset(A_1),P_5)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_1)) ) ) ) ).
fof(fact_2931_wset__less,axiom,
! [B_1,A_1,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),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_1),hAPP_i1948725293t_bool(wset(A_1),P_5)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,P_5),one_one_int))) ) ) ) ).
fof(fact_2932_wset__mem__imp__or,axiom,
! [B_1,P_5,A_1] :
( ( is_int(B_1)
& is_int(A_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),A_1))
=> ( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,A_1),one_one_int)),P_5)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(A_1),P_5)))
=> ( B_1 = A_1
| B_1 = hAPP_int_int(inv(P_5),A_1) ) ) ) ) ) ).
fof(fact_2933_le__Suc__ex__iff,axiom,
! [K,L] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L))
<=> ? [N_1] : L = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),N_1) ) ).
fof(fact_2934_wset__mem,axiom,
! [B_1,A_1,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),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_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(wset(A_1),P_5))) ) ) ) ) ).
fof(fact_2935_eq__diff__eq_H,axiom,
! [Xa,Ya,Z_2] :
( Xa = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Ya),Z_2)
<=> Ya = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Z_2) ) ).
fof(fact_2936_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_2937_phi__prime,axiom,
! [P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> 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_2938_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_2939_int__nat__abs,axiom,
! [X] : hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X))) = hAPP_int_int(abs_abs_int,X) ).
fof(fact_2940_EvenOdd_Oodd__times__odd,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya)),zOdd)) ) ) ).
fof(fact_2941_odd__mult__odd__prop,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya)),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) ) ).
fof(fact_2942_d22set__le__swap,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(d22set,A_1))) ) ).
fof(fact_2943_d22set__le,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(d22set,A_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1)) ) ).
fof(fact_2944_d22set__g__1,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(d22set,A_1)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_1)) ) ).
fof(fact_2945_power__preserves__odd,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Xa),Na)),zOdd))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) ) ) ).
fof(fact_2946_d22set__mem,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(d22set,A_1))) ) ) ).
fof(fact_2947_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_2948_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_2949_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_2950_EvenOdd_Oneg__one__odd__power,axiom,
! [Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Xa)) = hAPP_int_int(number_number_of_int,min) ) ) ).
fof(fact_2951_zOddI,axiom,
! [Xa,K] :
( Xa = 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)
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd)) ) ).
fof(fact_2952_divides__le,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
| N = zero_zero_nat ) ) ).
fof(fact_2953_best__odd__division__abs,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> ? [N_1] :
( is_int(N_1)
& 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,Ya),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N_1),Xa))))),Xa)) ) ) ) ).
fof(fact_2954_zOddE,axiom,
! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> ~ ! [K_1] :
( is_int(K_1)
=> Xa != 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) ) ) ) ).
fof(fact_2955_pow__divides__eq__int,axiom,
! [A_1,B_1,Na] :
( Na != 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_1),Na)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_1),Na)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_1),B_1)) ) ) ).
fof(fact_2956_pow__divides__pow__int,axiom,
! [A_3,N,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)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)))
=> ( N != zero_zero_nat
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,A_3),B_2)) ) ) ).
fof(fact_2957_pow__divides__eq__nat,axiom,
! [A_1,B_1,Na] :
( Na != 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_1),Na)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_1),B_1)) ) ) ).
fof(fact_2958_dvd__pos__nat,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M)) ) ) ).
fof(fact_2959_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_2960_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
! [A_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_1))
<=> A_1 = zero_zero_nat ) ).
fof(fact_2961_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
! [A_1] :
( A_1 != zero_zero_nat
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_1),zero_zero_nat))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,zero_zero_nat),A_1)) ) ) ).
fof(fact_2962_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_2963_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_2964_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_2965_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_2966_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
! [A_1] :
( A_1 != one_one_nat
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,one_one_nat),A_1))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_1),one_one_nat)) ) ) ).
fof(fact_2967_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
! [A_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_1),one_one_nat))
<=> A_1 = one_one_nat ) ).
fof(fact_2968_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_2969_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_2970_zmod__number__of__Bit1,axiom,
! [V_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_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,V_1))),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_1)),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_1))),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_1)),one_one_int)),hAPP_int_int(number_number_of_int,W)))),one_one_int) ) ) ).
fof(fact_2971_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_2972_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_2973_StandardRes__prop4,axiom,
! [X,Y,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)),hAPP_int_int(standardRes(M),Y))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),M)) ) ).
fof(fact_2974_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
! [Y,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),Y))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(nat_tsub(X),Y)) ) ) ).
fof(fact_2975_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_2976_Suc__mono,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N))) ) ).
fof(fact_2977_lessI,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,N))) ).
fof(fact_2978_zero__less__Suc,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N))) ).
fof(fact_2979_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_2980_StandardRes__SR__prop,axiom,
! [Xa,P_5] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(sr,P_5)))
=> hAPP_int_int(standardRes(P_5),Xa) = Xa ) ) ).
fof(fact_2981_StandardRes__def,axiom,
! [M,X] : hAPP_int_int(standardRes(M),X) = hAPP_int_int(div_mod_int(X),M) ).
fof(fact_2982_Suc__inject,axiom,
! [X,Y] :
( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y)
=> X = Y ) ).
fof(fact_2983_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_2984_Suc__n__not__n,axiom,
! [N] : hAPP_nat_nat(suc,N) != N ).
fof(fact_2985_n__not__Suc__n,axiom,
! [N] : N != hAPP_nat_nat(suc,N) ).
fof(fact_2986_zmod__self,axiom,
! [A_3] : hAPP_int_int(div_mod_int(A_3),A_3) = zero_zero_int ).
fof(fact_2987_zmod__zero,axiom,
! [B_2] : hAPP_int_int(div_mod_int(zero_zero_int),B_2) = zero_zero_int ).
fof(fact_2988_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_2989_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_2990_zdiff__zmod__left,axiom,
! [X,M,Y] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(div_mod_int(X),M)),Y)),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)),M) ).
fof(fact_2991_zdiff__zmod__right,axiom,
! [X,Y,M] : hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),hAPP_int_int(div_mod_int(Y),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)),M) ).
fof(fact_2992_Zero__not__Suc,axiom,
! [M] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
fof(fact_2993_nat_Osimps_I2_J,axiom,
! [Nat_3] : zero_zero_nat != hAPP_nat_nat(suc,Nat_3) ).
fof(fact_2994_Suc__not__Zero,axiom,
! [M] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
fof(fact_2995_nat_Osimps_I3_J,axiom,
! [Nat_2] : hAPP_nat_nat(suc,Nat_2) != zero_zero_nat ).
fof(fact_2996_Zero__neq__Suc,axiom,
! [M] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
fof(fact_2997_Suc__neq__Zero,axiom,
! [M] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
fof(fact_2998_zpower__zmod,axiom,
! [X,M,Y] : hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(div_mod_int(X),M)),Y)),M) = hAPP_int_int(div_mod_int(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),Y)),M) ).
fof(fact_2999_Residues_Oaux,axiom,
! [X,M,Y] :
( hAPP_int_int(div_mod_int(X),M) = hAPP_int_int(div_mod_int(Y),M)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),Y),M)) ) ).
fof(fact_3000_mod__mod__is__mod,axiom,
! [X,M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(div_mod_int(X),M)),M)) ).
fof(fact_3001_zcong__zmod,axiom,
! [A_1,B_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(div_mod_int(A_1),Ma)),hAPP_int_int(div_mod_int(B_1),Ma)),Ma)) ) ).
fof(fact_3002_zdvd__zmod__imp__zdvd,axiom,
! [K_2,M,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(div_mod_int(M),N)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).
fof(fact_3003_zdvd__zmod,axiom,
! [N,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))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,F_3),hAPP_int_int(div_mod_int(M),N))) ) ) ).
fof(fact_3004_Suc__less__SucD,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
fof(fact_3005_Suc__lessD,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
fof(fact_3006_less__SucE,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> M = N ) ) ).
fof(fact_3007_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_3008_Suc__lessI,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( hAPP_nat_nat(suc,M) != N
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N)) ) ) ).
fof(fact_3009_less__SucI,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))) ) ).
fof(fact_3010_less__antisym,axiom,
! [N,M] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M)))
=> M = N ) ) ).
fof(fact_3011_not__less__less__Suc__eq,axiom,
! [Na,Ma] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,Ma)))
<=> Na = Ma ) ) ).
fof(fact_3012_Suc__less__eq,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,Ma)),hAPP_nat_nat(suc,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_3013_less__Suc__eq,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
| Ma = Na ) ) ).
fof(fact_3014_not__less__eq,axiom,
! [Ma,Na] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,Ma))) ) ).
fof(fact_3015_add__Suc__shift,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N)) ).
fof(fact_3016_add__Suc,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
fof(fact_3017_add__Suc__right,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
fof(fact_3018_Suc__n__not__le__n,axiom,
! [N] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),N)) ).
fof(fact_3019_not__less__eq__eq,axiom,
! [Ma,Na] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),Ma)) ) ).
fof(fact_3020_le__Suc__eq,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),hAPP_nat_nat(suc,Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
| Ma = hAPP_nat_nat(suc,Na) ) ) ).
fof(fact_3021_Suc__le__mono,axiom,
! [Na,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),hAPP_nat_nat(suc,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),Ma)) ) ).
fof(fact_3022_le__SucI,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N))) ) ).
fof(fact_3023_le__SucE,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> M = hAPP_nat_nat(suc,N) ) ) ).
fof(fact_3024_Suc__leD,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
fof(fact_3025_Suc__mult__cancel1,axiom,
! [K,Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Ma) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na)
<=> Ma = Na ) ).
fof(fact_3026_diff__Suc__Suc,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).
fof(fact_3027_Suc__diff__diff,axiom,
! [M,N,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)),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)),K_2) ).
fof(fact_3028_exp__mono__eq,axiom,
! [Xa,Na,Ya] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),hAPP_nat_nat(suc,Na)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ya),hAPP_nat_nat(suc,Na))
<=> Xa = Ya ) ).
fof(fact_3029_StandardRes__prop1,axiom,
! [X,M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X),hAPP_int_int(standardRes(M),X)),M)) ).
fof(fact_3030_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_3031_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_3032_zmod__eq__0__iff,axiom,
! [Ma,D_1] :
( is_int(Ma)
=> ( hAPP_int_int(div_mod_int(Ma),D_1) = zero_zero_int
<=> ? [Q_4] :
( is_int(Q_4)
& Ma = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_1),Q_4) ) ) ) ).
fof(fact_3033_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_3034_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y,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),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(div_mod_int(X),Y))) ) ) ).
fof(fact_3035_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_3036_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_3037_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_3038_nat__lt__two__imp__zero__or__one,axiom,
! [X] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))
=> ( X = zero_zero_nat
| X = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_3039_less__Suc__eq__0__disj,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,Na)))
<=> ( 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),Na)) ) ) ) ).
fof(fact_3040_less__Suc0,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,zero_zero_nat)))
<=> Na = zero_zero_nat ) ).
fof(fact_3041_gr0__conv__Suc,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
<=> ? [M_1] : Na = hAPP_nat_nat(suc,M_1) ) ).
fof(fact_3042_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_3043_zmod__eq__dvd__iff,axiom,
! [Xa,Na,Ya] :
( hAPP_int_int(div_mod_int(Xa),Na) = hAPP_int_int(div_mod_int(Ya),Na)
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Na),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya))) ) ).
fof(fact_3044_one__is__add,axiom,
! [Ma,Na] :
( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na)
<=> ( ( Ma = hAPP_nat_nat(suc,zero_zero_nat)
& Na = zero_zero_nat )
| ( Ma = zero_zero_nat
& Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).
fof(fact_3045_add__is__1,axiom,
! [Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na) = hAPP_nat_nat(suc,zero_zero_nat)
<=> ( ( Ma = hAPP_nat_nat(suc,zero_zero_nat)
& Na = zero_zero_nat )
| ( Ma = zero_zero_nat
& Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ).
fof(fact_3046_mult__eq__1__iff,axiom,
! [Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na) = hAPP_nat_nat(suc,zero_zero_nat)
<=> ( Ma = hAPP_nat_nat(suc,zero_zero_nat)
& Na = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_3047_One__nat__def,axiom,
one_one_nat = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_3048_less__iff__Suc__add,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na))
<=> ? [K_1] : Na = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),K_1)) ) ).
fof(fact_3049_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_3050_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_3051_Suc__le__lessD,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
fof(fact_3052_le__less__Suc__eq,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_nat_nat(suc,Ma)))
<=> Na = Ma ) ) ).
fof(fact_3053_Suc__leI,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N)) ) ).
fof(fact_3054_le__imp__less__Suc,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))) ) ).
fof(fact_3055_Suc__le__eq,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),Na))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_3056_less__Suc__eq__le,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(suc,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_3057_less__eq__Suc__le,axiom,
! [Na,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),Ma)) ) ).
fof(fact_3058_Suc__mult__less__cancel1,axiom,
! [K,Ma,Na] :
( 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)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) ) ).
fof(fact_3059_diff__less__Suc,axiom,
! [M,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),hAPP_nat_nat(suc,M))) ).
fof(fact_3060_mult__Suc,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) ).
fof(fact_3061_mult__Suc__right,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),hAPP_nat_nat(suc,N)) = 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)) ).
fof(fact_3062_Suc__mult__le__cancel1,axiom,
! [K,Ma,Na] :
( 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)),Ma)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,K)),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ma),Na)) ) ).
fof(fact_3063_Suc__eq__plus1,axiom,
! [N] : hAPP_nat_nat(suc,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).
fof(fact_3064_Suc__eq__plus1__left,axiom,
! [N] : hAPP_nat_nat(suc,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).
fof(fact_3065_Suc__diff__le,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ).
fof(fact_3066_diff__Suc__1,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,N)),one_one_nat) = N ).
fof(fact_3067_diff__Suc__eq__diff__pred,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,N)) = 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) ).
fof(fact_3068_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_3069_power__Suc__0,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_nat_nat(suc,zero_zero_nat)),N) = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_3070_nat__power__eq__Suc__0__iff,axiom,
! [Xa,Ma] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),Ma) = hAPP_nat_nat(suc,zero_zero_nat)
<=> ( Ma = zero_zero_nat
| Xa = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_3071_exp__mono__lt,axiom,
! [Xa,Na,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),hAPP_nat_nat(suc,Na))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ya),hAPP_nat_nat(suc,Na))))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya)) ) ).
fof(fact_3072_divides__rexp,axiom,
! [N,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Y),hAPP_nat_nat(suc,N)))) ) ).
fof(fact_3073_exp__mono__le,axiom,
! [Xa,Na,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),hAPP_nat_nat(suc,Na))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Ya),hAPP_nat_nat(suc,Na))))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Xa),Ya)) ) ).
fof(fact_3074_StandardRes__ubound,axiom,
! [X,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)),P_3)) ) ).
fof(fact_3075_StandardRes__prop3,axiom,
! [Xa,P_5] :
( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),P_5))
<=> hAPP_int_int(standardRes(P_5),Xa) != zero_zero_int ) ).
fof(fact_3076_StandardRes__eq__zcong,axiom,
! [Ma,Xa] :
( hAPP_int_int(standardRes(Ma),Xa) = zero_zero_int
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),Ma)) ) ).
fof(fact_3077_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_3078_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_3079_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_3080_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_3081_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_3082_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_3083_zcong__zmod__eq,axiom,
! [A_1,B_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma))
<=> hAPP_int_int(div_mod_int(A_1),Ma) = hAPP_int_int(div_mod_int(B_1),Ma) ) ) ).
fof(fact_3084_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_3085_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_3086_zdvd__iff__zmod__eq__0__number__of,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(number_number_of_int,Xa)),hAPP_int_int(number_number_of_int,Ya)))
<=> hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,Ya)),hAPP_int_int(number_number_of_int,Xa)) = zero_zero_int ) ).
fof(fact_3087_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_3088_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_3089_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_3090_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_3091_zmult__div__cancel,axiom,
! [N,M] : hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,M),N)) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,M),hAPP_int_int(div_mod_int(M),N)) ).
fof(fact_3092_zmod__zdiv__equality_H,axiom,
! [M,N] : hAPP_int_int(div_mod_int(M),N) = 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)),N)) ).
fof(fact_3093_one__less__mult,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
=> ( 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))) ) ) ).
fof(fact_3094_n__less__n__mult__m,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
=> ( 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),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),M))) ) ) ).
fof(fact_3095_n__less__m__mult__n,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
=> ( 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),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N))) ) ) ).
fof(fact_3096_mn__eq__m__one,axiom,
! [N,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) = M
=> N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_3097_prod__mn__less__k,axiom,
! [M,K_2,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,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) = K_2
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),K_2)) ) ) ) ) ).
fof(fact_3098_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_3099_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_3100_diff__Suc__less,axiom,
! [I,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,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,I))),N)) ) ).
fof(fact_3101_Suc__pred,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))) = N ) ).
fof(fact_3102_one__le__mult__iff,axiom,
! [Ma,Na] :
( 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),Na)))
<=> ( 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)),Na)) ) ) ).
fof(fact_3103_int__Suc0__eq__1,axiom,
hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_int ).
fof(fact_3104_nat__1,axiom,
hAPP_int_nat(nat,one_one_int) = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_3105_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_3106_zless__iff__Suc__zadd,axiom,
! [Wa,Z_2] :
( is_int(Z_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Wa),Z_2))
<=> ? [N_1] : Z_2 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Wa),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))) ) ) ).
fof(fact_3107_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_3108_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_3109_nat__one__le__power,axiom,
! [N,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))) ) ).
fof(fact_3110_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_3111_xzgcda__linear__aux2,axiom,
! [R_2,S_1,T_1,R_4,S_3,M,T_2,N] :
( R_4 = 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))
=> ( R_2 = 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))
=> hAPP_int_int(div_mod_int(R_4),R_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(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_4),R_2)),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_4),R_2)),T_1))),N)) ) ) ).
fof(fact_3112_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_3113_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_3114_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_3115_Suc3__eq__add__3,axiom,
! [N] : hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N))) = 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) ).
fof(fact_3116_Suc__diff__1,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ).
fof(fact_3117_Suc__pred_H,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ).
fof(fact_3118_add__eq__if,axiom,
! [N,M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = N )
& ( M != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = 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)) ) ) ).
fof(fact_3119_StandardRes__lbound,axiom,
! [X,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))) ) ).
fof(fact_3120_div__if,axiom,
! [M,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,M),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = zero_zero_nat )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = 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)),N)) ) ) ) ).
fof(fact_3121_div__geq,axiom,
! [M,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,M),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = 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)),N)) ) ) ).
fof(fact_3122_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_3123_zmod__number__of__Bit0,axiom,
! [V_1,W] : hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,V_1))),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_1)),hAPP_int_int(number_number_of_int,W))) ).
fof(fact_3124_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [Y,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),Y))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(nat_tsub(X),Y))) ) ) ).
fof(fact_3125_split__zmod,axiom,
! [P_1,Na,K] :
( ( is_int(Na)
& is_int(K) )
=> ( hBOOL(hAPP_int_bool(P_1,hAPP_int_int(div_mod_int(Na),K)))
<=> ( ( K = zero_zero_int
=> hBOOL(hAPP_int_bool(P_1,Na)) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
=> ! [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))
& Na = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(P_1,J_1)) ) ) )
& ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),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),J_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
& Na = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(P_1,J_1)) ) ) ) ) ) ) ).
fof(fact_3126_zmult2__lemma__aux3,axiom,
! [Q_1,B_2,R_2,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),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_1),C_1))),R_2))) ) ) ) ).
fof(fact_3127_zmult2__lemma__aux4,axiom,
! [Q_1,B_2,R_2,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,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_1),C_1))),R_2)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_2),C_1))) ) ) ) ).
fof(fact_3128_zmult2__lemma__aux1,axiom,
! [Q_1,B_2,R_2,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),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_1),C_1))),R_2))) ) ) ) ).
fof(fact_3129_zmult2__lemma__aux2,axiom,
! [Q_1,B_2,R_2,C_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),C_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),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_1),C_1))),R_2)),zero_zero_int)) ) ) ) ).
fof(fact_3130_divmod__int__rel__mod__eq,axiom,
! [A_6,B_7,Q_6,Y] :
( ( is_int(B_7)
& is_int(Y) )
=> ( A_6 = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_7),Q_6)),Y)
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Y))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Y),B_7)) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_7))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_7),Y))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),zero_zero_int)) ) ) )
=> ( B_7 != zero_zero_int
=> hAPP_int_int(div_mod_int(A_6),B_7) = Y ) ) ) ) ).
fof(fact_3131_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_3132_tsub__eq,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> hAPP_int_int(nat_tsub(X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) ) ).
fof(fact_3133_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X,Y] : hAPP_int_int(nat_tsub(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,X),Y)) ).
fof(fact_3134_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_3135_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_3136_add__2__eq__Suc,axiom,
! [N] : 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) = hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)) ).
fof(fact_3137_add__2__eq__Suc_H,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),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)) ).
fof(fact_3138_Suc__diff__eq__diff__pred,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,pls)),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit1,pls)))) ) ).
fof(fact_3139_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_3140_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_3141_expand__Suc,axiom,
! [V_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_int_nat(number_number_of_nat,V_1)))
=> hAPP_int_nat(number_number_of_nat,V_1) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_1)),one_one_nat)) ) ).
fof(fact_3142_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_3143_Suc__div__eq__add3__div__number__of,axiom,
! [M,V_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)))),hAPP_int_nat(number_number_of_nat,V_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)),hAPP_int_nat(number_number_of_nat,V_1)) ).
fof(fact_3144_Suc__div__eq__add3__div,axiom,
! [M,N] : 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) = 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) ).
fof(fact_3145_div__Suc__eq__div__add3,axiom,
! [M,N] : 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)))) = 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)) ).
fof(fact_3146_split__div__lemma,axiom,
! [Q_2,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),Q_2)),Ma))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),hAPP_nat_nat(suc,Q_2)))) )
<=> Q_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),Na) ) ) ).
fof(fact_3147_split__div_H,axiom,
! [P_1,Ma,Na] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,Ma),Na)))
<=> ( ( Na = 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,Na),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,Na),hAPP_nat_nat(suc,Q_4))))
& hBOOL(hAPP_nat_bool(P_1,Q_4)) ) ) ) ).
fof(fact_3148_le__div__geq,axiom,
! [M,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,N),M))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = 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)),N)) ) ) ).
fof(fact_3149_neq__one__mod__two,axiom,
! [Xa] :
( hAPP_int_int(div_mod_int(Xa),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(Xa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = one_one_int ) ).
fof(fact_3150_split__pos__lemma,axiom,
! [P_1,Na,K] :
( is_int(Na)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),K))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Na),K)),hAPP_int_int(div_mod_int(Na),K)))
<=> ! [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))
& Na = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) ) ) ) ) ).
fof(fact_3151_split__neg__lemma,axiom,
! [P_1,Na,K] :
( is_int(Na)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Na),K)),hAPP_int_int(div_mod_int(Na),K)))
<=> ! [I_2,J_1] :
( ( is_int(I_2)
& is_int(J_1) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,K),J_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,J_1),zero_zero_int))
& Na = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),I_2)),J_1) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(P_1,I_2),J_1)) ) ) ) ) ) ).
fof(fact_3152_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_3153_less__2__cases,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
=> ( N = zero_zero_nat
| N = hAPP_nat_nat(suc,zero_zero_nat) ) ) ).
fof(fact_3154_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_3155_tsub__def,axiom,
! [Y,X] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> hAPP_int_int(nat_tsub(X),Y) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Y),X))
=> hAPP_int_int(nat_tsub(X),Y) = zero_zero_int ) ) ).
fof(fact_3156_lemma__Suc__Suc__4n__diff__2,axiom,
! [N] :
( N != 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)),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) ) ).
fof(fact_3157_Suc__Suc__mult__two__diff__two,axiom,
! [N] :
( N != 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)),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) ) ).
fof(fact_3158_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_3159_one__div__nat__number__of,axiom,
! [V_2] : 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_2)) = 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_2))) ).
fof(fact_3160_Suc__mult__two__diff__one,axiom,
! [N] :
( N != 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)),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) ) ).
fof(fact_3161_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_3162_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_3163_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_3164_SRStar__mult__prop2,axiom,
! [Xa,A_1,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_1),zero_zero_int),P_5))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),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_1),hAPP_int_int(multInv(P_5),Xa)))),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ) ).
fof(fact_3165_realpow__pos__nth2,axiom,
! [N,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)) = A_3 ) ) ).
fof(fact_3166_zmod__eq__0D,axiom,
! [M_5,D_4] :
( is_int(M_5)
=> ( hAPP_int_int(div_mod_int(M_5),D_4) = zero_zero_int
=> ? [Q_4] :
( is_int(Q_4)
& M_5 = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,D_4),Q_4) ) ) ) ).
fof(fact_3167_dvd_Olift__Suc__mono__le,axiom,
! [Na,N_6,F] :
( ! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,N_6))) ) ) ).
fof(fact_3168_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_3169_less__imp__Suc__add,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ? [K_1] : N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_1)) ) ).
fof(fact_3170_mod__Suc__eq__Suc__mod,axiom,
! [M,N] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N))),N) ).
fof(fact_3171_mod__less,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hAPP_nat_nat(div_mod_nat(M),N) = M ) ).
fof(fact_3172_mod__less__eq__dividend,axiom,
! [M,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(div_mod_nat(M),N)),M)) ).
fof(fact_3173_mod__mult__distrib2,axiom,
! [K_2,M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),hAPP_nat_nat(div_mod_nat(M),N)) = 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)) ).
fof(fact_3174_mod__mult__distrib,axiom,
! [M,N,K_2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(div_mod_nat(M),N)),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),K_2)) ).
fof(fact_3175_StandardRes__SRStar__prop3,axiom,
! [Xa,P_5] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(sRStar,P_5)))
=> hAPP_int_int(standardRes(P_5),Xa) = Xa ) ) ).
fof(fact_3176_mod__1,axiom,
! [M] : hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).
fof(fact_3177_mod__Suc,axiom,
! [M,N] :
( ( hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) = N
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N) = zero_zero_nat )
& ( hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) != N
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(div_mod_nat(M),N)) ) ) ).
fof(fact_3178_mod__less__divisor,axiom,
! [M,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,hAPP_nat_nat(div_mod_nat(M),N)),N)) ) ).
fof(fact_3179_mod__eq__0__iff,axiom,
! [Ma,D_1] :
( hAPP_nat_nat(div_mod_nat(Ma),D_1) = zero_zero_nat
<=> ? [Q_4] : Ma = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D_1),Q_4) ) ).
fof(fact_3180_mod__geq,axiom,
! [M,N] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hAPP_nat_nat(div_mod_nat(M),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ).
fof(fact_3181_mod__if,axiom,
! [M,N] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hAPP_nat_nat(div_mod_nat(M),N) = M )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hAPP_nat_nat(div_mod_nat(M),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).
fof(fact_3182_mod__mult__self3,axiom,
! [K_2,N,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)),M)),N) = hAPP_nat_nat(div_mod_nat(M),N) ).
fof(fact_3183_le__mod__geq,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hAPP_nat_nat(div_mod_nat(M),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ).
fof(fact_3184_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_3185_Divides_Otransfer__int__nat__functions_I2_J,axiom,
! [X,Y] : hAPP_int_int(div_mod_int(hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(div_mod_nat(X),Y)) ).
fof(fact_3186_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_3187_SRStar__SR__prop,axiom,
! [Xa,P_5] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(sRStar,P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(sr,P_5))) ) ).
fof(fact_3188_mod__le__divisor,axiom,
! [M,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(div_mod_nat(M),N)),N)) ) ).
fof(fact_3189_mod__mult__self4,axiom,
! [K_2,N,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)),M))),N) = hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,M)),N) ).
fof(fact_3190_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_3191_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_3192_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_3193_div__mod__equality_H,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N)) ).
fof(fact_3194_mult__div__cancel,axiom,
! [N,M] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(div_mod_nat(M),N)) ).
fof(fact_3195_Divides_Omod__div__equality_H,axiom,
! [M,N] : hAPP_nat_nat(div_mod_nat(M),N) = 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)),N)) ).
fof(fact_3196_StandardRes__SRStar__prop1a,axiom,
! [Xa,P_5] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),hAPP_i1948725293t_bool(sRStar,P_5)))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),P_5)) ) ).
fof(fact_3197_split__mod,axiom,
! [P_1,Na,K] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(div_mod_nat(Na),K)))
<=> ( ( K = zero_zero_nat
=> hBOOL(hAPP_nat_bool(P_1,Na)) )
& ( K != zero_zero_nat
=> ! [I_2,J_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),K))
=> ( Na = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K),I_2)),J_1)
=> hBOOL(hAPP_nat_bool(P_1,J_1)) ) ) ) ) ) ).
fof(fact_3198_mod__lemma,axiom,
! [Q_1,R_2,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_2),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_1),C_1))),R_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_2),C_1))) ) ) ).
fof(fact_3199_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_3200_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_3201_nat__mod__distrib,axiom,
! [Y,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),Y))
=> hAPP_int_nat(nat,hAPP_int_int(div_mod_int(X),Y)) = hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) ) ) ).
fof(fact_3202_Divides_Otransfer__nat__int__functions_I2_J,axiom,
! [Y,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),Y))
=> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(X),Y)) ) ) ).
fof(fact_3203_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_3204_mod__Suc__eq__mod__add3,axiom,
! [M,N] : hAPP_nat_nat(div_mod_nat(M),hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,N)))) = 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)) ).
fof(fact_3205_Suc__mod__eq__add3__mod,axiom,
! [M,N] : hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,hAPP_nat_nat(suc,hAPP_nat_nat(suc,M)))),N) = 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) ).
fof(fact_3206_Suc__mod__eq__add3__mod__number__of,axiom,
! [M,V_1] : 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_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)),hAPP_int_nat(number_number_of_nat,V_1)) ).
fof(fact_3207_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_3208_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_3209_StandardRes__SRStar__prop4,axiom,
! [Xa,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,Xa),hAPP_i1948725293t_bool(sRStar,P_5)))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(standardRes(P_5),Xa)),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ).
fof(fact_3210_StandardRes__SRStar__prop1,axiom,
! [Xa,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),Xa)),hAPP_i1948725293t_bool(sRStar,P_5)))
<=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(Xa),zero_zero_int),P_5)) ) ) ).
fof(fact_3211_SRStar__mult__prop1,axiom,
! [Ya,Xa,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,Xa),hAPP_i1948725293t_bool(sRStar,P_5)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),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,Xa),Ya))),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ) ).
fof(fact_3212_StandardRes__SRStar__prop2,axiom,
! [Xa,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,Xa),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),Xa))),hAPP_i1948725293t_bool(sRStar,P_5))) ) ) ) ).
fof(fact_3213_nat__mod__eq__lemma,axiom,
! [X,N,Y] :
( hAPP_nat_nat(div_mod_nat(X),N) = hAPP_nat_nat(div_mod_nat(Y),N)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y),X))
=> ? [Q_4] : X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),Q_4)) ) ) ).
fof(fact_3214_dvd_Olift__Suc__mono__less__iff,axiom,
! [Na,Ma,F] :
( ! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1))))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1))),hAPP_nat_nat(F,N_1))) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,Ma)))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Ma)),hAPP_nat_nat(F,Na))) )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) ) ) ).
fof(fact_3215_dvd_Olift__Suc__mono__less,axiom,
! [Na,N_6,F] :
( ! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1))))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,N_1))),hAPP_nat_nat(F,N_1))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),N_6))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,N_6)))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(F,N_6)),hAPP_nat_nat(F,Na))) ) ) ) ).
fof(fact_3216_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_3217_gr0__implies__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_3218_mod__eqD,axiom,
! [M,D_3,R_2] :
( hAPP_nat_nat(div_mod_nat(M),D_3) = R_2
=> ? [Q_4] : M = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,R_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Q_4),D_3)) ) ).
fof(fact_3219_nat__mod__eq__iff,axiom,
! [Xa,Na,Ya] :
( hAPP_nat_nat(div_mod_nat(Xa),Na) = hAPP_nat_nat(div_mod_nat(Ya),Na)
<=> ? [Q1,Q2] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Xa),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),Q1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ya),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),Q2)) ) ).
fof(fact_3220_mod__eq__0D,axiom,
! [M_5,D_4] :
( hAPP_nat_nat(div_mod_nat(M_5),D_4) = zero_zero_nat
=> ? [Q_4] : M_5 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,D_4),Q_4) ) ).
fof(fact_3221_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_3222_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_3223_zmult2__lemma,axiom,
! [C_1,A_3,B_2,Q_1,R_2] :
( 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_1),R_2)))
=> ( 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_1),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_1),C_1))),R_2)))) ) ) ) ) ).
fof(fact_3224_one__mod__nat__number__of,axiom,
! [V_2] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_nat_nat(suc,zero_zero_nat) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_2)))
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(suc,zero_zero_nat)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(one_one_int),hAPP_int_int(number_number_of_int,V_2))) ) ) ).
fof(fact_3225_two__times__odd__div__two__plus__one,axiom,
! [X] :
( is_int(X)
=> ( ~ hBOOL(hAPP_int_bool(even_odd_even_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(bit1,pls)))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))),one_one_int) = X ) ) ).
fof(fact_3226_lemma__interval,axiom,
! [B_2,A_3,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B_2))
=> ? [D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
& ! [Y_1] :
( 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),Y_1))),D_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_3),Y_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),B_2)) ) ) ) ) ) ).
fof(fact_3227_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_3228_not__neg__0,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,zero_zero_int)) ).
fof(fact_3229_even__zero__int,axiom,
hBOOL(hAPP_int_bool(even_odd_even_int,zero_zero_int)) ).
fof(fact_3230_not__neg__1,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,one_one_int)) ).
fof(fact_3231_odd__one__int,axiom,
~ hBOOL(hAPP_int_bool(even_odd_even_int,one_one_int)) ).
fof(fact_3232_Parity_Oodd__times__odd,axiom,
! [Y,X] :
( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
=> ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y))
=> ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).
fof(fact_3233_anything__times__even,axiom,
! [X,Y] :
( hBOOL(hAPP_int_bool(even_odd_even_int,Y))
=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ).
fof(fact_3234_even__times__anything,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(even_odd_even_int,X))
=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ).
fof(fact_3235_Parity_Oeven__product,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya)))
<=> ( hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
| hBOOL(hAPP_int_bool(even_odd_even_int,Ya)) ) ) ).
fof(fact_3236_Parity_Oodd__plus__odd,axiom,
! [Y,X] :
( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
=> ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y))
=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).
fof(fact_3237_odd__plus__even,axiom,
! [Y,X] :
( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
=> ( hBOOL(hAPP_int_bool(even_odd_even_int,Y))
=> ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).
fof(fact_3238_Parity_Oeven__plus__odd,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(even_odd_even_int,X))
=> ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Y))
=> ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).
fof(fact_3239_Parity_Oeven__plus__even,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(even_odd_even_int,X))
=> ( hBOOL(hAPP_int_bool(even_odd_even_int,Y))
=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).
fof(fact_3240_even__sum,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),Ya)))
<=> ( ( hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
& hBOOL(hAPP_int_bool(even_odd_even_int,Ya)) )
| ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
& ~ hBOOL(hAPP_int_bool(even_odd_even_int,Ya)) ) ) ) ).
fof(fact_3241_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_3242_not__neg__int,axiom,
! [N] : ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_nat_int(semiri1621563631at_int,N))) ).
fof(fact_3243_even__difference,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)))
<=> ( ( hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
& hBOOL(hAPP_int_bool(even_odd_even_int,Ya)) )
| ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
& ~ hBOOL(hAPP_int_bool(even_odd_even_int,Ya)) ) ) ) ).
fof(fact_3244_odd__pow,axiom,
! [N,X] :
( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
=> ~ hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) ) ).
fof(fact_3245_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_3246_neg__number__of__Bit1,axiom,
! [Wa] :
( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit1,Wa))))
<=> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Wa))) ) ).
fof(fact_3247_not__neg__eq__ge__0,axiom,
! [Xa] :
( ~ hBOOL(hAPP_int_bool(nat_neg,Xa))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) ) ).
fof(fact_3248_not__neg__number__of__Pls,axiom,
~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,pls))) ).
fof(fact_3249_neg__number__of__Bit0,axiom,
! [Wa] :
( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,Wa))))
<=> hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Wa))) ) ).
fof(fact_3250_neg__nat,axiom,
! [Z_1] :
( hBOOL(hAPP_int_bool(nat_neg,Z_1))
=> hAPP_int_nat(nat,Z_1) = zero_zero_nat ) ).
fof(fact_3251_even__power,axiom,
! [Xa,Na] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Xa),Na)))
<=> ( hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
& Na != zero_zero_nat ) ) ).
fof(fact_3252_neg__number__of__Min,axiom,
hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,min))) ).
fof(fact_3253_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_3254_unique__remainder,axiom,
! [Q_5,R_4,A_3,B_2,Q_1,R_2] :
( ( is_int(R_4)
& is_int(B_2)
& is_int(R_2) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_5),R_4)))
=> ( B_2 != zero_zero_int
=> R_2 = R_4 ) ) ) ) ).
fof(fact_3255_unique__quotient,axiom,
! [Q_5,R_4,A_3,B_2,Q_1,R_2] :
( ( is_int(Q_5)
& is_int(B_2)
& is_int(Q_1) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)))
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_5),R_4)))
=> ( B_2 != zero_zero_int
=> Q_1 = Q_5 ) ) ) ) ).
fof(fact_3256_self__remainder,axiom,
! [A_3,Q_1,R_2] :
( ( is_int(A_3)
& is_int(R_2) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)))
=> ( A_3 != zero_zero_int
=> R_2 = zero_zero_int ) ) ) ).
fof(fact_3257_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_3258_neg__imp__number__of__eq__0,axiom,
! [V_1] :
( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_int_nat(number_number_of_nat,V_1) = zero_zero_nat ) ).
fof(fact_3259_self__quotient,axiom,
! [A_3,Q_1,R_2] :
( ( is_int(A_3)
& is_int(Q_1) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,A_3),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)))
=> ( A_3 != zero_zero_int
=> Q_1 = one_one_int ) ) ) ).
fof(fact_3260_divmod__int__rel__div,axiom,
! [A_3,B_2,Q_1,R_2] :
( ( is_int(B_2)
& is_int(Q_1) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)))
=> ( B_2 != zero_zero_int
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_3),B_2) = Q_1 ) ) ) ).
fof(fact_3261_divmod__int__rel__mod,axiom,
! [A_3,B_2,Q_1,R_2] :
( ( is_int(B_2)
& is_int(R_2) )
=> ( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_3,B_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)))
=> ( B_2 != zero_zero_int
=> hAPP_int_int(div_mod_int(A_3),B_2) = R_2 ) ) ) ).
fof(fact_3262_eq__nat__number__of,axiom,
! [Va,V] :
( ( is_int(Va)
& is_int(V) )
=> ( hAPP_int_nat(number_number_of_nat,Va) = hAPP_int_nat(number_number_of_nat,V)
<=> ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Va)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,V)),zero_zero_int)) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Va)))
=> ( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V)))
=> hAPP_int_int(number_number_of_int,Va) = zero_zero_int )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V)))
=> Va = V ) ) ) ) ) ) ).
fof(fact_3263_nat__number__of__add__left,axiom,
! [V_2,K_2,V_1] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),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_1)))
=> ( ( 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_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),K_2)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),K_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_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_2)),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_1),V_2))),K_2) ) ) ) ) ).
fof(fact_3264_int__nat__number__of,axiom,
! [V_1] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_1)) = zero_zero_int )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_int(number_number_of_int,V_1) ) ) ).
fof(fact_3265_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_3266_even__equiv__def,axiom,
! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
<=> ? [Y_1] :
( is_int(Y_1)
& Xa = 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_1) ) ) ) ).
fof(fact_3267_int__even__iff__2__dvd,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
<=> 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)))),Xa)) ) ).
fof(fact_3268_Suc__nat__number__of,axiom,
! [V_1] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_1)) = one_one_nat )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(suc,hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_1)) ) ) ).
fof(fact_3269_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_3270_even__def,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
<=> hAPP_int_int(div_mod_int(Xa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int ) ).
fof(fact_3271_two__times__even__div__two,axiom,
! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(even_odd_even_int,X))
=> 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),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X ) ) ).
fof(fact_3272_power__nat__number__of,axiom,
! [N,V_1] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_1)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_1)),N) = hAPP_int_nat(nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,V_1)),N)) ) ) ).
fof(fact_3273_power__nat__number__of__number__of,axiom,
! [W,V_1] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_1)),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_1)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,hAPP_int_nat(number_number_of_nat,V_1)),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_1)),hAPP_int_nat(number_number_of_nat,W))) ) ) ).
fof(fact_3274_div__nat__number__of,axiom,
! [V_2,V_1] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(number_number_of_int,V_1)),hAPP_int_int(number_number_of_int,V_2))) ) ) ).
fof(fact_3275_mod__nat__number__of,axiom,
! [V_2,V_1] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(div_mod_nat(hAPP_int_nat(number_number_of_nat,V_1)),hAPP_int_nat(number_number_of_nat,V_2)) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> ( ( 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_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(number_number_of_nat,V_1) )
& ( ~ 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_1)),hAPP_int_nat(number_number_of_nat,V_2)) = hAPP_int_nat(nat,hAPP_int_int(div_mod_int(hAPP_int_int(number_number_of_int,V_1)),hAPP_int_int(number_number_of_int,V_2))) ) ) ) ) ).
fof(fact_3276_Suc__nat__number__of__add,axiom,
! [N,V_1] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,V_1)),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(number_number_of_nat,hAPP_int_int(succ,V_1))),N) ) ) ).
fof(fact_3277_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_3278_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_3279_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_3280_odd__equiv__def,axiom,
! [Xa] :
( is_int(Xa)
=> ( ~ hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
<=> ? [Y_1] :
( is_int(Y_1)
& Xa = 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_1)),one_one_int) ) ) ) ).
fof(fact_3281_divmod__int__relI,axiom,
! [A_3,B_2,Q_1,R_2] :
( 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_1)),R_2)
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),R_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,R_2),B_2)) ) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_2),R_2))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_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,Q_1),R_2))) ) ) ).
fof(fact_3282_odd__plus__one__div__two,axiom,
! [X] :
( ~ hBOOL(hAPP_int_bool(even_odd_even_int,X))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),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),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),one_one_int) ) ).
fof(fact_3283_even__plus__one__div__two,axiom,
! [X] :
( hBOOL(hAPP_int_bool(even_odd_even_int,X))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),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),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ).
fof(fact_3284_zmult1__lemma,axiom,
! [A_3,B_2,C_1,Q_1,R_2] :
( 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_1),R_2)))
=> ( 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_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),R_2)),C_1))),hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),R_2)),C_1)))) ) ) ) ).
fof(fact_3285_negDivAlg__eqn__number__of,axiom,
! [W,V_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_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,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_1))))
=> negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_1)) = 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_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,hAPP_int_int(number_number_of_int,W)),hAPP_int_int(number_number_of_int,V_1))))
=> negDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_1)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_1)),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_1)))) ) ) ) ).
fof(fact_3286_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_3287_lemma__interval__lt,axiom,
! [B_2,A_3,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),B_2))
=> ? [D_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_2))
& ! [Y_1] :
( 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),Y_1))),D_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),Y_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),B_2)) ) ) ) ) ) ).
fof(fact_3288_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_3289_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_3290_posDivAlg__eqn__number__of,axiom,
! [W,V_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(number_number_of_int,V_1)))
=> ( ( 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_1)))
=> posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_1)) = 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_1)))
=> posDivAlg(hAPP_int_int(number_number_of_int,W),hAPP_int_int(number_number_of_int,V_1)) = hAPP_P1975530577nt_int(adjust(hAPP_int_int(number_number_of_int,V_1)),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_1)))) ) ) ) ).
fof(fact_3291_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_3292_lemma__f__mono__add,axiom,
! [Ma,No,F] :
( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_1)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_1))))
=> 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_3293_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_3294_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_3295_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_3296_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_3297_natceiling__add__number__of,axiom,
! [X,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))
=> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(number267125858f_real,N))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),hAPP_int_nat(number_number_of_nat,N)) ) ) ).
fof(fact_3298_negDivAlg_Opsimps,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(negDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_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,A_1),B_1)))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) )
=> negDivAlg(A_1,B_1) = 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_1),B_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,A_1),B_1)))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) )
=> negDivAlg(A_1,B_1) = hAPP_P1975530577nt_int(adjust(B_1),negDivAlg(A_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)))),B_1))) ) ) ) ).
fof(fact_3299_natfloor__add__number__of,axiom,
! [X,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))
=> natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(number267125858f_real,N))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),hAPP_int_nat(number_number_of_nat,N)) ) ) ).
fof(fact_3300_natceiling__le__eq__number__of,axiom,
! [Xa,Na] :
( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Na)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(Xa)),hAPP_int_nat(number_number_of_nat,Na)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),hAPP_int_real(number267125858f_real,Na))) ) ) ) ).
fof(fact_3301_posDivAlg_Opsimps,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(posDivAlg_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_1)))
=> ( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) )
=> posDivAlg(A_1,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),A_1) )
& ( ~ ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1))
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),zero_zero_int)) )
=> posDivAlg(A_1,B_1) = hAPP_P1975530577nt_int(adjust(B_1),posDivAlg(A_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)))),B_1))) ) ) ) ).
fof(fact_3302_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_3303_natfloor__zero,axiom,
natfloor(zero_zero_real) = zero_zero_nat ).
fof(fact_3304_zero__le__natfloor,axiom,
! [X] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natfloor(X))) ).
fof(fact_3305_natfloor__mono,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natfloor(X)),natfloor(Y))) ) ).
fof(fact_3306_natfloor__number__of__eq,axiom,
! [N] : natfloor(hAPP_int_real(number267125858f_real,N)) = hAPP_int_nat(number_number_of_nat,N) ).
fof(fact_3307_natfloor__one,axiom,
natfloor(one_one_real) = one_one_nat ).
fof(fact_3308_natceiling__zero,axiom,
natceiling(zero_zero_real) = zero_zero_nat ).
fof(fact_3309_zero__le__natceiling,axiom,
! [X] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),natceiling(X))) ).
fof(fact_3310_natceiling__mono,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),natceiling(Y))) ) ).
fof(fact_3311_natceiling__number__of__eq,axiom,
! [N] : natceiling(hAPP_int_real(number267125858f_real,N)) = hAPP_int_nat(number_number_of_nat,N) ).
fof(fact_3312_natceiling__one,axiom,
natceiling(one_one_real) = one_one_nat ).
fof(fact_3313_natfloor__neg,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
=> natfloor(X) = zero_zero_nat ) ).
fof(fact_3314_natceiling__neg,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
=> natceiling(X) = zero_zero_nat ) ).
fof(fact_3315_le__natfloor__eq__one,axiom,
! [Xa] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),natfloor(Xa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Xa)) ) ).
fof(fact_3316_natceiling__le__eq__one,axiom,
! [Xa] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(Xa)),one_one_nat))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),one_one_real)) ) ).
fof(fact_3317_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_3318_natfloor__add__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),one_one_nat) ) ).
fof(fact_3319_natceiling__add__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),one_one_nat) ) ).
fof(fact_3320_le__natfloor__eq__number__of,axiom,
! [Xa,Na] :
( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,Na)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,Na)),natfloor(Xa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Na)),Xa)) ) ) ) ).
fof(fact_3321_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_3322_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_3323_MultInvPair__card__two,axiom,
! [J,A_1,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_1),zero_zero_int),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A_1))
=> ( ~ 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_1,P_5),J)) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ) ) ) ) ) ).
fof(fact_3324_SetS__elems__card,axiom,
! [A_1,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_1),zero_zero_int),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A_1))
=> ! [X_1] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),setS(A_1,P_5)))
=> hAPP_f957591787ol_nat(finite_card_int,X_1) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))) ) ) ) ) ) ).
fof(fact_3325_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_3326_int__diff__cases,axiom,
! [Z_1] :
( is_int(Z_1)
=> ~ ! [M_1,N_1] : 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_1)) ) ).
fof(fact_3327_code__numeral__zero__minus__one,axiom,
hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,zero_z126310315umeral),one_on1645066479umeral) = zero_z126310315umeral ).
fof(fact_3328_zero__code__numeral__code,axiom,
zero_z126310315umeral = hAPP_i769753017umeral(number1443263063umeral,pls) ).
fof(fact_3329_one__code__numeral__code,axiom,
one_on1645066479umeral = hAPP_i769753017umeral(number1443263063umeral,hAPP_int_int(bit1,pls)) ).
fof(fact_3330_SetS__card,axiom,
! [A_1,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_1),zero_zero_int),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A_1))
=> hAPP_nat_int(semiri1621563631at_int,hAPP_f521865025ol_nat(finite1876863882t_bool,setS(A_1,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_3331_MultInvPair__prop2,axiom,
! [A_1,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_1),zero_zero_int),P_5))
=> comple219730294t_bool(setS(A_1,P_5)) = hAPP_i1948725293t_bool(sRStar,P_5) ) ) ) ).
fof(fact_3332_EvenOdd_Oneg__one__even__power,axiom,
! [Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Xa)) = one_one_int ) ) ).
fof(fact_3333_le__Suc__ex,axiom,
! [K_2,L_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),L_2))
=> ? [N_1] : L_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_2),N_1) ) ).
fof(fact_3334_zero__induct__lemma,axiom,
! [I_1,P_1,K] :
( hBOOL(hAPP_nat_bool(P_1,K))
=> ( ! [N_1] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_1)))
=> hBOOL(hAPP_nat_bool(P_1,N_1)) )
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K),I_1))) ) ) ).
fof(fact_3335_abs__ln__one__plus__x__minus__x__bound__nonneg,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),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))),X))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3336_one__not__even,axiom,
~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,one_one_int),zEven)) ).
fof(fact_3337_even__times__either,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya)),zEven)) ) ).
fof(fact_3338_EvenOdd_Oeven__product,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya)),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
| hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven)) ) ) ).
fof(fact_3339_EvenOdd_Oeven__plus__even,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),Ya)),zEven)) ) ) ).
fof(fact_3340_even__diff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)),zEven))
<=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven)) ) ) ).
fof(fact_3341_even__minus__even,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)),zEven)) ) ) ).
fof(fact_3342_not__odd__impl__even,axiom,
! [Xa] :
( ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) ) ).
fof(fact_3343_even__odd__disj,axiom,
! [Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
| hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) ) ).
fof(fact_3344_odd__iff__not__even,axiom,
! [Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
<=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) ) ).
fof(fact_3345_even__odd__conj,axiom,
! [Xa] :
~ ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) ) ).
fof(fact_3346_ln__less__cancel__iff,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,Xa)),hAPP_real_real(ln,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) ) ) ) ).
fof(fact_3347_ln__inj__iff,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya))
=> ( hAPP_real_real(ln,Xa) = hAPP_real_real(ln,Ya)
<=> Xa = Ya ) ) ) ).
fof(fact_3348_ln__less__self,axiom,
! [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,hAPP_real_real(ln,X)),X)) ) ).
fof(fact_3349_ln__one,axiom,
hAPP_real_real(ln,one_one_real) = zero_zero_real ).
fof(fact_3350_even__plus__odd__prop2,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1)),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),zEven))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),zOdd)) ) ) ).
fof(fact_3351_even__plus__odd__prop1,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1)),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),zEven)) ) ) ).
fof(fact_3352_IntNatAux_Oeven__plus__odd,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1)),zOdd)) ) ) ).
fof(fact_3353_IntNatAux_Oodd__plus__odd,axiom,
! [B_1,A_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),B_1)),zEven)) ) ) ).
fof(fact_3354_even__minus__odd,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)),zOdd)) ) ) ).
fof(fact_3355_odd__minus__even,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)),zOdd)) ) ) ).
fof(fact_3356_odd__minus__odd,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),Ya)),zEven)) ) ) ).
fof(fact_3357_ln__eq__minus__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> ( hAPP_real_real(ln,X) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real)
=> X = one_one_real ) ) ).
fof(fact_3358_ln__le__cancel__iff,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,Xa)),hAPP_real_real(ln,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ) ) ).
fof(fact_3359_ln__ge__zero,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X))) ) ).
fof(fact_3360_ln__gt__zero__imp__gt__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,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,one_one_real),X)) ) ) ).
fof(fact_3361_ln__less__zero,axiom,
! [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),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,X)),zero_zero_real)) ) ) ).
fof(fact_3362_ln__less__zero__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(ln,Xa)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real)) ) ) ).
fof(fact_3363_ln__eq__zero__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( hAPP_real_real(ln,Xa) = zero_zero_real
<=> Xa = one_one_real ) ) ).
fof(fact_3364_ln__gt__zero__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,Xa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Xa)) ) ) ).
fof(fact_3365_ln__gt__zero,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(ln,X))) ) ).
fof(fact_3366_ln__le__minus__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real))) ) ).
fof(fact_3367_power__preserves__even,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Xa),Na)),zEven))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) ) ) ).
fof(fact_3368_odd__minus__one__even,axiom,
! [Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa),one_one_int)),zEven)) ) ).
fof(fact_3369_ln__ge__zero__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,Xa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Xa)) ) ) ).
fof(fact_3370_ln__ge__zero__imp__ge__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(ln,X)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X)) ) ) ).
fof(fact_3371_ln__mult,axiom,
! [Y,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,zero_zero_real),Y))
=> hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,Y)) ) ) ).
fof(fact_3372_ln__add__one__self__le__self,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,hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X))),X)) ) ).
fof(fact_3373_ln__add__one__self__le__self2,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),X))
=> 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))),X)) ) ).
fof(fact_3374_zEvenI,axiom,
! [Xa,K] :
( Xa = 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)
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven)) ) ).
fof(fact_3375_even__div__2__prop1,axiom,
! [Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
=> hAPP_int_int(div_mod_int(Xa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = zero_zero_int ) ).
fof(fact_3376_even__div__2__l,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,Xa),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,Ya),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3377_even__prod__div__2,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Xa),Ya)),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,Xa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Ya) ) ).
fof(fact_3378_even__div__2__prop2,axiom,
! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),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,Xa),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = Xa ) ) ).
fof(fact_3379_even__sum__div__2,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven))
=> hAPP_int_int(hAPP_int_fun_int_int(div_div_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa),Ya)),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,Xa),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,Ya),hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ) ).
fof(fact_3380_neg__one__power__parity,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ya),zEven)) )
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Xa)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(number_number_of_int,min)),hAPP_int_nat(nat,Ya)) ) ) ) ).
fof(fact_3381_ln__one__plus__pos__lower__bound,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),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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)))) ) ) ).
fof(fact_3382_zEvenE,axiom,
! [Xa] :
( is_int(Xa)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
=> ~ ! [K_1] :
( is_int(K_1)
=> Xa != 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) ) ) ) ).
fof(fact_3383_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),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))),X))),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_3384_nat__of__aux__code,axiom,
! [N,I] :
( ( I = zero_z126310315umeral
=> hAPP_nat_nat(code_nat_of_aux(I),N) = N )
& ( I != zero_z126310315umeral
=> hAPP_nat_nat(code_nat_of_aux(I),N) = hAPP_nat_nat(code_nat_of_aux(hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,I),one_on1645066479umeral)),hAPP_nat_nat(suc,N)) ) ) ).
fof(fact_3385_ln__bound,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(ln,X)),X)) ) ).
fof(fact_3386_aux4,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),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),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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))) ) ) ).
fof(fact_3387_exp__less__mono,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),hAPP_real_real(exp_real,Y))) ) ).
fof(fact_3388_exp__inj__iff,axiom,
! [Xa,Ya] :
( hAPP_real_real(exp_real,Xa) = hAPP_real_real(exp_real,Ya)
<=> Xa = Ya ) ).
fof(fact_3389_exp__le__cancel__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,Xa)),hAPP_real_real(exp_real,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ).
fof(fact_3390_exp__less__cancel,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),hAPP_real_real(exp_real,Y)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y)) ) ).
fof(fact_3391_exp__less__cancel__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,Xa)),hAPP_real_real(exp_real,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) ) ).
fof(fact_3392_real__divide__square__eq,axiom,
! [R_2,A_3] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),R_2)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,A_3),R_2) ).
fof(fact_3393_abs__exp__cancel,axiom,
! [X] : hAPP_real_real(abs_abs_real,hAPP_real_real(exp_real,X)) = hAPP_real_real(exp_real,X) ).
fof(fact_3394_ln__exp,axiom,
! [X] : hAPP_real_real(ln,hAPP_real_real(exp_real,X)) = X ).
fof(fact_3395_exp__ln__eq,axiom,
! [U_1,X] :
( hAPP_real_real(exp_real,U_1) = X
=> hAPP_real_real(ln,X) = U_1 ) ).
fof(fact_3396_ln__unique,axiom,
! [Y,X] :
( hAPP_real_real(exp_real,Y) = X
=> hAPP_real_real(ln,X) = Y ) ).
fof(fact_3397_ln__x__over__x__mono,axiom,
! [Y,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,one_one_real)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,Y)),Y)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X)),X))) ) ) ).
fof(fact_3398_not__exp__le__zero,axiom,
! [X] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X)),zero_zero_real)) ).
fof(fact_3399_exp__ge__zero,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(exp_real,X))) ).
fof(fact_3400_not__exp__less__zero,axiom,
! [X] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,X)),zero_zero_real)) ).
fof(fact_3401_exp__gt__zero,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(exp_real,X))) ).
fof(fact_3402_real__0__le__divide__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Xa),Ya)))
<=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),zero_zero_real))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ya)) )
& ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),zero_zero_real)) ) ) ) ).
fof(fact_3403_exp__eq__one__iff,axiom,
! [Xa] :
( hAPP_real_real(exp_real,Xa) = one_one_real
<=> Xa = zero_zero_real ) ).
fof(fact_3404_lemma__MVT,axiom,
! [F,A_1,B_1] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,A_1)),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_1)),hAPP_real_real(F,A_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1))),A_1)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B_1)),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_1)),hAPP_real_real(F,A_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1))),B_1)) ).
fof(fact_3405_exp__le__one__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,Xa)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),zero_zero_real)) ) ).
fof(fact_3406_one__le__exp__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(exp_real,Xa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa)) ) ).
fof(fact_3407_one__less__exp__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(exp_real,Xa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) ) ).
fof(fact_3408_exp__less__one__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(exp_real,Xa)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)) ) ).
fof(fact_3409_exp__gt__one,axiom,
! [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,one_one_real),hAPP_real_real(exp_real,X))) ) ).
fof(fact_3410_exp__ge__add__one__self,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_real_real(exp_real,X))) ).
fof(fact_3411_exp__ln,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(exp_real,hAPP_real_real(ln,X)) = X ) ).
fof(fact_3412_exp__ln__iff,axiom,
! [Xa] :
( hAPP_real_real(exp_real,hAPP_real_real(ln,Xa)) = Xa
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) ) ).
fof(fact_3413_exp__ge__add__one__self__aux,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,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_real_real(exp_real,X))) ) ).
fof(fact_3414_ln__div,axiom,
! [Y,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,zero_zero_real),Y))
=> hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,Y)) ) ) ).
fof(fact_3415_eq__divide__2__times__iff,axiom,
! [Xa,Ya,Z_2] :
( Xa = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Ya),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)))),Xa) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,Ya),Z_2) ) ).
fof(fact_3416_real__sum__of__halves,axiom,
! [X] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = X ).
fof(fact_3417_real__less__half__sum,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3418_real__gt__half__sum,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),Y)) ) ).
fof(fact_3419_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_3420_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_3421_exp__bound,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),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(exp_real,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),X)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ) ).
fof(fact_3422_lemma__exp__total,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Y))
=> ? [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(minus_minus_real,Y),one_one_real)))
& hAPP_real_real(exp_real,X_1) = Y ) ) ).
fof(fact_3423_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [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,one_one_real),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),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))),X))),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ) ).
fof(fact_3424_log__base__10__eq1,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> 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) = 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)) ) ).
fof(fact_3425_complex__mod__minus__le__complex__mod,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(norm_norm_complex,X))),hAPP_complex_real(norm_norm_complex,X))) ).
fof(fact_3426_real__norm__def,axiom,
! [R_2] : hAPP_real_real(norm_norm_real,R_2) = hAPP_real_real(abs_abs_real,R_2) ).
fof(fact_3427_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_3428_real__minus__mult__self__le,axiom,
! [U_1,X] : 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),X))) ).
fof(fact_3429_real__add__eq__0__iff,axiom,
! [Xa,Ya] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya) = zero_zero_real
<=> Ya = hAPP_real_real(uminus_uminus_real,Xa) ) ).
fof(fact_3430_real__add__minus__iff,axiom,
! [Xa,A_1] :
( hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),hAPP_real_real(uminus_uminus_real,A_1)) = zero_zero_real
<=> Xa = A_1 ) ).
fof(fact_3431_abs__le__interval__iff,axiom,
! [Xa,Ra] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Xa)),Ra))
<=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,Ra)),Xa))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ra)) ) ) ).
fof(fact_3432_real__diff__def,axiom,
! [R_2,S_1] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_2),S_1) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_2),hAPP_real_real(uminus_uminus_real,S_1)) ).
fof(fact_3433_minus__real__def,axiom,
! [X,Y] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,Y)) ).
fof(fact_3434_abs__minus__add__cancel,axiom,
! [X,Y] : hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(uminus_uminus_real,Y))) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y),hAPP_real_real(uminus_uminus_real,X))) ).
fof(fact_3435_log__one,axiom,
! [A_3] : hAPP_real_real(log(A_3),one_one_real) = zero_zero_real ).
fof(fact_3436_Log_Olog__def,axiom,
! [A_3,X] : hAPP_real_real(log(A_3),X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(ln,X)),hAPP_real_real(ln,A_3)) ).
fof(fact_3437_real__0__le__add__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,Xa)),Ya)) ) ).
fof(fact_3438_real__add__le__0__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),hAPP_real_real(uminus_uminus_real,Xa))) ) ).
fof(fact_3439_real__0__less__add__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(uminus_uminus_real,Xa)),Ya)) ) ).
fof(fact_3440_real__add__less__0__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),Ya)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),hAPP_real_real(uminus_uminus_real,Xa))) ) ).
fof(fact_3441_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_3442_real__abs__def,axiom,
! [R_2] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),zero_zero_real))
=> hAPP_real_real(abs_abs_real,R_2) = hAPP_real_real(uminus_uminus_real,R_2) )
& ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),zero_zero_real))
=> hAPP_real_real(abs_abs_real,R_2) = R_2 ) ) ).
fof(fact_3443_abs__sum__triangle__ineq,axiom,
! [X,Y,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),Y)),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),hAPP_real_real(uminus_uminus_real,L_2)))),hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Y),hAPP_real_real(uminus_uminus_real,M)))))) ).
fof(fact_3444_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_3445_log__less__cancel__iff,axiom,
! [Ya,Xa,A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(log(A_1),Xa)),hAPP_real_real(log(A_1),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) ) ) ) ) ).
fof(fact_3446_log__ln,axiom,
! [X] : hAPP_real_real(ln,X) = hAPP_real_real(log(hAPP_real_real(exp_real,one_one_real)),X) ).
fof(fact_3447_log__le__cancel__iff,axiom,
! [Ya,Xa,A_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),A_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(log(A_1),Xa)),hAPP_real_real(log(A_1),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ) ) ) ).
fof(fact_3448_realpow__square__minus__le,axiom,
! [U_1,X] : 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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3449_ln__one__minus__pos__upper__bound,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_real,X),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))),hAPP_real_real(uminus_uminus_real,X))) ) ) ).
fof(fact_3450_aux5,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),one_one_real))
=> hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X)) = 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),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),X))))) ) ).
fof(fact_3451_log__mult,axiom,
! [Y,X,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))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
=> hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(log(A_3),X)),hAPP_real_real(log(A_3),Y)) ) ) ) ) ).
fof(fact_3452_log__divide,axiom,
! [Y,X,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))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
=> hAPP_real_real(log(A_3),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(log(A_3),X)),hAPP_real_real(log(A_3),Y)) ) ) ) ) ).
fof(fact_3453_log__eq__div__ln__mult__log,axiom,
! [X,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))
=> hAPP_real_real(log(A_3),X) = 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)) ) ) ) ) ) ).
fof(fact_3454_log__base__10__eq2,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> 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) = 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)) ) ).
fof(fact_3455_ln__one__minus__pos__lower__bound,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_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)),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),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)))) ) ) ).
fof(fact_3456_negative__zle,axiom,
! [N,M] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))),hAPP_nat_int(semiri1621563631at_int,M))) ).
fof(fact_3457_negative__zless,axiom,
! [N,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)))),hAPP_nat_int(semiri1621563631at_int,M))) ).
fof(fact_3458_complex__diff__def,axiom,
! [X,Y] : hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),hAPP_complex_complex(uminus473333897omplex,Y)) ).
fof(fact_3459_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_3460_minus__Pls,axiom,
hAPP_int_int(uminus_uminus_int,pls) = pls ).
fof(fact_3461_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_3462_zminus__0,axiom,
hAPP_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).
fof(fact_3463_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_3464_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_3465_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_3466_zcong__zminus,axiom,
! [A_1,B_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),hAPP_int_int(uminus_uminus_int,Ma)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma)) ) ).
fof(fact_3467_uminus__dvd__conv_I1_J,axiom,
! [D_1,Ta] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),Ta))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(uminus_uminus_int,D_1)),Ta)) ) ).
fof(fact_3468_uminus__dvd__conv_I2_J,axiom,
! [D_1,Ta] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),Ta))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(uminus_uminus_int,Ta))) ) ).
fof(fact_3469_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_3470_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_3471_zminus__zmod,axiom,
! [X,M] : hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,hAPP_int_int(div_mod_int(X),M))),M) = hAPP_int_int(div_mod_int(hAPP_int_int(uminus_uminus_int,X)),M) ).
fof(fact_3472_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_3473_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_3474_even__neg,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_int_int(uminus_uminus_int,Xa)))
<=> hBOOL(hAPP_int_bool(even_odd_even_int,Xa)) ) ).
fof(fact_3475_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_3476_negative__eq__positive,axiom,
! [Na,Ma] :
( hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,Na)) = hAPP_nat_int(semiri1621563631at_int,Ma)
<=> ( Na = zero_zero_nat
& Ma = zero_zero_nat ) ) ).
fof(fact_3477_not__int__zless__negative,axiom,
! [N,M] : ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_nat_int(semiri1621563631at_int,N)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,M)))) ).
fof(fact_3478_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_3479_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_3480_Int_OMin__def,axiom,
min = hAPP_int_int(uminus_uminus_int,one_one_int) ).
fof(fact_3481_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_3482_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_3483_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_3484_minus__Min,axiom,
hAPP_int_int(uminus_uminus_int,min) = hAPP_int_int(bit1,pls) ).
fof(fact_3485_int__zle__neg,axiom,
! [Na,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_nat_int(semiri1621563631at_int,Na)),hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,Ma))))
<=> ( Na = zero_zero_nat
& Ma = zero_zero_nat ) ) ).
fof(fact_3486_negative__zle__0,axiom,
! [N] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))),zero_zero_int)) ).
fof(fact_3487_nat__zminus__int,axiom,
! [N] : hAPP_int_nat(nat,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N))) = zero_zero_nat ).
fof(fact_3488_minus__numeral__code_I6_J,axiom,
! [V_1,W] : hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(number_number_of_int,V_1)),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_1),hAPP_int_int(uminus_uminus_int,W))) ).
fof(fact_3489_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_3490_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_3491_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_3492_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_3493_neg__zminus__int,axiom,
! [N] : hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N))))) ).
fof(fact_3494_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_3495_negative__zless__0,axiom,
! [N] : 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)))),zero_zero_int)) ).
fof(fact_3496_not__zle__0__negative,axiom,
! [N] : ~ 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))))) ).
fof(fact_3497_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_3498_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_3499_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_3500_neg__even__power,axiom,
! [A_1,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zEven))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A_1)),hAPP_int_nat(nat,Xa)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),hAPP_int_nat(nat,Xa)) ) ) ).
fof(fact_3501_neg__odd__power,axiom,
! [A_1,Xa] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(uminus_uminus_int,A_1)),hAPP_int_nat(nat,Xa)) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_1),hAPP_int_nat(nat,Xa))) ) ) ).
fof(fact_3502_zminus1__lemma,axiom,
! [A_1,B_1,Q_2,Ra] :
( hBOOL(hAPP_P603027463t_bool(divmod_int_rel(A_1,B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),Ra)))
=> hBOOL(hAPP_P603027463t_bool(divmod_int_rel(hAPP_int_int(uminus_uminus_int,A_1),B_1),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,Ra),zero_zero_int)),hAPP_int_int(uminus_uminus_int,Q_2)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_int_int(uminus_uminus_int,Q_2)),one_one_int))),hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,Ra),zero_zero_int)),zero_zero_int),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,B_1),Ra))))) ) ).
fof(fact_3503_negD,axiom,
! [X] :
( is_int(X)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X),zero_zero_int))
=> ? [N_1] : X = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))) ) ) ).
fof(fact_3504_exp__total,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
=> ? [X_1] : hAPP_real_real(exp_real,X_1) = Y ) ).
fof(fact_3505_lemma__sqrt__hcomplex__capprox,axiom,
! [Y,X,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),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),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))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
=> 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),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)) ) ) ) ) ) ).
fof(fact_3506_arctan__add,axiom,
! [Y,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Y)),one_one_real))
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y)) = hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,one_one_real),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)))) ) ) ).
fof(fact_3507_real__sqrt__divide,axiom,
! [X,Y] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y)) ).
fof(fact_3508_real__sqrt__le__mono,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y))) ) ).
fof(fact_3509_real__sqrt__le__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,Xa)),hAPP_real_real(sqrt,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ).
fof(fact_3510_real__sqrt__eq__iff,axiom,
! [Xa,Ya] :
( hAPP_real_real(sqrt,Xa) = hAPP_real_real(sqrt,Ya)
<=> Xa = Ya ) ).
fof(fact_3511_real__sqrt__less__mono,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y))) ) ).
fof(fact_3512_real__sqrt__less__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,Xa)),hAPP_real_real(sqrt,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) ) ).
fof(fact_3513_real__sqrt__eq__0__iff,axiom,
! [Xa] :
( hAPP_real_real(sqrt,Xa) = zero_zero_real
<=> Xa = zero_zero_real ) ).
fof(fact_3514_real__sqrt__zero,axiom,
hAPP_real_real(sqrt,zero_zero_real) = zero_zero_real ).
fof(fact_3515_real__sqrt__one,axiom,
hAPP_real_real(sqrt,one_one_real) = one_one_real ).
fof(fact_3516_real__sqrt__eq__1__iff,axiom,
! [Xa] :
( hAPP_real_real(sqrt,Xa) = one_one_real
<=> Xa = one_one_real ) ).
fof(fact_3517_real__sqrt__power,axiom,
! [X,K_2] : hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),K_2) ).
fof(fact_3518_real__sqrt__mult,axiom,
! [X,Y] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sqrt,X)),hAPP_real_real(sqrt,Y)) ).
fof(fact_3519_real__sqrt__minus,axiom,
! [X] : hAPP_real_real(sqrt,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sqrt,X)) ).
fof(fact_3520_arctan__monotone_H,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y))) ) ).
fof(fact_3521_arctan__zero__zero,axiom,
hAPP_real_real(arctan,zero_zero_real) = zero_zero_real ).
fof(fact_3522_arctan__monotone,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,X)),hAPP_real_real(arctan,Y))) ) ).
fof(fact_3523_arctan__minus,axiom,
! [X] : hAPP_real_real(arctan,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(arctan,X)) ).
fof(fact_3524_real__sqrt__eq__zero__cancel,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( hAPP_real_real(sqrt,X) = zero_zero_real
=> X = zero_zero_real ) ) ).
fof(fact_3525_real__sqrt__ge__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,zero_zero_real),hAPP_real_real(sqrt,X))) ) ).
fof(fact_3526_real__sqrt__le__0__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,Xa)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),zero_zero_real)) ) ).
fof(fact_3527_real__sqrt__ge__0__iff,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sqrt,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ya)) ) ).
fof(fact_3528_real__sqrt__gt__0__iff,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sqrt,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya)) ) ).
fof(fact_3529_real__sqrt__lt__0__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,Xa)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)) ) ).
fof(fact_3530_real__sqrt__not__eq__zero,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(sqrt,X) != zero_zero_real ) ).
fof(fact_3531_real__sqrt__gt__zero,axiom,
! [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,zero_zero_real),hAPP_real_real(sqrt,X))) ) ).
fof(fact_3532_real__sqrt__ge__one,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,X))) ) ).
fof(fact_3533_real__sqrt__le__1__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sqrt,Xa)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),one_one_real)) ) ).
fof(fact_3534_real__sqrt__ge__1__iff,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(sqrt,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Ya)) ) ).
fof(fact_3535_real__sqrt__lt__1__iff,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sqrt,Xa)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real)) ) ).
fof(fact_3536_real__sqrt__gt__1__iff,axiom,
! [Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(sqrt,Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Ya)) ) ).
fof(fact_3537_real__sqrt__abs2,axiom,
! [X] : hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)) = hAPP_real_real(abs_abs_real,X) ).
fof(fact_3538_le__real__sqrt__sumsq,axiom,
! [X,Y] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),Y))))) ).
fof(fact_3539_real__sqrt__mult__self__sum__ge__zero,axiom,
! [X,Y] : 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),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),Y))))) ).
fof(fact_3540_arctan__half,axiom,
! [X] : hAPP_real_real(arctan,X) = 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),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) ).
fof(fact_3541_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_3542_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_3543_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_3544_real__sqrt__abs,axiom,
! [X] : hAPP_real_real(sqrt,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = hAPP_real_real(abs_abs_real,X) ).
fof(fact_3545_real__sqrt__pow2__iff,axiom,
! [Xa] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,Xa)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = Xa
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa)) ) ).
fof(fact_3546_real__sqrt__pow2,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X ) ).
fof(fact_3547_real__sqrt__unique,axiom,
! [Y,X] :
( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = X
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
=> hAPP_real_real(sqrt,X) = Y ) ) ).
fof(fact_3548_real__sqrt__pow2__gt__zero,axiom,
! [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,zero_zero_real),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sqrt,X)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3549_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_3550_real__sqrt__sum__squares__ge1,axiom,
! [X,Y] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3551_real__sqrt__sum__squares__ge2,axiom,
! [Y,X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3552_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_3553_real__sqrt__sum__squares__eq__cancel,axiom,
! [X,Y] :
( hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = X
=> Y = zero_zero_real ) ).
fof(fact_3554_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X,Y] :
( hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) = Y
=> X = zero_zero_real ) ).
fof(fact_3555_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X,Y,Xa_2,Ya_1] : 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),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),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_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_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),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),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_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3556_real__sqrt__sum__squares__ge__zero,axiom,
! [X,Y] : 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),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3557_real__sqrt__ge__abs1,axiom,
! [X,Y] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,X)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3558_real__sqrt__ge__abs2,axiom,
! [Y,X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3559_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X,Y,Xa_2,Ya_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(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),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_1),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).
fof(fact_3560_cos__x__y__le__one,axiom,
! [X,Y] : 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),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))),one_one_real)) ).
fof(fact_3561_real__sqrt__sum__squares__less,axiom,
! [Y,X,U_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),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)),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),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))),U_1)) ) ) ).
fof(fact_3562_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_3563_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_3564_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_3565_pi__neq__zero,axiom,
pi != zero_zero_real ).
fof(fact_3566_pi__ge__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),pi)) ).
fof(fact_3567_pi__gt__zero,axiom,
hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),pi)) ).
fof(fact_3568_pi__not__less__zero,axiom,
~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,pi),zero_zero_real)) ).
fof(fact_3569_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_3570_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_3571_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_3572_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_3573_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_3574_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_3575_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_3576_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_3577_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_3578_arctan__ubound,axiom,
! [Y] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,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)))))) ).
fof(fact_3579_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_3580_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_3581_arctan__lbound,axiom,
! [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)))))),hAPP_real_real(arctan,Y))) ).
fof(fact_3582_arctan__bounded,axiom,
! [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)))))),hAPP_real_real(arctan,Y)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,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)))))) ) ).
fof(fact_3583_arcsin__lt__bounded,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),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)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arcsin,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)))))) ) ) ) ).
fof(fact_3584_arcsin__lbound,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),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))) ) ) ).
fof(fact_3585_arcsin__bounded,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),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)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,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)))))) ) ) ) ).
fof(fact_3586_arcsin__ubound,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,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)))))) ) ) ).
fof(fact_3587_arctan__inverse,axiom,
! [X] :
( X != zero_zero_real
=> hAPP_real_real(arctan,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),X)) = 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)),pi)),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),hAPP_real_real(arctan,X)) ) ).
fof(fact_3588_cos__arcsin,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
=> hAPP_real_real(cos,hAPP_real_real(arcsin,X)) = 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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3589_arcsin,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),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)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,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,hAPP_real_real(arcsin,Y)) = Y ) ) ) ).
fof(fact_3590_sin__diff,axiom,
! [X,Y] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))) ).
fof(fact_3591_sin__diff2,axiom,
! [X,Y] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y)),hAPP_real_real(sin,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y)),hAPP_real_real(cos,X))) ).
fof(fact_3592_sin__minus,axiom,
! [X] : hAPP_real_real(sin,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) ).
fof(fact_3593_cos__minus,axiom,
! [X] : hAPP_real_real(cos,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(cos,X) ).
fof(fact_3594_sin__zero,axiom,
hAPP_real_real(sin,zero_zero_real) = zero_zero_real ).
fof(fact_3595_cos__one__sin__zero,axiom,
! [X] :
( hAPP_real_real(cos,X) = one_one_real
=> hAPP_real_real(sin,X) = zero_zero_real ) ).
fof(fact_3596_sin__add,axiom,
! [X,Y] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))) ).
fof(fact_3597_sin__cos__squared__add3,axiom,
! [X] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,X))) = one_one_real ).
fof(fact_3598_sin__zero__abs__cos__one,axiom,
! [X] :
( hAPP_real_real(sin,X) = zero_zero_real
=> hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X)) = one_one_real ) ).
fof(fact_3599_cos__diff2,axiom,
! [X,Y] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,Y)),hAPP_real_real(sin,X))) ).
fof(fact_3600_cos__diff,axiom,
! [X,Y] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))) ).
fof(fact_3601_cos__add,axiom,
! [X,Y] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))) ).
fof(fact_3602_sin__le__one,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,X)),one_one_real)) ).
fof(fact_3603_cos__le__one,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X)),one_one_real)) ).
fof(fact_3604_cos__zero,axiom,
hAPP_real_real(cos,zero_zero_real) = one_one_real ).
fof(fact_3605_sin__pi,axiom,
hAPP_real_real(sin,pi) = zero_zero_real ).
fof(fact_3606_cos__arctan__not__zero,axiom,
! [X] : hAPP_real_real(cos,hAPP_real_real(arctan,X)) != zero_zero_real ).
fof(fact_3607_sin__double,axiom,
! [X] : 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)) = 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))),hAPP_real_real(cos,X)) ).
fof(fact_3608_real__sgn__eq,axiom,
! [X] : hAPP_real_real(sgn_sgn_real,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(abs_abs_real,X)) ).
fof(fact_3609_sin__ge__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),pi))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).
fof(fact_3610_sin__gt__zero__pi,axiom,
! [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),pi))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).
fof(fact_3611_sin__ge__minus__one,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(sin,X))) ).
fof(fact_3612_cos__monotone__0__pi_H,axiom,
! [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,Y),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ) ).
fof(fact_3613_abs__sin__le__one,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(sin,X))),one_one_real)) ).
fof(fact_3614_cos__ge__minus__one,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),hAPP_real_real(cos,X))) ).
fof(fact_3615_abs__cos__le__one,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_real_real(cos,X))),one_one_real)) ).
fof(fact_3616_sin__periodic__pi2,axiom,
! [X] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,pi),X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) ).
fof(fact_3617_sin__periodic__pi,axiom,
! [X] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) ).
fof(fact_3618_cos__pi,axiom,
hAPP_real_real(cos,pi) = hAPP_int_real(number267125858f_real,min) ).
fof(fact_3619_cos__periodic__pi,axiom,
! [X] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(cos,X)) ).
fof(fact_3620_sin__cos__squared__add2__mult,axiom,
! [R_2,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_2),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_2),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_2),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ).
fof(fact_3621_sin__cos__squared__add,axiom,
! [X] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),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)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).
fof(fact_3622_sin__cos__squared__add2,axiom,
! [X] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),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)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = one_one_real ).
fof(fact_3623_sin__squared__eq,axiom,
! [X] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(sin,X)),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)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_3624_cos__squared__eq,axiom,
! [X] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),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)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_3625_cos__sin__eq,axiom,
! [X] : hAPP_real_real(cos,X) = 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)) ).
fof(fact_3626_sin__cos__eq,axiom,
! [X] : hAPP_real_real(sin,X) = 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)) ).
fof(fact_3627_real__sgn__pos,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(sgn_sgn_real,X) = one_one_real ) ).
fof(fact_3628_cos__double,axiom,
! [X] : 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)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,X)),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)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ).
fof(fact_3629_minus__sin__cos__eq,axiom,
! [X] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,X)) = hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_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)))))) ).
fof(fact_3630_cos__monotone__0__pi,axiom,
! [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_real,Y),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),pi))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ) ).
fof(fact_3631_cos__monotone__minus__pi__0_H,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))) ) ) ) ).
fof(fact_3632_sin__cos__add,axiom,
! [X,Y] : 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),Y))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(sin,Y))))),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),Y))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(sin,X)),hAPP_real_real(sin,Y))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).
fof(fact_3633_sin__cos__minus,axiom,
! [X] : 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))),hAPP_real_real(sin,X))),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))),hAPP_real_real(cos,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) = zero_zero_real ).
fof(fact_3634_real__sgn__neg,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real))
=> hAPP_real_real(sgn_sgn_real,X) = hAPP_int_real(number267125858f_real,min) ) ).
fof(fact_3635_real__sgn__def,axiom,
! [X] :
( ( X = zero_zero_real
=> hAPP_real_real(sgn_sgn_real,X) = zero_zero_real )
& ( X != zero_zero_real
=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(sgn_sgn_real,X) = one_one_real )
& ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(sgn_sgn_real,X) = hAPP_real_real(uminus_uminus_real,one_one_real) ) ) ) ) ).
fof(fact_3636_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_3637_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_3638_cos__monotone__minus__pi__0,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(cos,Y)),hAPP_real_real(cos,X))) ) ) ) ).
fof(fact_3639_sin__arcsin,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
=> hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = Y ) ) ).
fof(fact_3640_sin__gt__zero,axiom,
! [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_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))) ) ) ).
fof(fact_3641_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_3642_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_3643_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_3644_sin__periodic,axiom,
! [X] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),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) ).
fof(fact_3645_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_3646_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_3647_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_3648_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_3649_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_3650_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_3651_cos__periodic,axiom,
! [X] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),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) ).
fof(fact_3652_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_3653_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_3654_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_3655_sin__gt__zero2,axiom,
! [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,zero_zero_real),hAPP_real_real(sin,X))) ) ) ).
fof(fact_3656_cos__double__less__one,axiom,
! [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_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))),one_one_real)) ) ) ).
fof(fact_3657_sin__monotone__2pi_H,axiom,
! [X,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),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))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(sin,Y)),hAPP_real_real(sin,X))) ) ) ) ).
fof(fact_3658_cos__gt__zero,axiom,
! [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,zero_zero_real),hAPP_real_real(cos,X))) ) ) ).
fof(fact_3659_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_3660_sin__less__zero,axiom,
! [X] :
( 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))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(sin,X)),zero_zero_real)) ) ) ).
fof(fact_3661_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_3662_cos__ge__zero,axiom,
! [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))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(cos,X))) ) ) ).
fof(fact_3663_cos__gt__zero__pi,axiom,
! [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))))))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(cos,X))) ) ) ).
fof(fact_3664_arcsin__sin,axiom,
! [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(arcsin,hAPP_real_real(sin,X)) = X ) ) ).
fof(fact_3665_arcsin__pi,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),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)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arcsin,Y)),pi))
& hAPP_real_real(sin,hAPP_real_real(arcsin,Y)) = Y ) ) ) ).
fof(fact_3666_sin__total,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
=> ? [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(sin,X_1) = Y
& ! [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),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_1) = Y )
=> Y_1 = X_1 ) ) ) ) ).
fof(fact_3667_cos__total,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
=> ? [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(cos,X_1) = Y
& ! [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),pi))
& hAPP_real_real(cos,Y_1) = Y )
=> Y_1 = X_1 ) ) ) ) ).
fof(fact_3668_cos__is__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),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
& hAPP_real_real(cos,X_1) = zero_zero_real
& ! [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),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
& hAPP_real_real(cos,Y_1) = zero_zero_real )
=> Y_1 = X_1 ) ) ).
fof(fact_3669_sin__arccos,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),one_one_real))
=> hAPP_real_real(sin,hAPP_real_real(arccos,X)) = 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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3670_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_3671_arccos__cos,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),pi))
=> hAPP_real_real(arccos,hAPP_real_real(cos,X)) = X ) ) ).
fof(fact_3672_cos__arccos__abs,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),one_one_real))
=> hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y ) ).
fof(fact_3673_arccos__lbound,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y))) ) ) ).
fof(fact_3674_cos__arccos,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
=> hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y ) ) ).
fof(fact_3675_arccos__ubound,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi)) ) ) ).
fof(fact_3676_arccos__cos2,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),zero_zero_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(uminus_uminus_real,pi)),X))
=> hAPP_real_real(arccos,hAPP_real_real(cos,X)) = hAPP_real_real(uminus_uminus_real,X) ) ) ).
fof(fact_3677_arccos__bounded,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi)) ) ) ) ).
fof(fact_3678_arccos__lt__bounded,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(arccos,Y)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arccos,Y)),pi)) ) ) ) ).
fof(fact_3679_arccos,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,min)),Y))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y),one_one_real))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(arccos,Y)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(arccos,Y)),pi))
& hAPP_real_real(cos,hAPP_real_real(arccos,Y)) = Y ) ) ) ).
fof(fact_3680_cos__arccos__lemma1,axiom,
! [X_3,Y_3] : hAPP_real_real(cos,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_3),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_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,Y_3),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_3),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_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,Y_3),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ).
fof(fact_3681_sin__arccos__abs,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Y)),one_one_real))
=> hAPP_real_real(sin,hAPP_real_real(arccos,Y)) = 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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3682_sin__arccos__lemma1,axiom,
! [X_3,Y_3] : hAPP_real_real(sin,hAPP_real_real(arccos,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X_3),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_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,Y_3),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_3),hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X_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,Y_3),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_3683_tan__half,axiom,
! [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) = 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))),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))),one_one_real)) ) ) ).
fof(fact_3684_tan__double,axiom,
! [X] :
( hAPP_real_real(cos,X) != 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)) != 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)) = 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))),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)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ) ).
fof(fact_3685_sin__cos__npi,axiom,
! [N] : 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)))),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) ).
fof(fact_3686_cos__expansion__lemma,axiom,
! [X,M] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),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),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_3687_real__of__nat__ge__zero,axiom,
! [N] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(real_nat,N))) ).
fof(fact_3688_tan__minus,axiom,
! [X] : hAPP_real_real(tan,hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(tan,X)) ).
fof(fact_3689_natceiling__real__of__nat,axiom,
! [N] : natceiling(hAPP_nat_real(real_nat,N)) = N ).
fof(fact_3690_real__of__nat__inject,axiom,
! [Na,Ma] :
( hAPP_nat_real(real_nat,Na) = hAPP_nat_real(real_nat,Ma)
<=> Na = Ma ) ).
fof(fact_3691_tan__zero,axiom,
hAPP_real_real(tan,zero_zero_real) = zero_zero_real ).
fof(fact_3692_abs__real__of__nat__cancel,axiom,
! [X] : hAPP_real_real(abs_abs_real,hAPP_nat_real(real_nat,X)) = hAPP_nat_real(real_nat,X) ).
fof(fact_3693_natfloor__real__of__nat,axiom,
! [N] : natfloor(hAPP_nat_real(real_nat,N)) = N ).
fof(fact_3694_tan__arctan,axiom,
! [Y] : hAPP_real_real(tan,hAPP_real_real(arctan,Y)) = Y ).
fof(fact_3695_tan__npi,axiom,
! [N] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi)) = zero_zero_real ).
fof(fact_3696_tan__periodic__nat,axiom,
! [X,N] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi))) = hAPP_real_real(tan,X) ).
fof(fact_3697_real__of__nat__def,axiom,
real_nat = semiri132038758t_real ).
fof(fact_3698_real__eq__of__nat,axiom,
real_nat = semiri132038758t_real ).
fof(fact_3699_real__of__nat__zero,axiom,
hAPP_nat_real(real_nat,zero_zero_nat) = zero_zero_real ).
fof(fact_3700_real__of__nat__zero__iff,axiom,
! [Na] :
( hAPP_nat_real(real_nat,Na) = zero_zero_real
<=> Na = zero_zero_nat ) ).
fof(fact_3701_not__real__of__nat__less__zero,axiom,
! [N] : ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),zero_zero_real)) ).
fof(fact_3702_real__of__nat__less__iff,axiom,
! [Na,Ma] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(real_nat,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),Ma)) ) ).
fof(fact_3703_real__of__nat__le__iff,axiom,
! [Na,Ma] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(real_nat,Ma)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),Ma)) ) ).
fof(fact_3704_real__of__nat__mult,axiom,
! [M,N] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N)) ).
fof(fact_3705_real__of__nat__add,axiom,
! [M,N] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N)) ).
fof(fact_3706_real__of__nat__1,axiom,
hAPP_nat_real(real_nat,one_one_nat) = one_one_real ).
fof(fact_3707_power__real__of__nat,axiom,
! [M,N] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M)),N) = hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) ).
fof(fact_3708_real__of__nat__power,axiom,
! [M,N] : hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,M),N)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_nat_real(real_nat,M)),N) ).
fof(fact_3709_real__natceiling__ge,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,natceiling(X)))) ).
fof(fact_3710_tan__pi,axiom,
hAPP_real_real(tan,pi) = zero_zero_real ).
fof(fact_3711_tan__periodic__pi,axiom,
! [X] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),pi)) = hAPP_real_real(tan,X) ).
fof(fact_3712_real__of__nat__le__zero__cancel__iff,axiom,
! [Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,Na)),zero_zero_real))
<=> Na = zero_zero_nat ) ).
fof(fact_3713_real__of__nat__Suc__gt__zero,axiom,
! [N] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)))) ).
fof(fact_3714_real__of__nat__one,axiom,
hAPP_nat_real(real_nat,hAPP_nat_nat(suc,zero_zero_nat)) = one_one_real ).
fof(fact_3715_real__of__nat__Suc,axiom,
! [N] : hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real) ).
fof(fact_3716_real__of__nat__diff,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_nat_real(real_nat,M)),hAPP_nat_real(real_nat,N)) ) ).
fof(fact_3717_real__of__nat__div4,axiom,
! [N,X] : 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),X))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,X)))) ).
fof(fact_3718_real__natfloor__le,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,hAPP_nat_real(real_nat,natfloor(X))),X)) ) ).
fof(fact_3719_le__natfloor,axiom,
! [X,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,X)),A_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),natfloor(A_3))) ) ).
fof(fact_3720_exp__real__of__nat__mult,axiom,
! [N,X] : hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),X)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(exp_real,X)),N) ).
fof(fact_3721_natceiling__le,axiom,
! [X,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_nat_real(real_nat,A_3)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(X)),A_3)) ) ).
fof(fact_3722_natfloor__power,axiom,
! [N,X] :
( X = hAPP_nat_real(real_nat,natfloor(X))
=> natfloor(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,natfloor(X)),N) ) ).
fof(fact_3723_tan__def,axiom,
! [X] : hAPP_real_real(tan,X) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,X)),hAPP_real_real(cos,X)) ).
fof(fact_3724_real__of__nat__gt__zero__cancel__iff,axiom,
! [Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(real_nat,Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) ) ).
fof(fact_3725_nat__less__real__le,axiom,
! [Na,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),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,Na)),one_one_real)),hAPP_nat_real(real_nat,Ma))) ) ).
fof(fact_3726_nat__le__real__less,axiom,
! [Na,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),Ma))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,Na)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,Ma)),one_one_real))) ) ).
fof(fact_3727_sin__npi,axiom,
! [N] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi)) = zero_zero_real ).
fof(fact_3728_sin__npi2,axiom,
! [N] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N))) = zero_zero_real ).
fof(fact_3729_le__natfloor__eq,axiom,
! [A_1,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),natfloor(Xa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_1)),Xa)) ) ) ).
fof(fact_3730_natceiling__le__eq,axiom,
! [A_1,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,natceiling(Xa)),A_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),hAPP_nat_real(real_nat,A_1))) ) ) ).
fof(fact_3731_real__natfloor__add__one__gt,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,natfloor(X))),one_one_real))) ).
fof(fact_3732_natfloor__subtract,axiom,
! [A_3,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X))
=> natfloor(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natfloor(X)),A_3) ) ).
fof(fact_3733_real__natfloor__gt__diff__one,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),one_one_real)),hAPP_nat_real(real_nat,natfloor(X)))) ).
fof(fact_3734_natceiling__subtract,axiom,
! [A_3,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,A_3)),X))
=> natceiling(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,natceiling(X)),A_3) ) ).
fof(fact_3735_tan__periodic__n,axiom,
! [X,N] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(number267125858f_real,N)),pi))) = hAPP_real_real(tan,X) ).
fof(fact_3736_real__of__nat__div,axiom,
! [N,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))
=> hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,N)),hAPP_nat_real(real_nat,D_3)) ) ) ).
fof(fact_3737_ln__realpow,axiom,
! [N,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(ln,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_real_real(ln,X)) ) ).
fof(fact_3738_less__natfloor,axiom,
! [N,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_nat_real(real_nat,N)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,natfloor(X)),N)) ) ) ).
fof(fact_3739_real__of__nat__div2,axiom,
! [N,X] : 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)),hAPP_nat_real(real_nat,X))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),X))))) ).
fof(fact_3740_real__of__nat__number__of,axiom,
! [V_1] :
( ( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_1)) = zero_zero_real )
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,V_1)))
=> hAPP_nat_real(real_nat,hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_int_real(number267125858f_real,V_1) ) ) ).
fof(fact_3741_real__of__nat__div3,axiom,
! [N,X] : 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)),hAPP_nat_real(real_nat,X))),hAPP_nat_real(real_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,N),X)))),one_one_real)) ).
fof(fact_3742_natfloor__add,axiom,
! [A_3,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> natfloor(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natfloor(X)),A_3) ) ).
fof(fact_3743_ge__natfloor__plus__one__imp__gt,axiom,
! [Z_1,N] :
( 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))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z_1),hAPP_nat_real(real_nat,N))) ) ).
fof(fact_3744_log__nat__power,axiom,
! [B_2,N,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(log(B_2),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),hAPP_real_real(log(B_2),X)) ) ).
fof(fact_3745_natfloor__eq,axiom,
! [N,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real)))
=> natfloor(X) = N ) ) ).
fof(fact_3746_natceiling__add,axiom,
! [A_3,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> natceiling(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_nat_real(real_nat,A_3))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,natceiling(X)),A_3) ) ).
fof(fact_3747_two__realpow__gt,axiom,
! [N] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),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_3748_LIMSEQ__inverse__realpow__zero__lemma,axiom,
! [N,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,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),X)),one_one_real)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),one_one_real)),N))) ) ).
fof(fact_3749_real__of__nat__div__aux,axiom,
! [X,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)),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),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,hAPP_nat_nat(div_mod_nat(X),D_3))),hAPP_nat_real(real_nat,D_3))) ) ).
fof(fact_3750_cos__npi,axiom,
! [N] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),pi)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N) ).
fof(fact_3751_cos__npi2,axiom,
! [N] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,pi),hAPP_nat_real(real_nat,N))) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,min)),N) ).
fof(fact_3752_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_3753_add__tan__eq,axiom,
! [Y,X] :
( hAPP_real_real(cos,X) != zero_zero_real
=> ( hAPP_real_real(cos,Y) != zero_zero_real
=> hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ).
fof(fact_3754_fact__lemma,axiom,
! [N] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),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)) ).
fof(fact_3755_natfloor__div__nat,axiom,
! [Y,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y))
=> natfloor(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_nat_real(real_nat,Y))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,natfloor(X)),Y) ) ) ).
fof(fact_3756_natceiling__eq,axiom,
! [N,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(real_nat,N)),one_one_real)))
=> natceiling(X) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ) ) ).
fof(fact_3757_tan__periodic,axiom,
! [X] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),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) ).
fof(fact_3758_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_3759_tan__add,axiom,
! [Y,X] :
( hAPP_real_real(cos,X) != zero_zero_real
=> ( hAPP_real_real(cos,Y) != zero_zero_real
=> ( hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) != zero_zero_real
=> hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(tan,X)),hAPP_real_real(tan,Y))),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)),hAPP_real_real(tan,Y)))) ) ) ) ).
fof(fact_3760_lemma__tan__add1,axiom,
! [Y,X] :
( hAPP_real_real(cos,X) != zero_zero_real
=> ( hAPP_real_real(cos,Y) != 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)),hAPP_real_real(tan,Y))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,X)),hAPP_real_real(cos,Y))) ) ) ).
fof(fact_3761_sin__2npi,axiom,
! [N] : 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))),pi)) = zero_zero_real ).
fof(fact_3762_cos__2npi,axiom,
! [N] : 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))),pi)) = one_one_real ).
fof(fact_3763_tan__gt__zero,axiom,
! [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,zero_zero_real),hAPP_real_real(tan,X))) ) ) ).
fof(fact_3764_tan__monotone_H,axiom,
! [Xa,Ya] :
( 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)))))),Ya))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),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)))))),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),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,Ya),Xa))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,Ya)),hAPP_real_real(tan,Xa))) ) ) ) ) ) ).
fof(fact_3765_tan__monotone,axiom,
! [X,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),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,hAPP_real_real(tan,Y)),hAPP_real_real(tan,X))) ) ) ) ).
fof(fact_3766_tan__inverse,axiom,
! [Y] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),hAPP_real_real(tan,Y)) = 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)) ).
fof(fact_3767_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_3768_tan__less__zero,axiom,
! [X] :
( 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))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),zero_zero_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(tan,X)),zero_zero_real)) ) ) ).
fof(fact_3769_arctan,axiom,
! [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)))))),hAPP_real_real(arctan,Y)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(arctan,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,hAPP_real_real(arctan,Y)) = Y ) ).
fof(fact_3770_arctan__tan,axiom,
! [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(arctan,hAPP_real_real(tan,X)) = X ) ) ).
fof(fact_3771_sin__expansion__lemma,axiom,
! [X,M] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),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),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_3772_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_3773_tan__total__pi4,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,X)),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 ) ) ).
fof(fact_3774_tan__total__pos,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
=> ? [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_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) = Y ) ) ).
fof(fact_3775_lemma__tan__total1,axiom,
! [Y] :
? [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) = Y ) ).
fof(fact_3776_tan__total,axiom,
! [Y] :
? [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) = Y
& ! [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),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_1) = Y )
=> Y_1 = X_1 ) ) ).
fof(fact_3777_lemma__tan__total,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
=> ? [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,Y),hAPP_real_real(tan,X_1))) ) ) ).
fof(fact_3778_reals__Archimedean6,axiom,
! [R_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_2))
=> ? [N_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(minus_minus_nat,N_1),one_one_nat))),R_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_nat_real(real_nat,N_1))) ) ) ).
fof(fact_3779_reals__Archimedean4,axiom,
! [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_eq_real,zero_zero_real),X))
=> ? [N_1] :
( 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_1)),Y)),X))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1))),Y))) ) ) ) ).
fof(fact_3780_reals__Archimedean6a,axiom,
! [R_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_2))
=> ? [N_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N_1)),R_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N_1)))) ) ) ).
fof(fact_3781_reals__Archimedean3,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> ! [Y_1] :
? [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_1)),X))) ) ).
fof(fact_3782_lemma__STAR__cos,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> 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),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_3783_sin__zero__iff,axiom,
! [Xa] :
( hAPP_real_real(sin,Xa) = zero_zero_real
<=> ( ? [N_1] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
& Xa = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_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))))) )
| ? [N_1] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
& Xa = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_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_3784_cos__zero__iff,axiom,
! [Xa] :
( hAPP_real_real(cos,Xa) = zero_zero_real
<=> ( ? [N_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
& Xa = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_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))))) )
| ? [N_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
& Xa = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_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_3785_odd__add,axiom,
! [Ma,Na] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na)))
<=> ~ ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma))
<=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) ) ) ).
fof(fact_3786_even__add,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ma),Na)))
<=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Ma))
<=> hBOOL(hAPP_nat_bool(even_odd_even_nat,Na)) ) ) ).
fof(fact_3787_even__sum__nat,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Xa),Ya)))
<=> ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
& hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)) )
| ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
& ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)) ) ) ) ).
fof(fact_3788_even__zero__nat,axiom,
hBOOL(hAPP_nat_bool(even_odd_even_nat,zero_zero_nat)) ).
fof(fact_3789_even__Suc,axiom,
! [Xa] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(suc,Xa)))
<=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) ) ).
fof(fact_3790_odd__1__nat,axiom,
~ hBOOL(hAPP_nat_bool(even_odd_even_nat,one_one_nat)) ).
fof(fact_3791_even__product__nat,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Xa),Ya)))
<=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
| hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)) ) ) ).
fof(fact_3792_real__of__nat__fact__not__zero,axiom,
! [N] : hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,N)) != zero_zero_real ).
fof(fact_3793_odd__pos,axiom,
! [N] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
fof(fact_3794_even__difference__nat,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Xa),Ya)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Xa),Ya))
| ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
& hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)) )
| ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
& ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Ya)) ) ) ) ).
fof(fact_3795_even__nat__def,axiom,
! [Xa] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
<=> hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,Xa))) ) ).
fof(fact_3796_Parity_Otransfer__int__nat__relations,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(even_odd_even_int,hAPP_nat_int(semiri1621563631at_int,Xa)))
<=> hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa)) ) ).
fof(fact_3797_real__of__nat__fact__ge__zero,axiom,
! [N] : 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)))) ).
fof(fact_3798_real__of__nat__fact__gt__zero,axiom,
! [N] : 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)))) ).
fof(fact_3799_even__nat__equiv__def2,axiom,
! [Xa] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
<=> ? [Y_1] : Xa = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))),Y_1) ) ).
fof(fact_3800_odd__nat__equiv__def2,axiom,
! [Xa] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
<=> ? [Y_1] : Xa = 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_1)) ) ).
fof(fact_3801_odd__nat__plus__one__div__two,axiom,
! [X] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X)),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),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) ) ).
fof(fact_3802_even__nat__plus__one__div__two,axiom,
! [X] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) ) ).
fof(fact_3803_odd__nat__mod__two__eq__one,axiom,
! [X] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
=> hAPP_nat_nat(div_mod_nat(X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,zero_zero_nat) ) ).
fof(fact_3804_even__nat__mod__two__eq__zero,axiom,
! [X] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
=> hAPP_nat_nat(div_mod_nat(X),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat ) ).
fof(fact_3805_odd__nat__equiv__def,axiom,
! [Xa] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
<=> hAPP_nat_nat(div_mod_nat(Xa),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = hAPP_nat_nat(suc,zero_zero_nat) ) ).
fof(fact_3806_even__nat__equiv__def,axiom,
! [Xa] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
<=> hAPP_nat_nat(div_mod_nat(Xa),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))) = zero_zero_nat ) ).
fof(fact_3807_even__power__nat,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,Xa),Ya)))
<=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Ya)) ) ) ).
fof(fact_3808_fact__diff__Suc,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),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)) = 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)),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ).
fof(fact_3809_even__num__iff,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))
<=> ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).
fof(fact_3810_odd__nat__div__two__times__two__plus__one,axiom,
! [X] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
=> 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),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat))))) = X ) ).
fof(fact_3811_even__nat__div__two__times__two,axiom,
! [X] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,X))
=> 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),hAPP_nat_nat(suc,hAPP_nat_nat(suc,zero_zero_nat)))) = X ) ).
fof(fact_3812_pos__int__even__equiv__nat__even,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(even_odd_even_int,Xa))
<=> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_int_nat(nat,Xa))) ) ) ).
fof(fact_3813_nat__even__iff__2__dvd,axiom,
! [Xa] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,Xa))
<=> 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)))),Xa)) ) ).
fof(fact_3814_even__dvd,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))
<=> 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)))),Na)) ) ).
fof(fact_3815_even__mult__two__ex,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))
<=> ? [M_1] : Na = 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_3816_even__even__mod__4__iff,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))
<=> hBOOL(hAPP_nat_bool(even_odd_even_nat,hAPP_nat_nat(div_mod_nat(Na),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_3817_odd__Suc__mult__two__ex,axiom,
! [Na] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))
<=> ? [M_1] : Na = 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_3818_lemma__even__div2,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ).
fof(fact_3819_lemma__STAR__sin,axiom,
! [Na] : 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,Na)),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,Na),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,Na))))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,zero_zero_real),Na)) = zero_zero_real ).
fof(fact_3820_lemma__not__even__div2,axiom,
! [N] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))) ) ).
fof(fact_3821_lemma__even__mod__4__div__2,axiom,
! [N] :
( 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))))) = 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),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3822_lemma__odd__mod__4__div__2,axiom,
! [N] :
( 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))))) = 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),one_one_nat)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3823_sin__zero__lemma,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( hAPP_real_real(sin,X) = zero_zero_real
=> ? [N_1] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
& X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_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_3824_cos__zero__lemma,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( hAPP_real_real(cos,X) = zero_zero_real
=> ? [N_1] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N_1))
& X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N_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_3825_odd__square,axiom,
! [N] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> ? [X_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N),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_1)),one_one_nat) ) ).
fof(fact_3826_sin__coeff__def,axiom,
! [X_1] :
( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
=> hAPP_nat_real(sin_coeff,X_1) = zero_zero_real )
& ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
=> hAPP_nat_real(sin_coeff,X_1) = 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_1),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_1))) ) ) ).
fof(fact_3827_even__square,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> ? [X_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,N),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_1) ) ).
fof(fact_3828_cos__coeff__def,axiom,
! [X_1] :
( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
=> hAPP_nat_real(cos_coeff,X_1) = 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_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,X_1))) )
& ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,X_1))
=> hAPP_nat_real(cos_coeff,X_1) = zero_zero_real ) ) ).
fof(fact_3829_fact__add__num__eq__if__nat,axiom,
! [M,N] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat
=> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = one_one_nat )
& ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) != zero_zero_nat
=> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),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)),one_one_nat))) ) ) ).
fof(fact_3830_fact__add__num__eq__if2__nat,axiom,
! [N,M] :
( ( M = zero_zero_nat
=> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(fact_fact_nat,N) )
& ( M != zero_zero_nat
=> hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),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))) ) ) ).
fof(fact_3831_fact__nonzero__nat,axiom,
! [N] : hAPP_nat_nat(fact_fact_nat,N) != zero_zero_nat ).
fof(fact_3832_fact__mono__nat,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N))) ) ).
fof(fact_3833_fact__1__nat,axiom,
hAPP_nat_nat(fact_fact_nat,one_one_nat) = one_one_nat ).
fof(fact_3834_Fact_Ofact__0__nat,axiom,
hAPP_nat_nat(fact_fact_nat,zero_zero_nat) = hAPP_nat_nat(suc,zero_zero_nat) ).
fof(fact_3835_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_3836_fact__gt__zero__nat,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact_fact_nat,N))) ).
fof(fact_3837_fact__less__mono__nat,axiom,
! [N,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))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(fact_fact_nat,M)),hAPP_nat_nat(fact_fact_nat,N))) ) ) ).
fof(fact_3838_Fact_Ofact__Suc,axiom,
! [X] : hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(suc,X)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,X)),hAPP_nat_nat(fact_fact_nat,X)) ).
fof(fact_3839_fact__dvd,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(fact_fact_nat,N)),hAPP_nat_nat(fact_fact_nat,M))) ) ).
fof(fact_3840_fact__ge__one__nat,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact_fact_nat,N))) ).
fof(fact_3841_fact__ge__Suc__0__nat,axiom,
! [N] : 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))) ).
fof(fact_3842_dvd__fact__nat,axiom,
! [N,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))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),hAPP_nat_nat(fact_fact_nat,N))) ) ) ).
fof(fact_3843_fact__plus__one__nat,axiom,
! [N] : hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),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),one_one_nat)),hAPP_nat_nat(fact_fact_nat,N)) ).
fof(fact_3844_fact__mod,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(fact_fact_nat,N)),hAPP_nat_nat(fact_fact_nat,M)) = zero_zero_nat ) ).
fof(fact_3845_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_3846_fact__reduce__nat,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_nat_nat(fact_fact_nat,N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,N),hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).
fof(fact_3847_Ln_Oaux1,axiom,
! [N,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),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),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),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),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),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)))) ) ) ).
fof(fact_3848_tan__sec,axiom,
! [X] :
( hAPP_real_real(cos,X) != 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)),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))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ).
fof(fact_3849_polar__ex2,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real))
=> ? [R_1,A_2] :
( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2))
& Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) ) ) ).
fof(fact_3850_polar__ex1,axiom,
! [X,Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
=> ? [R_1,A_2] :
( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2))
& Y = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(sin,A_2)) ) ) ).
fof(fact_3851_real__sqrt__inverse,axiom,
! [X] : hAPP_real_real(sqrt,hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X)) ).
fof(fact_3852_INVERSE__ZERO,axiom,
hAPP_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).
fof(fact_3853_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_3854_divide__real__def,axiom,
! [X,Y] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),hAPP_real_real(inverse_inverse_real,Y)) ).
fof(fact_3855_real__mult__inverse__cancel2,axiom,
! [Y,U_1,X1_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,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)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),U_1)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(inverse_inverse_real,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,U_1),hAPP_real_real(inverse_inverse_real,X1_1)))) ) ) ) ).
fof(fact_3856_real__mult__inverse__cancel,axiom,
! [Y,U_1,X1_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,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)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),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)),Y)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X1_1)),U_1))) ) ) ) ).
fof(fact_3857_real__mult__inverse__left,axiom,
! [X] :
( X != zero_zero_real
=> hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(inverse_inverse_real,X)),X) = one_one_real ) ).
fof(fact_3858_inv__real__of__nat__fact__ge__zero,axiom,
! [N] : 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))))) ).
fof(fact_3859_sqrt__divide__self__eq,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(sqrt,X)),X) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X)) ) ).
fof(fact_3860_inv__real__of__nat__fact__gt__zero,axiom,
! [N] : 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))))) ).
fof(fact_3861_ln__inverse,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(ln,hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(ln,X)) ) ).
fof(fact_3862_log__inverse,axiom,
! [X,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))
=> hAPP_real_real(log(A_3),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(log(A_3),X)) ) ) ) ).
fof(fact_3863_real__inv__sqrt__pow2,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,X))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) = hAPP_real_real(inverse_inverse_real,X) ) ).
fof(fact_3864_reals__Archimedean,axiom,
! [X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> ? [N_1] : 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_1)))),X)) ) ).
fof(fact_3865_DERIV__arccos,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real))
=> hBOOL(hAPP_real_bool(deriv_real(arccos,Xa),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,Xa),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))))) ) ) ).
fof(fact_3866_DERIV__arcsin,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real))
=> hBOOL(hAPP_real_bool(deriv_real(arcsin,Xa),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,Xa),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ) ) ).
fof(fact_3867_DERIV__sin,axiom,
! [Xa] : hBOOL(hAPP_real_bool(deriv_real(sin,Xa),hAPP_real_real(cos,Xa))) ).
fof(fact_3868_DERIV__cos,axiom,
! [Xa] : hBOOL(hAPP_real_bool(deriv_real(cos,Xa),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,Xa)))) ).
fof(fact_3869_DERIV__ln,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> hBOOL(hAPP_real_bool(deriv_real(ln,Xa),hAPP_real_real(inverse_inverse_real,Xa))) ) ).
fof(fact_3870_DERIV__ln__divide,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> hBOOL(hAPP_real_bool(deriv_real(ln,Xa),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),Xa))) ) ).
fof(fact_3871_DERIV__real__sqrt,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> hBOOL(hAPP_real_bool(deriv_real(sqrt,Xa),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,Xa))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ) ).
fof(fact_3872_DERIV__tan,axiom,
! [Xa] :
( hAPP_real_real(cos,Xa) != zero_zero_real
=> hBOOL(hAPP_real_bool(deriv_real(tan,Xa),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,Xa)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_3873_DERIV__arctan,axiom,
! [Xa] : hBOOL(hAPP_real_bool(deriv_real(arctan,Xa),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,Xa),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))))) ).
fof(fact_3874_DERIV__real__sqrt__generic,axiom,
! [D,Xa] :
( Xa != zero_zero_real
=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> D = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(inverse_inverse_real,hAPP_real_real(sqrt,Xa))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real))
=> D = 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,Xa)))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) )
=> hBOOL(hAPP_real_bool(deriv_real(sqrt,Xa),D)) ) ) ) ).
fof(fact_3875_DERIV__const__average,axiom,
! [Va,K,A_1,B_1] :
( A_1 != B_1
=> ( ! [X_1] : hBOOL(hAPP_real_bool(deriv_real(Va,X_1),K))
=> hAPP_real_real(Va,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,A_1),B_1)),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(Va,A_1)),hAPP_real_real(Va,B_1))),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) ) ) ).
fof(fact_3876_DERIV__local__max,axiom,
! [D_1,F,Xa,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,Xa),L))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1))
=> ( ! [Y_1] :
( 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,Xa),Y_1))),D_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Y_1)),hAPP_real_real(F,Xa))) )
=> L = zero_zero_real ) ) ) ).
fof(fact_3877_DERIV__local__min,axiom,
! [D_1,F,Xa,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,Xa),L))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1))
=> ( ! [Y_1] :
( 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,Xa),Y_1))),D_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,Xa)),hAPP_real_real(F,Y_1))) )
=> L = zero_zero_real ) ) ) ).
fof(fact_3878_DERIV__local__const,axiom,
! [D_1,F,Xa,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,Xa),L))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1))
=> ( ! [Y_1] :
( 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,Xa),Y_1))),D_1))
=> hAPP_real_real(F,Xa) = hAPP_real_real(F,Y_1) )
=> L = zero_zero_real ) ) ) ).
fof(fact_3879_DERIV__neg__dec__left,axiom,
! [F,Xa,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,Xa),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,Xa)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Xa),H_1)))) ) ) ) ) ) ).
fof(fact_3880_DERIV__pos__inc__left,axiom,
! [F,Xa,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,Xa),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,Xa),H_1))),hAPP_real_real(F,Xa))) ) ) ) ) ) ).
fof(fact_3881_DERIV__neg__dec__right,axiom,
! [F,Xa,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,Xa),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,Xa),H_1))),hAPP_real_real(F,Xa))) ) ) ) ) ) ).
fof(fact_3882_DERIV__pos__inc__right,axiom,
! [F,Xa,L] :
( hBOOL(hAPP_real_bool(deriv_real(F,Xa),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,Xa)),hAPP_real_real(F,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),H_1)))) ) ) ) ) ) ).
fof(fact_3883_DERIV__const__ratio__const2,axiom,
! [F,K,A_1,B_1] :
( A_1 != B_1
=> ( ! [X_1] : hBOOL(hAPP_real_bool(deriv_real(F,X_1),K))
=> hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1))),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1)) = K ) ) ).
fof(fact_3884_DERIV__const__ratio__const,axiom,
! [F,K,A_1,B_1] :
( A_1 != B_1
=> ( ! [X_1] : hBOOL(hAPP_real_bool(deriv_real(F,X_1),K))
=> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1)),K) ) ) ).
fof(fact_3885_DERIV__real__root__generic,axiom,
! [D,Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( Xa != zero_zero_real
=> ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> D = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat))))) ) )
=> ( ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real))
=> D = 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,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat)))))) ) )
=> ( ( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))
=> D = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat))))) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,Na),Xa),D)) ) ) ) ) ) ).
fof(fact_3886_DERIV__even__real__root,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,Na),Xa),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,Na))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ) ).
fof(fact_3887_DERIV__real__root,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,Na),Xa),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ).
fof(fact_3888_MVT2,axiom,
! [F,F_1,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1))
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) )
=> hBOOL(hAPP_real_bool(deriv_real(F,X_1),hAPP_real_real(F_1,X_1))) )
=> ? [Z] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),Z))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Z),B_1))
& hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1)),hAPP_real_real(F_1,Z)) ) ) ) ).
fof(fact_3889_real__root__zero,axiom,
! [N] : hAPP_real_real(hAPP_n546711566l_real(root,N),zero_zero_real) = zero_zero_real ).
fof(fact_3890_real__root__Suc__0,axiom,
! [X] : hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,zero_zero_nat)),X) = X ).
fof(fact_3891_real__root__eq__iff,axiom,
! [Xa,Ya,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) = hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)
<=> Xa = Ya ) ) ).
fof(fact_3892_real__root__commute,axiom,
! [X,N,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))
=> hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)) = hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_n546711566l_real(root,M),X)) ) ) ).
fof(fact_3893_odd__real__root__unique,axiom,
! [Y,X,N] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N) = X
=> hAPP_real_real(hAPP_n546711566l_real(root,N),X) = Y ) ) ).
fof(fact_3894_odd__real__root__pow,axiom,
! [X,N] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N) = X ) ).
fof(fact_3895_odd__real__root__power__cancel,axiom,
! [X,N] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,N))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = X ) ).
fof(fact_3896_real__root__le__iff,axiom,
! [Xa,Ya,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ) ).
fof(fact_3897_real__root__le__mono,axiom,
! [X,Y,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,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y))) ) ) ).
fof(fact_3898_real__root__eq__0__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) = zero_zero_real
<=> Xa = zero_zero_real ) ) ).
fof(fact_3899_real__root__less__iff,axiom,
! [Xa,Ya,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya)) ) ) ).
fof(fact_3900_real__root__less__mono,axiom,
! [X,Y,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),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y))) ) ) ).
fof(fact_3901_real__root__mult,axiom,
! [X,Y,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)) ) ).
fof(fact_3902_real__root__mult__exp,axiom,
! [X,N,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))
=> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),X) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ) ).
fof(fact_3903_real__root__power,axiom,
! [X,K_2,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),K_2)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),K_2) ) ).
fof(fact_3904_real__root__one,axiom,
! [N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),one_one_real) = one_one_real ) ).
fof(fact_3905_real__root__eq__1__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) = one_one_real
<=> Xa = one_one_real ) ) ).
fof(fact_3906_real__root__divide,axiom,
! [X,Y,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)) ) ).
fof(fact_3907_real__root__minus,axiom,
! [X,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(uminus_uminus_real,X)) = hAPP_real_real(uminus_uminus_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ).
fof(fact_3908_real__root__abs,axiom,
! [X,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(abs_abs_real,X)) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ).
fof(fact_3909_real__root__inverse,axiom,
! [X,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ).
fof(fact_3910_real__root__ge__0__iff,axiom,
! [Ya,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Ya)) ) ) ).
fof(fact_3911_real__root__le__0__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),zero_zero_real)) ) ) ).
fof(fact_3912_real__root__ge__zero,axiom,
! [X,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),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),X))) ) ) ).
fof(fact_3913_real__root__gt__0__iff,axiom,
! [Ya,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ya)) ) ) ).
fof(fact_3914_real__root__lt__0__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),zero_zero_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real)) ) ) ).
fof(fact_3915_real__root__gt__zero,axiom,
! [X,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))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N),X))) ) ) ).
fof(fact_3916_real__root__ge__1__iff,axiom,
! [Ya,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),Ya)) ) ) ).
fof(fact_3917_real__root__le__1__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),one_one_real)) ) ) ).
fof(fact_3918_real__root__decreasing,axiom,
! [X,N_4,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),N_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N_4),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),X))) ) ) ) ).
fof(fact_3919_real__root__gt__1__iff,axiom,
! [Ya,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),hAPP_real_real(hAPP_n546711566l_real(root,Na),Ya)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Ya)) ) ) ).
fof(fact_3920_real__root__lt__1__iff,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),one_one_real))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real)) ) ) ).
fof(fact_3921_real__root__strict__decreasing,axiom,
! [X,N_4,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),N_4))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N_4),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),X))) ) ) ) ).
fof(fact_3922_real__root__pos,axiom,
! [N,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(suc,N)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),hAPP_nat_nat(suc,N))) = X ) ).
fof(fact_3923_real__root__pos__pos,axiom,
! [X_3,N_5] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_5))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_n546711566l_real(root,N_5),X_3))) ) ) ).
fof(fact_3924_real__root__less__mono__lemma,axiom,
! [Y,X,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),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y))) ) ) ) ).
fof(fact_3925_real__root__mult__lemma,axiom,
! [Y,X,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),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N),Y)) ) ) ) ).
fof(fact_3926_real__root__pos__mult__exp,axiom,
! [X,N,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))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(hAPP_n546711566l_real(root,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),X) = hAPP_real_real(hAPP_n546711566l_real(root,M),hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ) ) ).
fof(fact_3927_real__root__pos2,axiom,
! [X,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),X))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = X ) ) ).
fof(fact_3928_real__root__pow__pos2,axiom,
! [X,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),X))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N) = X ) ) ).
fof(fact_3929_real__root__pos__unique,axiom,
! [X,Y,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),Y))
=> ( hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Y),N) = X
=> hAPP_real_real(hAPP_n546711566l_real(root,N),X) = Y ) ) ) ).
fof(fact_3930_real__root__pow__pos,axiom,
! [X,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))
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),N) = X ) ) ).
fof(fact_3931_real__root__increasing,axiom,
! [X,N_4,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),N_4))
=> ( 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),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_4),X))) ) ) ) ) ).
fof(fact_3932_real__root__strict__increasing,axiom,
! [X,N_4,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),N_4))
=> ( 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),one_one_real))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)),hAPP_real_real(hAPP_n546711566l_real(root,N_4),X))) ) ) ) ) ).
fof(fact_3933_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_3934_real__root__inverse__lemma,axiom,
! [X,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),X))
=> hAPP_real_real(hAPP_n546711566l_real(root,N),hAPP_real_real(inverse_inverse_real,X)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_n546711566l_real(root,N),X)) ) ) ).
fof(fact_3935_DERIV__odd__real__root,axiom,
! [Xa,Na] :
( ~ hBOOL(hAPP_nat_bool(even_odd_even_nat,Na))
=> ( Xa != zero_zero_real
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(root,Na),Xa),hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat))))))) ) ) ).
fof(fact_3936_DERIV__isconst__all,axiom,
! [Xa,Ya,F] :
( ! [X_1] : hBOOL(hAPP_real_bool(deriv_real(F,X_1),zero_zero_real))
=> hAPP_real_real(F,Xa) = hAPP_real_real(F,Ya) ) ).
fof(fact_3937_reals__Archimedean2,axiom,
! [X] :
? [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,N_1))) ).
fof(fact_3938_Union__SetS__finite,axiom,
! [A_1,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_1,P_5)))) ) ).
fof(fact_3939_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_3940_Suc__le__D,axiom,
! [N,M_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),M_4))
=> ? [M_1] : M_4 = hAPP_nat_nat(suc,M_1) ) ).
fof(fact_3941_wset__fin,axiom,
! [A_1,P_5] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(wset(A_1),P_5))) ).
fof(fact_3942_d22set__fin,axiom,
! [A_1] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(d22set,A_1))) ).
fof(fact_3943_SetS__elems__finite,axiom,
! [A_1,P_5,X_1] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),setS(A_1,P_5)))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_1)) ) ).
fof(fact_3944_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_3945_SetS__finite,axiom,
! [A_1,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_1,P_5))) ) ).
fof(fact_3946_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_3947_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_3948_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_3949_complex__norm,axiom,
! [X,Y] : hAPP_complex_real(norm_norm_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)) = hAPP_real_real(sqrt,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_3950_exists__least__lemma,axiom,
! [P_1] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( ? [X1] : hBOOL(hAPP_nat_bool(P_1,X1))
=> ? [N_1] :
( ~ hBOOL(hAPP_nat_bool(P_1,N_1))
& hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_1))) ) ) ) ).
fof(fact_3951_Complex__eq__0,axiom,
! [A_1,B_1] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,A_1),B_1) = zero_zero_complex
<=> ( A_1 = zero_zero_real
& B_1 = zero_zero_real ) ) ).
fof(fact_3952_complex__zero__def,axiom,
zero_zero_complex = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),zero_zero_real) ).
fof(fact_3953_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_3954_Complex__eq__number__of,axiom,
! [A_1,B_1,Wa] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,A_1),B_1) = hAPP_int_complex(number528085621omplex,Wa)
<=> ( A_1 = hAPP_int_real(number267125858f_real,Wa)
& B_1 = zero_zero_real ) ) ).
fof(fact_3955_Complex__eq__1,axiom,
! [A_1,B_1] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,A_1),B_1) = one_one_complex
<=> ( A_1 = one_one_real
& B_1 = zero_zero_real ) ) ).
fof(fact_3956_complex__one__def,axiom,
one_one_complex = hAPP_real_complex(hAPP_r265291036omplex(complex,one_one_real),zero_zero_real) ).
fof(fact_3957_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_3958_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_3959_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_3960_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_3961_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_3962_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_3963_complex_Osize_I1_J,axiom,
! [Real1,Real2] : complex_size(hAPP_real_complex(hAPP_r265291036omplex(complex,Real1),Real2)) = zero_zero_nat ).
fof(fact_3964_cos__arg__i__mult__zero__pos,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Y))
=> hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y))) = zero_zero_real ) ).
fof(fact_3965_cos__arg__i__mult__zero__neg,axiom,
! [Y] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y),zero_zero_real))
=> hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y))) = zero_zero_real ) ).
fof(fact_3966_complex__divide__def,axiom,
! [X,Y] : hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X),Y) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),hAPP_complex_complex(invers1449016382omplex,Y)) ).
fof(fact_3967_cos__arg__i__mult__zero,axiom,
! [Y] :
( Y != zero_zero_real
=> hAPP_real_real(cos,arg(hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),Y))) = zero_zero_real ) ).
fof(fact_3968_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_3969_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_3970_complex__i__not__zero,axiom,
ii != zero_zero_complex ).
fof(fact_3971_complex__i__not__one,axiom,
ii != one_one_complex ).
fof(fact_3972_complex__i__not__number__of,axiom,
! [W] : ii != hAPP_int_complex(number528085621omplex,W) ).
fof(fact_3973_complex__i__mult__minus,axiom,
! [X] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),X)) = hAPP_complex_complex(uminus473333897omplex,X) ).
fof(fact_3974_inverse__i,axiom,
hAPP_complex_complex(invers1449016382omplex,ii) = hAPP_complex_complex(uminus473333897omplex,ii) ).
fof(fact_3975_i__def,axiom,
ii = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),one_one_real) ).
fof(fact_3976_Complex__eq__i,axiom,
! [Xa,Ya] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,Xa),Ya) = ii
<=> ( Xa = zero_zero_real
& Ya = one_one_real ) ) ).
fof(fact_3977_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_3978_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_3979_i__squared,axiom,
hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = hAPP_int_complex(number528085621omplex,min) ).
fof(fact_3980_i__mult__eq2,axiom,
hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),ii) = hAPP_complex_complex(uminus473333897omplex,one_one_complex) ).
fof(fact_3981_complex__inverse__complex__split,axiom,
! [X,Y] : hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(X)),hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(Y)))) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,of_real_complex(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),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),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))))) ).
fof(fact_3982_not0__implies__Suc,axiom,
! [N] :
( N != zero_zero_nat
=> ? [M_1] : N = hAPP_nat_nat(suc,M_1) ) ).
fof(fact_3983_complex__eq__cancel__iff2,axiom,
! [Xa,Ya,Xaa] :
( hAPP_real_complex(hAPP_r265291036omplex(complex,Xa),Ya) = of_real_complex(Xaa)
<=> ( Xa = Xaa
& Ya = zero_zero_real ) ) ).
fof(fact_3984_complex__of__real__def,axiom,
! [R_2] : of_real_complex(R_2) = hAPP_real_complex(hAPP_r265291036omplex(complex,R_2),zero_zero_real) ).
fof(fact_3985_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_3986_complex__of__real__mult__Complex,axiom,
! [R_2,X,Y] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),X)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),Y)) ).
fof(fact_3987_Complex__mult__complex__of__real,axiom,
! [X,Y,R_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)),of_real_complex(R_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),R_2)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),R_2)) ).
fof(fact_3988_Complex__add__complex__of__real,axiom,
! [X,Y,R_2] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)),of_real_complex(R_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),R_2)),Y) ).
fof(fact_3989_complex__of__real__add__Complex,axiom,
! [R_2,X,Y] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,of_real_complex(R_2)),hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_2),X)),Y) ).
fof(fact_3990_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_3991_i__complex__of__real,axiom,
! [R_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,ii),of_real_complex(R_2)) = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_2) ).
fof(fact_3992_complex__of__real__i,axiom,
! [R_2] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_2)),ii) = hAPP_real_complex(hAPP_r265291036omplex(complex,zero_zero_real),R_2) ).
fof(fact_3993_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_3994_cmod__complex__polar,axiom,
! [R_2,A_3] : hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_2)),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_2) ).
fof(fact_3995_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_3996_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_3997_expi__zero,axiom,
expi(zero_zero_complex) = one_one_complex ).
fof(fact_3998_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_3999_nat__induct,axiom,
! [Na,P_1] :
( hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( ! [N_1] :
( hBOOL(hAPP_nat_bool(P_1,N_1))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_1))) )
=> hBOOL(hAPP_nat_bool(P_1,Na)) ) ) ).
fof(fact_4000_nat_Oexhaust,axiom,
! [Y] :
( Y != zero_zero_nat
=> ~ ! [Nat_1] : Y != hAPP_nat_nat(suc,Nat_1) ) ).
fof(fact_4001_zero__induct,axiom,
! [P_1,K] :
( hBOOL(hAPP_nat_bool(P_1,K))
=> ( ! [N_1] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_1)))
=> hBOOL(hAPP_nat_bool(P_1,N_1)) )
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ).
fof(fact_4002_DERIV__neg__imp__decreasing,axiom,
! [F,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1))
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) )
=> ? [Y_1] :
( hBOOL(hAPP_real_bool(deriv_real(F,X_1),Y_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),zero_zero_real)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1))) ) ) ).
fof(fact_4003_DERIV__pos__imp__increasing,axiom,
! [F,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1))
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) )
=> ? [Y_1] :
( hBOOL(hAPP_real_bool(deriv_real(F,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_real,hAPP_real_real(F,A_1)),hAPP_real_real(F,B_1))) ) ) ).
fof(fact_4004_int__induct,axiom,
! [I_1,P_1,K] :
( hBOOL(hAPP_int_bool(P_1,K))
=> ( ! [I_2] :
( is_int(I_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),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))
=> ( 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_minusinfinity,axiom,
! [P_1,P1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ! [X_1,K_1] :
( ( is_int(X_1)
& is_int(K_1) )
=> ( hBOOL(hAPP_int_bool(P1,X_1))
<=> hBOOL(hAPP_int_bool(P1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D_1)))) ) )
=> ( ? [Z] :
! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Z))
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
<=> hBOOL(hAPP_int_bool(P1,X_1)) ) ) )
=> ( ? [X1] : hBOOL(hAPP_int_bool(P1,X1))
=> ? [X1] :
( is_int(X1)
& hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ) ).
fof(fact_4006_plusinfinity,axiom,
! [P_1,P_2,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ! [X_1,K_1] :
( ( is_int(X_1)
& is_int(K_1) )
=> ( hBOOL(hAPP_int_bool(P_2,X_1))
<=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D_1)))) ) )
=> ( ? [Z] :
! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_1))
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
<=> hBOOL(hAPP_int_bool(P_2,X_1)) ) ) )
=> ( ? [X1] : hBOOL(hAPP_int_bool(P_2,X1))
=> ? [X1] :
( is_int(X1)
& hBOOL(hAPP_int_bool(P_1,X1)) ) ) ) ) ) ).
fof(fact_4007_coprime__sos,axiom,
! [X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),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),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_4008_coprime__exp,axiom,
! [N,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))) ) ).
fof(fact_4009_coprime__exp__imp,axiom,
! [N,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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N))) ) ).
fof(fact_4010_coprime__commute,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,B_1),A_1)) ) ).
fof(fact_4011_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_4012_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_4013_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_4014_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_4015_coprime__mul__eq,axiom,
! [D_1,A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),B_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_1),A_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_1),B_1)) ) ) ).
fof(fact_4016_coprime__1_H,axiom,
! [A_3] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,one_one_nat),A_3)) ).
fof(fact_4017_coprime__1,axiom,
! [A_3] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),one_one_nat)) ).
fof(fact_4018_coprime__refl,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,Na),Na))
<=> Na = one_one_nat ) ).
fof(fact_4019_floor__real__of__nat,axiom,
! [N] : archim1246769320r_real(hAPP_nat_real(real_nat,N)) = hAPP_nat_int(semiri1621563631at_int,N) ).
fof(fact_4020_natfloor__def,axiom,
! [X] : natfloor(X) = hAPP_int_nat(nat,archim1246769320r_real(X)) ).
fof(fact_4021_coprime__Suc0,axiom,
! [A_3] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_3),hAPP_nat_nat(suc,zero_zero_nat))) ).
fof(fact_4022_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_4023_coprime__0,axiom,
! [D_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,D_1),zero_zero_nat))
<=> D_1 = one_one_nat ) ).
fof(fact_4024_coprime__0_H,axiom,
! [D_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,zero_zero_nat),D_1))
<=> D_1 = one_one_nat ) ).
fof(fact_4025_divides__mul,axiom,
! [N,M,R_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),R_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M),N)),R_2)) ) ) ) ).
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_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1))
<=> ! [D_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B_1)) )
<=> D_2 = one_one_nat ) ) ).
fof(fact_4028_coprime__plus1,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),N)) ).
fof(fact_4029_coprime__exp2,axiom,
! [A_1,Na,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,A_1),hAPP_nat_nat(suc,Na))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_1),hAPP_nat_nat(suc,Na))))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1)) ) ).
fof(fact_4030_coprime__minus1,axiom,
! [N] :
( N != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N)) ) ).
fof(fact_4031_floor__eq3,axiom,
! [N,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(real_nat,N)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))))
=> hAPP_int_nat(nat,archim1246769320r_real(X)) = N ) ) ).
fof(fact_4032_floor__minus__real__of__nat,axiom,
! [N] : archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_nat_real(real_nat,N))) = hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,N)) ).
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,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(real_nat,N)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))))
=> hAPP_int_nat(nat,archim1246769320r_real(X)) = N ) ) ).
fof(fact_4035_coprime__bezout,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1))
<=> ? [X_1,Y_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),Y_1)) = 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_1),X_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),Y_1)) = 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_1,Y_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_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,B_2),Y_1)),one_one_nat) ) ) ).
fof(fact_4037_nat__number__of__diff__1,axiom,
! [V_1] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_1)),one_one_nat) = zero_zero_nat )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,V_1),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,V_1)),one_one_nat) = hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_1)) ) ) ).
fof(fact_4038_succ__pred,axiom,
! [X] :
( is_int(X)
=> hAPP_int_int(succ,hAPP_int_int(pred,X)) = X ) ).
fof(fact_4039_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_4040_pred__Bit1,axiom,
! [K_2] : hAPP_int_int(pred,hAPP_int_int(bit1,K_2)) = hAPP_int_int(bit0,K_2) ).
fof(fact_4041_le__iff__pred__less,axiom,
! [K,L] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,K),L))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(pred,K)),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,
! [Va] :
( hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
<=> hAPP_int_nat(number_number_of_nat,Va) = zero_zero_nat ) ).
fof(fact_4050_Suc__diff__number__of,axiom,
! [M,V_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,pls),V_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_int_nat(number_number_of_nat,V_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_int_nat(number_number_of_nat,hAPP_int_int(pred,V_1))) ) ).
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_2,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_2))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_2)) ) ) ).
fof(fact_4053_ResSet__def,axiom,
! [Ma,X_2] :
( hBOOL(hAPP_f448129468l_bool(resSet(Ma),X_2))
<=> ! [Y1,Y2] :
( ( is_int(Y1)
& is_int(Y2) )
=> ( ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y1),X_2))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y2),X_2))
& 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] : hAPP_nat_nat(size_size_nat,N) = N ).
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__complex__of__real,axiom,
! [X] : hAPP_complex_complex(cnj,of_real_complex(X)) = of_real_complex(X) ).
fof(fact_4064_complex__cnj__divide,axiom,
! [X,Y] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(invers1025623611omplex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).
fof(fact_4065_complex__cnj__power,axiom,
! [X,N] : hAPP_complex_complex(cnj,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,X),N)) = hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,hAPP_complex_complex(cnj,X)),N) ).
fof(fact_4066_cnj_Oadd,axiom,
! [X,Y] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).
fof(fact_4067_cnj_Ominus,axiom,
! [X] : hAPP_complex_complex(cnj,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_complex_complex(uminus473333897omplex,hAPP_complex_complex(cnj,X)) ).
fof(fact_4068_complex__cnj__cnj,axiom,
! [Z_1] : hAPP_complex_complex(cnj,hAPP_complex_complex(cnj,Z_1)) = Z_1 ).
fof(fact_4069_complex__cnj__cancel__iff,axiom,
! [Xa,Ya] :
( hAPP_complex_complex(cnj,Xa) = hAPP_complex_complex(cnj,Ya)
<=> Xa = Ya ) ).
fof(fact_4070_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_4071_complex__cnj__of__nat,axiom,
! [N] : hAPP_complex_complex(cnj,hAPP_nat_complex(semiri2020571505omplex,N)) = hAPP_nat_complex(semiri2020571505omplex,N) ).
fof(fact_4072_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_4073_cnj_Odiff,axiom,
! [X,Y] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).
fof(fact_4074_complex__cnj__inverse,axiom,
! [X] : hAPP_complex_complex(cnj,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_complex_complex(invers1449016382omplex,hAPP_complex_complex(cnj,X)) ).
fof(fact_4075_cnj_Ozero,axiom,
hAPP_complex_complex(cnj,zero_zero_complex) = zero_zero_complex ).
fof(fact_4076_complex__cnj__zero__iff,axiom,
! [Z_2] :
( hAPP_complex_complex(cnj,Z_2) = zero_zero_complex
<=> Z_2 = zero_zero_complex ) ).
fof(fact_4077_complex__cnj__number__of,axiom,
! [W] : hAPP_complex_complex(cnj,hAPP_int_complex(number528085621omplex,W)) = hAPP_int_complex(number528085621omplex,W) ).
fof(fact_4078_complex__cnj__mult,axiom,
! [X,Y] : hAPP_complex_complex(cnj,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,hAPP_complex_complex(cnj,X)),hAPP_complex_complex(cnj,Y)) ).
fof(fact_4079_complex__cnj__one,axiom,
hAPP_complex_complex(cnj,one_one_complex) = one_one_complex ).
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] : hAPP_C498520661umeral(hAPP_C1594335432umeral(minus_1690775515umeral,code_S1047413653umeral(N)),one_on1645066479umeral) = N ).
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_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),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_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),K_4))) ) ).
fof(fact_4087_cnj_Obounded,axiom,
? [K_4] :
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(cnj,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),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] :
( Y != zero_z126310315umeral
=> ~ ! [Code_numeral] : Y != code_S1047413653umeral(Code_numeral) ) ).
fof(fact_4091_Im,axiom,
! [X,Y] : hAPP_complex_real(im,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)) = Y ).
fof(fact_4092_complex__Im__cnj,axiom,
! [X] : hAPP_complex_real(im,hAPP_complex_complex(cnj,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X)) ).
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,Y] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)) ).
fof(fact_4098_complex__Im__number__of,axiom,
! [V_1] : hAPP_complex_real(im,hAPP_int_complex(number528085621omplex,V_1)) = zero_zero_real ).
fof(fact_4099_Im_Ominus,axiom,
! [X] : hAPP_complex_real(im,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X)) ).
fof(fact_4100_Im_Odiff,axiom,
! [X,Y] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)) ).
fof(fact_4101_complex__Im__of__nat,axiom,
! [N] : hAPP_complex_real(im,hAPP_nat_complex(semiri2020571505omplex,N)) = zero_zero_real ).
fof(fact_4102_abs__Im__le__cmod,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(im,X))),hAPP_complex_real(norm_norm_complex,X))) ).
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_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),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_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),K_4))) ) ).
fof(fact_4108_Im_Obounded,axiom,
? [K_4] :
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(im,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),K_4))) ).
fof(fact_4109_complex__inverse__def,axiom,
! [X] : hAPP_complex_complex(invers1449016382omplex,X) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),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)),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))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),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)),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] : hAPP_complex_real(im,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),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)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_4112_Re__complex__of__real,axiom,
! [Z_1] : hAPP_complex_real(re,of_real_complex(Z_1)) = Z_1 ).
fof(fact_4113_Re,axiom,
! [X,Y] : hAPP_complex_real(re,hAPP_real_complex(hAPP_r265291036omplex(complex,X),Y)) = X ).
fof(fact_4114_complex__Re__cnj,axiom,
! [X] : hAPP_complex_real(re,hAPP_complex_complex(cnj,X)) = hAPP_complex_real(re,X) ).
fof(fact_4115_complex__eqI,axiom,
! [X,Y] :
( hAPP_complex_real(re,X) = hAPP_complex_real(re,Y)
=> ( hAPP_complex_real(im,X) = hAPP_complex_real(im,Y)
=> X = Y ) ) ).
fof(fact_4116_complex__eq__iff,axiom,
! [Xa,Ya] :
( Xa = Ya
<=> ( hAPP_complex_real(re,Xa) = hAPP_complex_real(re,Ya)
& hAPP_complex_real(im,Xa) = hAPP_complex_real(im,Ya) ) ) ).
fof(fact_4117_complex__Re__le__cmod,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_complex_real(re,X)),hAPP_complex_real(norm_norm_complex,X))) ).
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,Y] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y)) ).
fof(fact_4122_complex__Re__one,axiom,
hAPP_complex_real(re,one_one_complex) = one_one_real ).
fof(fact_4123_Re_Ominus,axiom,
! [X] : hAPP_complex_real(re,hAPP_complex_complex(uminus473333897omplex,X)) = hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X)) ).
fof(fact_4124_complex__Re__number__of,axiom,
! [V_1] : hAPP_complex_real(re,hAPP_int_complex(number528085621omplex,V_1)) = hAPP_int_real(number267125858f_real,V_1) ).
fof(fact_4125_Re_Odiff,axiom,
! [X,Y] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y)) ).
fof(fact_4126_complex__Re__of__nat,axiom,
! [N] : hAPP_complex_real(re,hAPP_nat_complex(semiri2020571505omplex,N)) = hAPP_nat_real(semiri132038758t_real,N) ).
fof(fact_4127_abs__Re__le__cmod,axiom,
! [X] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_complex_real(re,X))),hAPP_complex_real(norm_norm_complex,X))) ).
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,Y] : hAPP_complex_real(im,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(im,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(re,Y))) ).
fof(fact_4130_complex__Re__mult,axiom,
! [X,Y] : hAPP_complex_real(re,hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y))) ).
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,Y] : hAPP_complex_complex(hAPP_c172932010omplex(plus_plus_complex,X),Y) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(re,X)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y))) ).
fof(fact_4134_complex__minus__def,axiom,
! [X] : hAPP_complex_complex(uminus473333897omplex,X) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(uminus_uminus_real,hAPP_complex_real(re,X))),hAPP_real_real(uminus_uminus_real,hAPP_complex_real(im,X))) ).
fof(fact_4135_complex__mult__def,axiom,
! [X,Y] : hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,X),Y) = 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)),hAPP_complex_real(re,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(im,Y)))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(re,X)),hAPP_complex_real(im,Y))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(im,X)),hAPP_complex_real(re,Y)))) ).
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] : hAPP_complex_real(re,hAPP_complex_complex(invers1449016382omplex,X)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_complex_real(re,X)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_complex_real(re,X)),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)),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_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),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_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),K_4))) ) ).
fof(fact_4141_Re_Obounded,axiom,
? [K_4] :
! [X_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(norm_norm_real,hAPP_complex_real(re,X_1))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_complex_real(norm_norm_complex,X_1)),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,A_3] : hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3)) = hAPP_complex_real(im,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N)) ).
fof(fact_4144_cos__n__Re__cis__pow__n,axiom,
! [N,A_3] : hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3)) = hAPP_complex_real(re,hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N)) ).
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] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,cis(A_3)),N) = cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3)) ).
fof(fact_4153_cis__real__of__nat__Suc__mult,axiom,
! [N,A_3] : cis(hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,N))),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)),A_3))) ).
fof(fact_4154_bezw__0,axiom,
! [X] : bezw(X,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_I3_J,axiom,
! [Z_1,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).
fof(fact_4156_termination__basic__simps_I2_J,axiom,
! [Y,X,Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).
fof(fact_4157_termination__basic__simps_I1_J,axiom,
! [Z_1,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_1))) ) ).
fof(fact_4158_termination__basic__simps_I5_J,axiom,
! [X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).
fof(fact_4159_termination__basic__simps_I4_J,axiom,
! [Y,X,Z_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),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_primes__eq,axiom,
! [Q_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(prime,Q_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),Q_1))
=> P_3 = Q_1 ) ) ) ).
fof(fact_4163_prime__1,axiom,
~ hBOOL(hAPP_nat_bool(prime,one_one_nat)) ).
fof(fact_4164_prime__0,axiom,
~ hBOOL(hAPP_nat_bool(prime,zero_zero_nat)) ).
fof(fact_4165_distinct__prime__coprime,axiom,
! [Q_1,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(prime,Q_1))
=> ( P_3 != Q_1
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),Q_1)) ) ) ) ).
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_1,B_1,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_1),B_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),A_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_5),B_1)) ) ) ) ).
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,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)))
=> ( 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)) ) ) ) ).
fof(fact_4171_prime__product,axiom,
! [P_3,Q_1] :
( hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,P_3),Q_1)))
=> ( P_3 = one_one_nat
| Q_1 = one_one_nat ) ) ).
fof(fact_4172_prime__dvd__power,axiom,
! [A_3,N,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)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),A_3)) ) ) ).
fof(fact_4173_prime__divexp__n,axiom,
! [X,N,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),N)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N))) ) ) ).
fof(fact_4174_prime__divexp,axiom,
! [X,N,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),N)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),X)) ) ) ).
fof(fact_4175_prime__exp,axiom,
! [P_5,Na] :
( hBOOL(hAPP_nat_bool(prime,hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),Na)))
<=> ( hBOOL(hAPP_nat_bool(prime,P_5))
& Na = one_one_nat ) ) ).
fof(fact_4176_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_4177_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_4178_prime__coprime__strong,axiom,
! [N,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N)) ) ) ).
fof(fact_4179_coprime__prime__eq,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,A_1),B_1))
<=> ! [P_4] :
~ ( hBOOL(hAPP_nat_bool(prime,P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),A_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),B_1)) ) ) ).
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,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( N != zero_zero_nat
=> ( N = 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)) ) ) ) ).
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_1,K,P_5] :
( hBOOL(hAPP_nat_bool(prime,P_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_5),K)))
<=> ? [I_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_2),K))
& D_1 = 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,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)),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)),A_3)) ) ) ) ).
fof(fact_4186_prime__coprime__lt,axiom,
! [X,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),X))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),P_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),P_3)) ) ) ) ).
fof(fact_4187_prime__coprime,axiom,
! [N,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( N = one_one_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,P_3),N)) ) ) ).
fof(fact_4188_negateSnd__eq,axiom,
! [Q_1,R_2] : hAPP_P1975530577nt_int(negateSnd,hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),R_2)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_1),hAPP_int_int(uminus_uminus_int,R_2)) ).
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,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)),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)),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)),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_1] : divmod_int(P_3,Q_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,P_3),Q_1)),hAPP_int_int(div_mod_int(P_3),Q_1)) ).
fof(fact_4197_div__mod__code__numeral__def,axiom,
! [N,M] : code_d418564891umeral(N,M) = hAPP_C2132160860umeral(hAPP_C1614127039umeral(produc2136830103umeral,hAPP_C498520661umeral(hAPP_C1594335432umeral(div_di1218280263umeral,N),M)),hAPP_C498520661umeral(div_mo1740067990umeral(N),M)) ).
fof(fact_4198_xzgcd__linear,axiom,
! [M,R_2,S_1,T_1,N] :
( is_int(R_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> ( xzgcd(M,N) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_2),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,S_1),T_1))
=> R_2 = 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)) ) ) ) ).
fof(fact_4199_xzgcda_Osimps,axiom,
! [M,N,R_4,S_3,S_1,T_2,T_1,R_2] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,R_2),zero_zero_int))
=> xzgcda(M,N,R_4,R_2,S_3,S_1,T_2,T_1) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,R_4),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_2),zero_zero_int))
=> xzgcda(M,N,R_4,R_2,S_3,S_1,T_2,T_1) = xzgcda(M,N,R_2,hAPP_int_int(div_mod_int(R_4),R_2),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_4),R_2)),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_4),R_2)),T_1))) ) ) ).
fof(fact_4200_xzgcd__def,axiom,
! [M,N] : xzgcd(M,N) = xzgcda(M,N,M,N,one_one_int,zero_zero_int,zero_zero_int,one_one_int) ).
fof(fact_4201_xzgcda__linear,axiom,
! [M,N,R_4,S_3,S_1,T_2,T_1,Rn,Sn_1,Tn_1,R_2] :
( is_int(Rn)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_2))
=> ( xzgcda(M,N,R_4,R_2,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_4 = 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))
=> ( R_2 = 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))
=> 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)) ) ) ) ) ) ).
fof(fact_4202_coprime__prime__dvd__ex,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(coprime,X),Y))
=> ? [P_4] :
( hBOOL(hAPP_nat_bool(prime,P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),X))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_4),Y)) ) ) ).
fof(fact_4203_prime__power__exp,axiom,
! [X,K_2,N,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( N != zero_zero_nat
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,X),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2)
=> ? [I_2] : X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2) ) ) ) ).
fof(fact_4204_prime__factor,axiom,
! [N] :
( N != 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)) ) ) ).
fof(fact_4205_divmod__nat__step,axiom,
! [M,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,N),M))
=> divmod_nat(M,N) = 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)),N))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ).
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_1,Y_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_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,P_3),Y_1)),one_one_nat) ) ) ).
fof(fact_4207_not__prime__ex__mk,axiom,
! [N] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,zero_zero_nat)),N))
& ~ hBOOL(hAPP_nat_bool(prime,N)) )
=> ? [M_1,K_1] :
( 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_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K_1),N))
& N = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,M_1),K_1) ) ) ).
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] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> divmod_nat(M,N) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),M) ) ).
fof(fact_4210_divmod__nat__div__mod,axiom,
! [M,N] : divmod_nat(M,N) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),hAPP_nat_nat(div_mod_nat(M),N)) ).
fof(fact_4211_divmod__nat__rel__mult2__eq,axiom,
! [C_1,A_3,B_2,Q_1,R_2] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(A_3,B_2),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_1),R_2)))
=> ( 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_1),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_1),C_1))),R_2)))) ) ) ) ).
fof(fact_4212_divmod__nat__rel__mult1__eq,axiom,
! [A_3,B_2,C_1,Q_1,R_2] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(B_2,C_1),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_1),R_2)))
=> ( 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_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),R_2)),C_1))),hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_3),R_2)),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,Qr_1] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr_1))
=> ( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr_2))
=> Qr_1 = Qr_2 ) ) ).
fof(fact_4215_divmod__nat__rel__divmod__nat,axiom,
! [M,N] : hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),divmod_nat(M,N))) ).
fof(fact_4216_divmod__nat__eq,axiom,
! [M,N,Qr_1] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),Qr_1))
=> divmod_nat(M,N) = Qr_1 ) ).
fof(fact_4217_div__eq,axiom,
! [M,N,Q_1,R_2] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_1),R_2)))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = Q_1 ) ).
fof(fact_4218_mod__eq,axiom,
! [M,N,Q_1,R_2] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_1),R_2)))
=> hAPP_nat_nat(div_mod_nat(M),N) = R_2 ) ).
fof(fact_4219_divmod__nat__rel,axiom,
! [M,N] : hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N)),hAPP_nat_nat(div_mod_nat(M),N)))) ).
fof(fact_4220_divmod__nat__rel__ex,axiom,
! [M,N] :
~ ! [Q_4,R_1] : ~ hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(M,N),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Q_4),R_1))) ).
fof(fact_4221_euclid,axiom,
! [N] :
? [P_4] :
( hBOOL(hAPP_nat_bool(prime,P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),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,Y] :
( ( Y = zero_zero_nat
=> bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,one_one_int),zero_zero_int) )
& ( Y != zero_zero_nat
=> bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y))))) ) ) ).
fof(fact_4230_bezw__non__0,axiom,
! [X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Y))
=> bezw(X,Y) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,hAPP_P1175774780nt_int(product_fst_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(Y,hAPP_nat_nat(div_mod_nat(X),Y)))),hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,X),Y))))) ) ).
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] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,M),N) = hAPP_P1654798560at_nat(product_fst_nat_nat,divmod_nat(M,N)) ).
fof(fact_4239_mod__nat__def,axiom,
! [M,N] : hAPP_nat_nat(div_mod_nat(M),N) = hAPP_P1654798560at_nat(product_snd_nat_nat,divmod_nat(M,N)) ).
fof(fact_4240_divmod__nat__rel__def,axiom,
! [Ma,Na,Qr] :
( hBOOL(hAPP_P1555980039t_bool(divmod_nat_rel(Ma,Na),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)),Na)),hAPP_P1654798560at_nat(product_snd_nat_nat,Qr))
& ( Na = zero_zero_nat
=> hAPP_P1654798560at_nat(product_fst_nat_nat,Qr) = zero_zero_nat )
& ( Na != zero_zero_nat
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( 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)),Na)) ) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),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,Na,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),Na))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),Na))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,B_1),A_1)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Na)) ) ).
fof(fact_4242_Bolzano__bisect__le,axiom,
! [P_1,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1)))) ) ).
fof(fact_4243_Bolzano__bisect_Osimps_I1_J,axiom,
! [P_1,A_1,B_1] : hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),zero_zero_nat) = hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_1) ).
fof(fact_4244_Bolzano__bisect__fst__le__Suc,axiom,
! [P_1,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1))),hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),hAPP_nat_nat(suc,N_1))))) ) ).
fof(fact_4245_Bolzano__bisect__Suc__le__snd,axiom,
! [P_1,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),hAPP_nat_nat(suc,N_1)))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1)))) ) ).
fof(fact_4246_not__P__Bolzano__bisect,axiom,
! [Na,A_1,B_1,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_1),B_1)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> ~ 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_1,B_1),Na))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),Na))))) ) ) ) ).
fof(fact_4247_not__P__Bolzano__bisect_H,axiom,
! [A_1,B_1,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_1),B_1)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> ! [N_1] : ~ 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_1,B_1),N_1))),hAPP_P731461727l_real(produc556554744l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1))))) ) ) ) ).
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_2,A_3,N] : hAPP_nat_complex(hAPP_c1088319240omplex(power_power_complex,rcis(R_2,A_3)),N) = rcis(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,R_2),N),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,N)),A_3)) ).
fof(fact_4251_less__eq__code__int__def,axiom,
! [Na,Ma] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le258702272de_int,Na),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,quickcheck_int_of(Na)),quickcheck_int_of(Ma))) ) ).
fof(fact_4252_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_4253_less__code__int__def,axiom,
! [Na,Ma] :
( hBOOL(hAPP_Q2096512830t_bool(hAPP_Q1010094925t_bool(ord_le1860547276de_int,Na),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,quickcheck_int_of(Na)),quickcheck_int_of(Ma))) ) ).
fof(fact_4254_Quickcheck__Narrowing_Oint__of__inject,axiom,
! [K,L] :
( quickcheck_int_of(K) = quickcheck_int_of(L)
<=> K = L ) ).
fof(fact_4255_code__int_Oint__of__inject,axiom,
! [Xa,Ya] :
( quickcheck_int_of(Xa) = quickcheck_int_of(Ya)
<=> Xa = Ya ) ).
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_2] : rcis(R_2,zero_zero_real) = of_real_complex(R_2) ).
fof(fact_4258_complex__mod__rcis,axiom,
! [R_2,A_3] : hAPP_complex_real(norm_norm_complex,rcis(R_2,A_3)) = hAPP_real_real(abs_abs_real,R_2) ).
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_2,A_3] : hAPP_complex_real(re,rcis(R_2,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_real_real(cos,A_3)) ).
fof(fact_4262_Im__rcis,axiom,
! [R_2,A_3] : hAPP_complex_real(im,rcis(R_2,A_3)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),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_2,A_3] : rcis(R_2,A_3) = hAPP_complex_complex(hAPP_c172932010omplex(times_times_complex,of_real_complex(R_2)),cis(A_3)) ).
fof(fact_4266_rcis__inverse,axiom,
! [R_2,A_3] : hAPP_complex_complex(invers1449016382omplex,rcis(R_2,A_3)) = rcis(hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,one_one_real),R_2),hAPP_real_real(uminus_uminus_real,A_3)) ).
fof(fact_4267_div__mod__code__int__def,axiom,
! [N,M] : quickc495462417de_int(N,M) = hAPP_Q1256397320de_int(hAPP_Q952608535de_int(produc1318306967de_int,hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N),M)),hAPP_Q1762011733de_int(div_mo231679042de_int(N),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] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hAPP_int_int(fact_fact_int,X) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X))) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hAPP_int_int(fact_fact_int,X) = zero_zero_int ) ) ).
fof(fact_4271_fact__mono__int,axiom,
! [M,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N))
=> 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))) ) ).
fof(fact_4272_fact__1__int,axiom,
hAPP_int_int(fact_fact_int,one_one_int) = one_one_int ).
fof(fact_4273_fact__0__int,axiom,
hAPP_int_int(fact_fact_int,zero_zero_int) = one_one_int ).
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__less__mono__int,axiom,
! [N,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))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(fact_fact_int,M)),hAPP_int_int(fact_fact_int,N))) ) ) ).
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] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
=> hAPP_int_int(fact_fact_int,N) != zero_zero_int ) ).
fof(fact_4278_transfer__nat__int__factorial__closure,axiom,
! [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(fact_fact_int,X))) ) ).
fof(fact_4279_transfer__int__nat__factorial,axiom,
! [X] : hAPP_int_int(fact_fact_int,hAPP_nat_int(semiri1621563631at_int,X)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(fact_fact_nat,X)) ).
fof(fact_4280_fact__ge__one__int,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,one_one_int),hAPP_int_int(fact_fact_int,N))) ) ).
fof(fact_4281_fact__gt__zero__int,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_int_int(fact_fact_int,N))) ) ).
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,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))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),hAPP_int_int(fact_fact_int,N))) ) ) ).
fof(fact_4284_transfer__nat__int__factorial,axiom,
! [X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hAPP_nat_nat(fact_fact_nat,hAPP_int_nat(nat,X)) = hAPP_int_nat(nat,hAPP_int_int(fact_fact_int,X)) ) ).
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] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),N))
=> hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),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),one_one_int)),hAPP_int_int(fact_fact_int,N)) ) ).
fof(fact_4287_fact__reduce__int,axiom,
! [N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> hAPP_int_int(fact_fact_int,N) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,N),hAPP_int_int(fact_fact_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),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_1,P_5] :
( hBOOL(hAPP_int_bool(zprime,P_5))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),P_5))
=> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A_1),P_5)) = hAPP_int_nat(nat,A_1) ) ) ).
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_1] :
? [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_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),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,axiom,
! [B_1,A_1,Ma] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(bnorRset(A_1),Ma)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1)) ) ).
fof(fact_4292_Bnor__mem__zle__swap,axiom,
! [Ma,A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1))
=> ~ hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(bnorRset(A_1),Ma))) ) ).
fof(fact_4293_Bnor__fin,axiom,
! [A_1,Ma] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(bnorRset(A_1),Ma))) ).
fof(fact_4294_Bnor__mem__zg,axiom,
! [B_1,A_1,Ma] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(bnorRset(A_1),Ma)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1)) ) ).
fof(fact_4295_lemma__BOLZANO,axiom,
! [A_1,B_1,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_1] :
? [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_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),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_1),B_1))
=> hBOOL(hAPP_P1333854989l_bool(P_1,hAPP_r1195171167l_real(hAPP_r2019696015l_real(produc865579683l_real,A_1),B_1))) ) ) ) ).
fof(fact_4296_bezw__aux,axiom,
! [X,Y] : 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,Y))),hAPP_nat_int(semiri1621563631at_int,X))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_snd_int_int,bezw(X,Y))),hAPP_nat_int(semiri1621563631at_int,Y))) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) ).
fof(fact_4297_gcd__dvd1__nat,axiom,
! [M,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)),M)) ).
fof(fact_4298_gcd__dvd2__nat,axiom,
! [M,N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N)),N)) ).
fof(fact_4299_gcd__dvd__prod__nat,axiom,
! [M,N,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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,K_2),N))) ).
fof(fact_4300_coprime__exp2__nat,axiom,
! [N,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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),M)) = one_one_nat ) ).
fof(fact_4301_gcd__mult__distrib__nat,axiom,
! [K_2,M,N] : 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)) = 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)) ).
fof(fact_4302_gcd__semilattice__nat_Oinf__assoc,axiom,
! [X,Y,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),Y)),Z_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)) ).
fof(fact_4303_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_4304_gcd__semilattice__nat_Oinf__left__commute,axiom,
! [X,Y,Z_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Z_1)) ).
fof(fact_4305_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_4306_gcd__semilattice__nat_Oinf__left__idem,axiom,
! [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) ).
fof(fact_4307_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_4308_gcd__semilattice__nat_Oinf__commute,axiom,
! [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),X) ).
fof(fact_4309_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_4310_gcd__idem__nat,axiom,
! [X] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),X) = X ).
fof(fact_4311_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_4312_gcd__semilattice__nat_Oinf__le1,axiom,
! [X,Y] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),X)) ).
fof(fact_4313_gcd__semilattice__nat_Oinf__le2,axiom,
! [X,Y] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)),Y)) ).
fof(fact_4314_gcd__semilattice__nat_Ole__iff__inf,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
<=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Xa),Ya) = Xa ) ).
fof(fact_4315_gcd__greatest__iff__nat,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),Na)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),Ma))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),Na)) ) ) ).
fof(fact_4316_gcd__semilattice__nat_Ole__inf__iff,axiom,
! [Xa,Ya,Z_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ya),Z_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Ya))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Xa),Z_2)) ) ) ).
fof(fact_4317_gcd__unique__nat,axiom,
! [B_1,D_1,A_1] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),A_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_1),B_1))
& ! [E] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),A_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),B_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,E),D_1)) ) )
<=> D_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_1),B_1) ) ).
fof(fact_4318_gcd__semilattice__nat_Ole__infI1,axiom,
! [B_2,A_3,X] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X))
=> 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)) ) ).
fof(fact_4319_gcd__semilattice__nat_Ole__infI2,axiom,
! [A_3,B_2,X] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),X))
=> 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)) ) ).
fof(fact_4320_gcd__proj1__if__dvd__nat,axiom,
! [X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X ) ).
fof(fact_4321_gcd__proj2__if__dvd__nat,axiom,
! [Y,X] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,Y),X))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = Y ) ).
fof(fact_4322_gcd__greatest__nat,axiom,
! [N,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))
=> 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))) ) ) ).
fof(fact_4323_gcd__semilattice__nat_Ole__infI,axiom,
! [B_2,X,A_3] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),A_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) ) ) ).
fof(fact_4324_gcd__semilattice__nat_Oinf__greatest,axiom,
! [Z_1,X,Y] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Y))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),Z_1))) ) ) ).
fof(fact_4325_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_4326_dvd__gcd__D1__nat,axiom,
! [K_2,M,N] :
( 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)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),M)) ) ).
fof(fact_4327_dvd__gcd__D2__nat,axiom,
! [K_2,M,N] :
( 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)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K_2),N)) ) ).
fof(fact_4328_gcd__semilattice__nat_Ole__infE,axiom,
! [X,A_3,B_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),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),A_3))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2)) ) ) ).
fof(fact_4329_gcd__semilattice__nat_Oless__infI1,axiom,
! [B_2,A_3,X] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,A_3),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),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))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) ) ) ).
fof(fact_4330_gcd__semilattice__nat_Oless__infI2,axiom,
! [A_3,B_2,X] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,B_2),X))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),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))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2))) ) ) ).
fof(fact_4331_gcd__add2__nat,axiom,
! [M,N] : 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)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ).
fof(fact_4332_gcd__add1__nat,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ).
fof(fact_4333_gcd__exp__nat,axiom,
! [A_3,N,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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,B_2),N)) = 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) ).
fof(fact_4334_gcd__red__nat,axiom,
! [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) ).
fof(fact_4335_gcd__lcm__complete__lattice__nat_Oinf__bot__right,axiom,
! [X] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),one_one_nat) = one_one_nat ).
fof(fact_4336_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_4337_gcd__lcm__complete__lattice__nat_Oinf__bot__left,axiom,
! [X] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,one_one_nat),X) = one_one_nat ).
fof(fact_4338_gcd__0__left__nat,axiom,
! [X] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),X) = X ).
fof(fact_4339_gcd__0__nat,axiom,
! [X] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),zero_zero_nat) = X ).
fof(fact_4340_gcd__zero__nat,axiom,
! [Ma,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Ma),Na) = zero_zero_nat
<=> ( Ma = zero_zero_nat
& Na = zero_zero_nat ) ) ).
fof(fact_4341_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff,axiom,
! [Xa,Ya] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Xa),Ya) = zero_zero_nat
<=> ( Xa = zero_zero_nat
& Ya = zero_zero_nat ) ) ).
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,Na] :
( 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),Na)))
<=> ( Ma != zero_zero_nat
| Na != 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] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(suc,N)),N) = one_one_nat ).
fof(fact_4347_gcd__add__mult__nat,axiom,
! [M,K_2,N] : 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)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ).
fof(fact_4348_coprime__common__divisor__nat,axiom,
! [X,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),A_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,X),B_2))
=> X = one_one_nat ) ) ) ).
fof(fact_4349_coprime__nat,axiom,
! [A_1,B_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_1),B_1) = one_one_nat
<=> ! [D_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B_1)) )
<=> D_2 = one_one_nat ) ) ).
fof(fact_4350_coprime__plus__one__nat,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),N) = one_one_nat ).
fof(fact_4351_coprime__mul__eq__nat,axiom,
! [D_1,A_1,B_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),B_1)) = one_one_nat
<=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_1),A_1) = one_one_nat
& hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,D_1),B_1) = one_one_nat ) ) ).
fof(fact_4352_gcd__mult__cancel__nat,axiom,
! [M,K_2,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N) = 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) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ).
fof(fact_4353_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_4354_coprime__crossproduct__nat,axiom,
! [B_1,C,A_1,D_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_1),D_1) = one_one_nat
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,B_1),C) = one_one_nat
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_1),C) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,B_1),D_1)
<=> ( A_1 = B_1
& C = D_1 ) ) ) ) ).
fof(fact_4355_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_4356_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_4357_gcd__diff2__nat,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ).
fof(fact_4358_gcd__diff1__nat,axiom,
! [N,M] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),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)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ).
fof(fact_4359_gcd__nat_Osimps,axiom,
! [X,Y] :
( ( Y = zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = X )
& ( Y != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) ) ) ).
fof(fact_4360_gcd__non__0__nat,axiom,
! [X,Y] :
( Y != zero_zero_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,Y),hAPP_nat_nat(div_mod_nat(X),Y)) ) ).
fof(fact_4361_coprime__exp__nat,axiom,
! [N,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)) = one_one_nat ) ).
fof(fact_4362_ceiling__real__of__nat,axiom,
! [N] : archim856651990g_real(hAPP_nat_real(real_nat,N)) = hAPP_nat_int(semiri1621563631at_int,N) ).
fof(fact_4363_natceiling__def,axiom,
! [X] : natceiling(X) = hAPP_int_nat(nat,archim856651990g_real(X)) ).
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] :
( N != 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),one_one_nat)),N) = one_one_nat ) ).
fof(fact_4366_divides__mult__nat,axiom,
! [N,M,R_2] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,M),R_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,N),R_2))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M),N) = 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)),R_2)) ) ) ) ).
fof(fact_4367_coprime__dvd__mult__iff__nat,axiom,
! [Ma,K,Na] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K),Na) = one_one_nat
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Ma),Na)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,K),Ma)) ) ) ).
fof(fact_4368_coprime__dvd__mult__nat,axiom,
! [M,K_2,N] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,K_2),N) = 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)))
=> 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,Y,M] :
( hAPP_nat_nat(div_mod_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),M) = one_one_nat
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),M) = one_one_nat ) ).
fof(fact_4371_coprime__Suc__0__nat,axiom,
! [A_1,B_1] :
( hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_1),B_1) = one_one_nat
<=> ! [D_2] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),A_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,D_2),B_1)) )
<=> 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_1,Y_1] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,A_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,B_2),Y_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,A_3),B_2)) ) ).
fof(fact_4374_Cauchy__iff2,axiom,
! [X_2] :
( hBOOL(cauchy_real(X_2))
<=> ! [J_1] :
? [M_3] :
! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),M_1))
=> ! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),N_1))
=> 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_2,M_1)),hAPP_nat_real(X_2,N_1)))),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(suc,J_1))))) ) ) ) ).
fof(fact_4375_ceiling__eq2,axiom,
! [N,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)))
=> archim856651990g_real(X) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ).
fof(fact_4376_ln__powr__bound,axiom,
! [A_3,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
=> ( 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)),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),A_3))) ) ) ).
fof(fact_4377_real__number__of,axiom,
! [V_1] : hAPP_int_real(real_int,hAPP_int_int(number_number_of_int,V_1)) = hAPP_int_real(number267125858f_real,V_1) ).
fof(fact_4378_real__of__int__zero__cancel,axiom,
! [Xa] :
( is_int(Xa)
=> ( hAPP_int_real(real_int,Xa) = zero_zero_real
<=> Xa = zero_zero_int ) ) ).
fof(fact_4379_real__of__int__zero,axiom,
hAPP_int_real(real_int,zero_zero_int) = zero_zero_real ).
fof(fact_4380_powr__not__zero,axiom,
! [X,A_3] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3) != zero_zero_real ).
fof(fact_4381_powr__one__eq__one,axiom,
! [A_3] : hAPP_real_real(hAPP_r1250527377l_real(powr,one_one_real),A_3) = one_one_real ).
fof(fact_4382_real__of__int__floor__cancel,axiom,
! [Xa] :
( hAPP_int_real(real_int,archim1246769320r_real(Xa)) = Xa
<=> ? [N_1] :
( is_int(N_1)
& Xa = hAPP_int_real(real_int,N_1) ) ) ).
fof(fact_4383_floor__real__of__int,axiom,
! [N] :
( is_int(N)
=> archim1246769320r_real(hAPP_int_real(real_int,N)) = N ) ).
fof(fact_4384_real__of__int__inject,axiom,
! [Xa,Ya] :
( ( is_int(Xa)
& is_int(Ya) )
=> ( hAPP_int_real(real_int,Xa) = hAPP_int_real(real_int,Ya)
<=> Xa = Ya ) ) ).
fof(fact_4385_powr__powr__swap,axiom,
! [X,A_3,B_2] : hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),B_2) = hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)),A_3) ).
fof(fact_4386_powr__powr,axiom,
! [X,A_3,B_2] : hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),B_2) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),B_2)) ).
fof(fact_4387_real__of__int__less__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Xa)),hAPP_int_real(real_int,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Xa),Ya)) ) ).
fof(fact_4388_real__of__int__le__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Xa)),hAPP_int_real(real_int,Ya)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),Ya)) ) ).
fof(fact_4389_real__of__one,axiom,
hAPP_int_real(real_int,one_one_int) = one_one_real ).
fof(fact_4390_real__of__int__mult,axiom,
! [X,Y] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).
fof(fact_4391_real__of__int__diff,axiom,
! [X,Y] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).
fof(fact_4392_real__of__int__power,axiom,
! [X,N] : hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X)),N) ).
fof(fact_4393_power__real__of__int,axiom,
! [X,N] : hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_int_real(real_int,X)),N) = hAPP_int_real(real_int,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N)) ).
fof(fact_4394_real__of__int__of__nat__eq,axiom,
! [N] : hAPP_int_real(real_int,hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_real(real_nat,N) ).
fof(fact_4395_real__of__int__add,axiom,
! [X,Y] : hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,X)),hAPP_int_real(real_int,Y)) ).
fof(fact_4396_real__of__int__ceiling__cancel,axiom,
! [Xa] :
( hAPP_int_real(real_int,archim856651990g_real(Xa)) = Xa
<=> ? [N_1] :
( is_int(N_1)
& Xa = hAPP_int_real(real_int,N_1) ) ) ).
fof(fact_4397_ceiling__real__of__int,axiom,
! [N] :
( is_int(N)
=> archim856651990g_real(hAPP_int_real(real_int,N)) = N ) ).
fof(fact_4398_real__of__int__minus,axiom,
! [X] : hAPP_int_real(real_int,hAPP_int_int(uminus_uminus_int,X)) = hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,X)) ).
fof(fact_4399_real__of__int__abs,axiom,
! [X] : hAPP_int_real(real_int,hAPP_int_int(abs_abs_int,X)) = hAPP_real_real(abs_abs_real,hAPP_int_real(real_int,X)) ).
fof(fact_4400_real__of__int__floor__le,axiom,
! [R_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim1246769320r_real(R_2))),R_2)) ).
fof(fact_4401_real__of__int__ceiling__ge,axiom,
! [R_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_2),hAPP_int_real(real_int,archim856651990g_real(R_2)))) ).
fof(fact_4402_powr__ge__pzero,axiom,
! [X,Y] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y))) ).
fof(fact_4403_powr__gt__zero,axiom,
! [X,A_3] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3))) ).
fof(fact_4404_powr__less__mono2,axiom,
! [Y,X,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))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3))) ) ) ) ).
fof(fact_4405_powr__less__mono2__neg,axiom,
! [Y,X,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))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3))) ) ) ) ).
fof(fact_4406_powr__mono,axiom,
! [X,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))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2))) ) ) ).
fof(fact_4407_powr__zero__eq__one,axiom,
! [X] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),zero_zero_real) = one_one_real ).
fof(fact_4408_powr__less__cancel,axiom,
! [X,A_3,B_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_3),B_2)) ) ) ).
fof(fact_4409_powr__less__mono,axiom,
! [X,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))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2))) ) ) ).
fof(fact_4410_powr__less__cancel__iff,axiom,
! [A_1,B_1,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(powr,Xa),A_1)),hAPP_real_real(hAPP_r1250527377l_real(powr,Xa),B_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1)) ) ) ).
fof(fact_4411_powr__add,axiom,
! [X,A_3,B_2] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),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),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)) ).
fof(fact_4412_powr__divide2,axiom,
! [X,A_3,B_2] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,X),B_2)) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,A_3),B_2)) ).
fof(fact_4413_powr__minus,axiom,
! [X,A_3] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_real_real(uminus_uminus_real,A_3)) = hAPP_real_real(inverse_inverse_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)) ).
fof(fact_4414_real__of__int__div4,axiom,
! [N,X] : 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),X))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,X)))) ).
fof(fact_4415_floor__less__eq,axiom,
! [Xa,A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim1246769320r_real(Xa)),A_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),hAPP_int_real(real_int,A_1))) ) ).
fof(fact_4416_le__floor,axiom,
! [A_3,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X))) ) ).
fof(fact_4417_real__le__floor,axiom,
! [A_3,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_3),archim1246769320r_real(X)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_3)),X)) ) ).
fof(fact_4418_le__floor__eq,axiom,
! [A_1,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),archim1246769320r_real(Xa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,A_1)),Xa)) ) ).
fof(fact_4419_floor__add,axiom,
! [X,A_3] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim1246769320r_real(X)),A_3) ).
fof(fact_4420_less__ceiling__eq,axiom,
! [A_1,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),archim856651990g_real(Xa)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,A_1)),Xa)) ) ).
fof(fact_4421_ceiling__le__eq,axiom,
! [Xa,A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(Xa)),A_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),hAPP_int_real(real_int,A_1))) ) ).
fof(fact_4422_ceiling__le,axiom,
! [X,A_3] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A_3)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A_3)) ) ).
fof(fact_4423_ceiling__le__real,axiom,
! [X,A_3] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim856651990g_real(X)),A_3))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,A_3))) ) ).
fof(fact_4424_floor__power,axiom,
! [N,X] :
( X = hAPP_int_real(real_int,archim1246769320r_real(X))
=> archim1246769320r_real(hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N)) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,archim1246769320r_real(X)),N) ) ).
fof(fact_4425_floor__subtract,axiom,
! [X,A_3] : archim1246769320r_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim1246769320r_real(X)),A_3) ).
fof(fact_4426_ceiling__add,axiom,
! [X,A_3] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,archim856651990g_real(X)),A_3) ).
fof(fact_4427_floor__minus__real__of__int,axiom,
! [N] : archim1246769320r_real(hAPP_real_real(uminus_uminus_real,hAPP_int_real(real_int,N))) = hAPP_int_int(uminus_uminus_int,N) ).
fof(fact_4428_ceiling__subtract,axiom,
! [X,A_3] : archim856651990g_real(hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),hAPP_int_real(real_int,A_3))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,archim856651990g_real(X)),A_3) ).
fof(fact_4429_powr__mono2,axiom,
! [Y,X,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))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),Y))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3))) ) ) ) ).
fof(fact_4430_powr__mult,axiom,
! [A_3,Y,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,zero_zero_real),Y))
=> hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,X),Y)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3)) ) ) ).
fof(fact_4431_ge__one__powr__ge__zero,axiom,
! [A_3,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,one_one_real),X))
=> ( 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),A_3))) ) ) ).
fof(fact_4432_powr__le__cancel__iff,axiom,
! [A_1,B_1,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(powr,Xa),A_1)),hAPP_real_real(hAPP_r1250527377l_real(powr,Xa),B_1)))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1)) ) ) ).
fof(fact_4433_powr__one__gt__zero__iff,axiom,
! [Xa] :
( hAPP_real_real(hAPP_r1250527377l_real(powr,Xa),one_one_real) = Xa
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa)) ) ).
fof(fact_4434_powr__divide,axiom,
! [A_3,Y,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,zero_zero_real),Y))
=> hAPP_real_real(hAPP_r1250527377l_real(powr,hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,X),Y)),A_3) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3)),hAPP_real_real(hAPP_r1250527377l_real(powr,Y),A_3)) ) ) ).
fof(fact_4435_powr__minus__divide,axiom,
! [X,A_3] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),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),A_3)) ).
fof(fact_4436_powr__def,axiom,
! [X,A_3] : hAPP_real_real(hAPP_r1250527377l_real(powr,X),A_3) = hAPP_real_real(exp_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),hAPP_real_real(ln,X))) ).
fof(fact_4437_real__of__int__lt__zero__cancel__iff,axiom,
! [Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Na)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Na),zero_zero_int)) ) ).
fof(fact_4438_real__of__int__gt__zero__cancel__iff,axiom,
! [Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_int_real(real_int,Na)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Na)) ) ).
fof(fact_4439_real__of__int__ge__zero__cancel__iff,axiom,
! [Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_int_real(real_int,Na)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na)) ) ).
fof(fact_4440_real__of__int__le__zero__cancel__iff,axiom,
! [Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Na)),zero_zero_real))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Na),zero_zero_int)) ) ).
fof(fact_4441_number__of__less__real__of__int__iff2,axiom,
! [Ma,Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,Na)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ma),hAPP_int_int(number_number_of_int,Na))) ) ).
fof(fact_4442_number__of__less__real__of__int__iff,axiom,
! [Na,Ma] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,Na)),hAPP_int_real(real_int,Ma)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(number_number_of_int,Na)),Ma)) ) ).
fof(fact_4443_number__of__le__real__of__int__iff,axiom,
! [Na,Ma] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(number267125858f_real,Na)),hAPP_int_real(real_int,Ma)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(number_number_of_int,Na)),Ma)) ) ).
fof(fact_4444_number__of__le__real__of__int__iff2,axiom,
! [Ma,Na] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,Ma)),hAPP_int_real(number267125858f_real,Na)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ma),hAPP_int_int(number_number_of_int,Na))) ) ).
fof(fact_4445_int__less__real__le,axiom,
! [Na,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Na),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,Na)),one_one_real)),hAPP_int_real(real_int,Ma))) ) ).
fof(fact_4446_int__le__real__less,axiom,
! [Na,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Na),Ma))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,Na)),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,Ma)),one_one_real))) ) ).
fof(fact_4447_lemma__floor2,axiom,
! [N,X] :
( 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,X)),one_one_real)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,N),X)) ) ).
fof(fact_4448_real__nat__eq__real,axiom,
! [X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X))
=> hAPP_nat_real(real_nat,hAPP_int_nat(nat,X)) = hAPP_int_real(real_int,X) ) ).
fof(fact_4449_real__of__int__div,axiom,
! [N,D_3] :
( is_int(D_3)
=> ( D_3 != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_3),N))
=> hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),D_3)) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,N)),hAPP_int_real(real_int,D_3)) ) ) ) ).
fof(fact_4450_real__of__int__floor__add__one__ge,axiom,
! [R_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,R_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_2))),one_one_real))) ).
fof(fact_4451_real__of__int__floor__add__one__gt,axiom,
! [R_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,archim1246769320r_real(R_2))),one_one_real))) ).
fof(fact_4452_floor__eq,axiom,
! [N,X] :
( is_int(N)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)))
=> archim1246769320r_real(X) = N ) ) ) ).
fof(fact_4453_tan__periodic__int,axiom,
! [X,I] : hAPP_real_real(tan,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,X),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_int_real(real_int,I)),pi))) = hAPP_real_real(tan,X) ).
fof(fact_4454_real__of__int__floor__ge__diff__one,axiom,
! [R_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_2),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_2)))) ).
fof(fact_4455_real__of__int__floor__gt__diff__one,axiom,
! [R_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,R_2),one_one_real)),hAPP_int_real(real_int,archim1246769320r_real(R_2)))) ).
fof(fact_4456_real__of__int__ceiling__le__add__one,axiom,
! [R_2] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,archim856651990g_real(R_2))),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,R_2),one_one_real))) ).
fof(fact_4457_real__of__int__ceiling__diff__one__le,axiom,
! [R_2] : 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_2))),one_one_real)),R_2)) ).
fof(fact_4458_powr__realpow,axiom,
! [N,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_nat_real(real_nat,N)) = hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N) ) ).
fof(fact_4459_ln__powr,axiom,
! [Y,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,zero_zero_real),Y))
=> hAPP_real_real(ln,hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(ln,X)) ) ) ).
fof(fact_4460_powr__log__cancel,axiom,
! [X,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))
=> hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),hAPP_real_real(log(A_3),X)) = X ) ) ) ).
fof(fact_4461_log__powr__cancel,axiom,
! [Y,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)) = Y ) ) ).
fof(fact_4462_lemma__floor,axiom,
! [N,M,R_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,M)),R_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,M),N)) ) ) ).
fof(fact_4463_real__of__int__div2,axiom,
! [N,X] : 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)),hAPP_int_real(real_int,X))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),X))))) ).
fof(fact_4464_real__of__int__div3,axiom,
! [N,X] : 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)),hAPP_int_real(real_int,X))),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,N),X)))),one_one_real)) ).
fof(fact_4465_real__of__int__div__aux,axiom,
! [X,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)),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),D_3))),hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(real_int,hAPP_int_int(div_mod_int(X),D_3))),hAPP_int_real(real_int,D_3))) ) ) ).
fof(fact_4466_floor__eq2,axiom,
! [N,X] :
( is_int(N)
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)))
=> archim1246769320r_real(X) = N ) ) ) ).
fof(fact_4467_less__floor__eq,axiom,
! [A_1,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),archim1246769320r_real(Xa)))
<=> 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_1)),one_one_real)),Xa)) ) ).
fof(fact_4468_floor__le__eq,axiom,
! [Xa,A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,archim1246769320r_real(Xa)),A_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,A_1)),one_one_real))) ) ).
fof(fact_4469_ceiling__eq3,axiom,
! [N,X] :
( is_int(N)
=> ( 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)),one_one_real)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X),hAPP_int_real(real_int,N)))
=> archim856651990g_real(X) = N ) ) ) ).
fof(fact_4470_ceiling__less__eq,axiom,
! [Xa,A_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,archim856651990g_real(Xa)),A_1))
<=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,hAPP_int_real(real_int,A_1)),one_one_real))) ) ).
fof(fact_4471_le__ceiling__eq,axiom,
! [A_1,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),archim856651990g_real(Xa)))
<=> 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_1)),one_one_real)),Xa)) ) ).
fof(fact_4472_log__powr,axiom,
! [B_2,Y,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Y))
=> hAPP_real_real(log(B_2),hAPP_real_real(hAPP_r1250527377l_real(powr,X),Y)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,Y),hAPP_real_real(log(B_2),X)) ) ) ).
fof(fact_4473_ceiling__eq,axiom,
! [N,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(real_int,N)),X))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_int_real(real_int,N)),one_one_real)))
=> archim856651990g_real(X) = hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ).
fof(fact_4474_powr__realpow2,axiom,
! [N,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),X))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( ( X = zero_zero_real
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N) = zero_zero_real )
& ( X != zero_zero_real
=> hAPP_nat_real(hAPP_r474017924t_real(power_power_real,X),N) = hAPP_real_real(hAPP_r1250527377l_real(powr,X),hAPP_nat_real(real_nat,N)) ) ) ) ) ).
fof(fact_4475_ln__powr__bound2,axiom,
! [A_3,X] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),X))
=> ( 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)),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(powr,A_3),A_3)),X))) ) ) ).
fof(fact_4476_reals__Archimedean__6b__int,axiom,
! [R_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),R_2))
=> ? [N_1] :
( is_int(N_1)
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),R_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)))) ) ) ).
fof(fact_4477_reals__Archimedean__6c__int,axiom,
! [R_2] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),zero_zero_real))
=> ? [N_1] :
( is_int(N_1)
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),R_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)))) ) ) ).
fof(fact_4478_real__lb__ub__int,axiom,
! [R_2] :
? [N_1] :
( is_int(N_1)
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_int_real(real_int,N_1)),R_2))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_2),hAPP_int_real(real_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N_1),one_one_int)))) ) ).
fof(fact_4479_coprime__pow,axiom,
! [C_1,N,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)
=> ? [R_1,S_2] :
( A_3 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,R_1),N)
& B_2 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,S_2),N) ) ) ) ).
fof(fact_4480_prime__power__mult,axiom,
! [X,Y,K_2,P_3] :
( hBOOL(hAPP_nat_bool(prime,P_3))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),K_2)
=> ? [I_2,J_1] :
( X = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),I_2)
& Y = hAPP_nat_nat(hAPP_nat_fun_nat_nat(power_power_nat,P_3),J_1) ) ) ) ).
fof(fact_4481_Euler__Fermat,axiom,
! [X,M] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),M))
=> ( hAPP_int_int(legacy_zgcd(X),M) = one_one_int
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),phi(M))),one_one_int),M)) ) ) ).
fof(fact_4482_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_4483_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_4484_zgcd__zdvd__zgcd__zmult,axiom,
! [M,N,K_2] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(legacy_zgcd(M),N)),hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N))) ).
fof(fact_4485_zgcd__zmult__cancel,axiom,
! [M,K_2,N] :
( hAPP_int_int(legacy_zgcd(K_2),N) = one_one_int
=> hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),M)),N) = hAPP_int_int(legacy_zgcd(M),N) ) ).
fof(fact_4486_zgcd__zgcd__zmult,axiom,
! [N,K_2,M] :
( hAPP_int_int(legacy_zgcd(K_2),M) = one_one_int
=> ( hAPP_int_int(legacy_zgcd(N),M) = one_one_int
=> hAPP_int_int(legacy_zgcd(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_2),N)),M) = one_one_int ) ) ).
fof(fact_4487_zgcd__1,axiom,
! [M] : hAPP_int_int(legacy_zgcd(M),one_one_int) = one_one_int ).
fof(fact_4488_zgcd__1__left,axiom,
! [M] : hAPP_int_int(legacy_zgcd(one_one_int),M) = one_one_int ).
fof(fact_4489_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_4490_zgcd__eq,axiom,
! [M,N] : hAPP_int_int(legacy_zgcd(M),N) = hAPP_int_int(legacy_zgcd(N),hAPP_int_int(div_mod_int(M),N)) ).
fof(fact_4491_zgcd__power__distrib,axiom,
! [A_3,B_2,N] : hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(legacy_zgcd(A_3),B_2)),N) = hAPP_int_int(legacy_zgcd(hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_3),N)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)) ).
fof(fact_4492_zgcd__greatest,axiom,
! [N,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))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),hAPP_int_int(legacy_zgcd(M),N))) ) ) ).
fof(fact_4493_zgcd__greatest__iff,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),hAPP_int_int(legacy_zgcd(Ma),Na)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Ma))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Na)) ) ) ).
fof(fact_4494_zgcd__commute,axiom,
! [I,J_2] : hAPP_int_int(legacy_zgcd(I),J_2) = hAPP_int_int(legacy_zgcd(J_2),I) ).
fof(fact_4495_zgcd__left__commute,axiom,
! [K_2,M,N] : hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N)) = hAPP_int_int(legacy_zgcd(M),hAPP_int_int(legacy_zgcd(K_2),N)) ).
fof(fact_4496_zgcd__assoc,axiom,
! [K_2,M,N] : hAPP_int_int(legacy_zgcd(hAPP_int_int(legacy_zgcd(K_2),M)),N) = hAPP_int_int(legacy_zgcd(K_2),hAPP_int_int(legacy_zgcd(M),N)) ).
fof(fact_4497_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_4498_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_4499_zgcd__zadd__zmult,axiom,
! [M,N,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),K_2))),N) = hAPP_int_int(legacy_zgcd(M),N) ).
fof(fact_4500_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_4501_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_4502_zgcd__geq__zero,axiom,
! [X,Y] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(legacy_zgcd(X),Y))) ).
fof(fact_4503_zgcd__1__power__distrib,axiom,
! [N,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)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)) = one_one_int ) ).
fof(fact_4504_zgcd__1__power__left__distrib,axiom,
! [N,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)),B_2) = one_one_int ) ).
fof(fact_4505_zgcd__zmult__distrib2__abs,axiom,
! [K_2,M,N] : 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)) = 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)) ).
fof(fact_4506_zgcd__0__left,axiom,
! [M] : hAPP_int_int(legacy_zgcd(zero_zero_int),M) = hAPP_int_int(abs_abs_int,M) ).
fof(fact_4507_zgcd__0,axiom,
! [M] : hAPP_int_int(legacy_zgcd(M),zero_zero_int) = hAPP_int_int(abs_abs_int,M) ).
fof(fact_4508_zgcd__zmult__eq__self2,axiom,
! [N,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)),K_2) = K_2 ) ) ).
fof(fact_4509_zgcd__zmult__distrib2,axiom,
! [M,N,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)) = 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)) ) ).
fof(fact_4510_zgcd__zmult__eq__self,axiom,
! [N,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)) = K_2 ) ) ).
fof(fact_4511_zdvd__iff__zgcd,axiom,
! [Na,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),Na))
<=> hAPP_int_int(legacy_zgcd(Na),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_zcong__zgcd__zmult__zmod,axiom,
! [N,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))
=> ( hAPP_int_int(legacy_zgcd(M),N) = 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))) ) ) ) ).
fof(fact_4514_zgcd__non__0,axiom,
! [M,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> hAPP_int_int(legacy_zgcd(M),N) = hAPP_int_int(legacy_zgcd(N),hAPP_int_int(div_mod_int(M),N)) ) ).
fof(fact_4515_zgcd__zmult__zdvd__zgcd,axiom,
! [M,K_2,N] :
( hAPP_int_int(legacy_zgcd(K_2),N) = 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)),hAPP_int_int(legacy_zgcd(M),N))) ) ).
fof(fact_4516_zrelprime__zdvd__zmult,axiom,
! [M,N,K_2] :
( hAPP_int_int(legacy_zgcd(N),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)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).
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,L] : hAPP_int_int(legacy_zgcd(K),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),hAPP_int_int(legacy_zgcd(L),hAPP_int_int(div_mod_int(hAPP_int_int(abs_abs_int,K)),hAPP_int_int(abs_abs_int,L))))) ).
fof(fact_4520_zprime__imp__zrelprime,axiom,
! [N,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_3),N))
=> hAPP_int_int(legacy_zgcd(N),P_3) = one_one_int ) ) ).
fof(fact_4521_zgcd1__iff__no__common__primedivisor,axiom,
! [A_1,B_1] :
( hAPP_int_int(legacy_zgcd(A_1),B_1) = 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_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,P_4),B_1)) ) ) ).
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,P_3] :
( hBOOL(hAPP_int_bool(zprime,P_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,N),P_3))
=> hAPP_int_int(legacy_zgcd(N),P_3) = one_one_int ) ) ) ).
fof(fact_4524_zcong__cancel2,axiom,
! [A_1,B_1,K,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(K),Ma) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),A_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K),B_1)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma)) ) ) ) ).
fof(fact_4525_zcong__cancel,axiom,
! [A_1,B_1,K,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(K),Ma) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),K)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),K)),Ma))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma)) ) ) ) ).
fof(fact_4526_zrelprime__zdvd__zmult__aux,axiom,
! [M,N,K_2] :
( hAPP_int_int(legacy_zgcd(N),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)))
=> ( 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_1,B_1,Ma] :
( hAPP_int_int(legacy_zgcd(B_1),Ma) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),hAPP_i1948725293t_bool(bnorRset(A_1),Ma))) ) ) ) ).
fof(fact_4528_int__relprime__odd__power__divisors,axiom,
! [A_1,B_1,C,Xa] :
( is_int(A_1)
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),B_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,C),hAPP_int_nat(nat,Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat,Xa)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> ( hAPP_int_int(legacy_zgcd(A_1),B_1) = one_one_int
=> ? [K_1] :
( is_int(K_1)
& A_1 = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_1),hAPP_int_nat(nat,Xa)) ) ) ) ) ) ) ).
fof(fact_4529_xzgcd__correct,axiom,
! [Ma,K,Na] :
( is_int(K)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Na))
=> ( hAPP_int_int(legacy_zgcd(Ma),Na) = K
<=> ? [S_2,T] :
( is_int(S_2)
& is_int(T)
& xzgcd(Ma,Na) = hAPP_P408881810nt_int(hAPP_i1730167831nt_int(produc282740534nt_int,K),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] :
( 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)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N))
=> ( hAPP_int_int(legacy_zgcd(A_3),B_2) = one_one_int
=> ? [K_1] :
( is_int(K_1)
& hAPP_int_int(abs_abs_int,A_3) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_1),N) ) ) ) ) ).
fof(fact_4531_xzgcd__correct__aux2,axiom,
! [M,N,R_4,R_2,S_3,S_1,T_2,T_1,K_2] :
( is_int(K_2)
=> ( ? [Sn,Tn] : xzgcda(M,N,R_4,R_2,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_2))
=> hAPP_int_int(legacy_zgcd(R_4),R_2) = K_2 ) ) ) ).
fof(fact_4532_zcong__lineq__unique,axiom,
! [B_2,A_3,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> ( hAPP_int_int(legacy_zgcd(A_3),N) = one_one_int
=> ? [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),N))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X_1)),B_2),N))
& ! [Y_1] :
( is_int(Y_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_int,Y_1),N))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),Y_1)),B_2),N)) )
=> Y_1 = X_1 ) ) ) ) ) ).
fof(fact_4533_zcong__lineq__ex,axiom,
! [A_3,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> ( hAPP_int_int(legacy_zgcd(A_3),N) = one_one_int
=> ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_3),X_1)),one_one_int),N)) ) ) ) ).
fof(fact_4534_xzgcd__correct__aux1,axiom,
! [M,N,S_3,S_1,T_2,T_1,R_4,R_2,K_2] :
( hAPP_int_int(legacy_zgcd(R_4),R_2) = K_2
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),R_2))
=> ? [Sn,Tn] :
( is_int(Sn)
& is_int(Tn)
& xzgcda(M,N,R_4,R_2,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_1,B_1,C,D_1,Xa] :
( ( is_int(A_1)
& is_int(B_1)
& 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_1),B_1)),C) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,D_1),hAPP_int_nat(nat,Xa))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),hAPP_int_nat(nat,Xa)))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),zOdd))
=> ( hAPP_int_int(legacy_zgcd(A_1),B_1) = one_one_int
=> ( hAPP_int_int(legacy_zgcd(B_1),C) = one_one_int
=> ( hAPP_int_int(legacy_zgcd(C),A_1) = one_one_int
=> ? [K_1,L_1,M_1] :
( is_int(K_1)
& is_int(L_1)
& is_int(M_1)
& A_1 = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,K_1),hAPP_int_nat(nat,Xa))
& B_1 = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),hAPP_int_nat(nat,Xa))
& C = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),hAPP_int_nat(nat,Xa)) ) ) ) ) ) ) ) ) ).
fof(fact_4536_zgcd__ex__linear,axiom,
! [M,K_2,N] :
( is_int(K_2)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),N))
=> ( hAPP_int_int(legacy_zgcd(M),N) = 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)) ) ) ) ) ).
fof(fact_4537_int__triple__relprime__power__divisors,axiom,
! [A_3,B_2,C_1,D_3,N] :
( 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)
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,one_one_nat),N))
=> ( 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_1,L_1,M_1] :
( is_int(K_1)
& 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_1),N)
& hAPP_int_int(abs_abs_int,B_2) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,L_1),N)
& hAPP_int_int(abs_abs_int,C_1) = hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,M_1),N) ) ) ) ) ) ) ).
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,
! [Sb,Ta,A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Sb),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_1),Sb)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B_1),Ta))),pair_less)) ) ) ).
fof(fact_4540_pair__lessI1,axiom,
! [Sb,Ta,A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1))
=> 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_1),Sb)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B_1),Ta))),pair_less)) ) ).
fof(fact_4541_pair__leqI2,axiom,
! [Sb,Ta,A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_1),B_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Sb),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_1),Sb)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B_1),Ta))),pair_leq)) ) ) ).
fof(fact_4542_pair__leqI1,axiom,
! [Sb,Ta,A_1,B_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_1),B_1))
=> 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_1),Sb)),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,B_1),Ta))),pair_leq)) ) ).
fof(fact_4543_polar__Ex,axiom,
! [Y,X] :
? [R_1,A_2] :
( X = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_1),hAPP_real_real(cos,A_2))
& Y = 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,Y] : hAPP_real_real(dist_dist_real(X),Y) = hAPP_real_real(abs_abs_real,hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,X),Y)) ).
fof(fact_4545_dist__complex__def,axiom,
! [X,Y] : hAPP_complex_real(dist_dist_complex(X),Y) = hAPP_complex_real(norm_norm_complex,hAPP_complex_complex(hAPP_c172932010omplex(minus_minus_complex,X),Y)) ).
fof(fact_4546_DERIV__nonneg__imp__nonincreasing,axiom,
! [F,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) )
=> ? [Y_1] :
( hBOOL(hAPP_real_bool(deriv_real(F,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,hAPP_real_real(F,A_1)),hAPP_real_real(F,B_1))) ) ) ).
fof(fact_4547_DERIV__nonpos__imp__nonincreasing,axiom,
! [F,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) )
=> ? [Y_1] :
( hBOOL(hAPP_real_bool(deriv_real(F,X_1),Y_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Y_1),zero_zero_real)) ) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(F,B_1)),hAPP_real_real(F,A_1))) ) ) ).
fof(fact_4548_int__cases,axiom,
! [Z_1] :
( is_int(Z_1)
=> ( ! [N_1] : Z_1 != hAPP_nat_int(semiri1621563631at_int,N_1)
=> ~ ! [N_1] : Z_1 != hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1))) ) ) ).
fof(fact_4549_int__of__nat__induct,axiom,
! [Z_2,P_1] :
( ! [N_1] : hBOOL(hAPP_int_bool(P_1,hAPP_nat_int(semiri1621563631at_int,N_1)))
=> ( ! [N_1] : hBOOL(hAPP_int_bool(P_1,hAPP_int_int(uminus_uminus_int,hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(suc,N_1)))))
=> hBOOL(hAPP_int_bool(P_1,Z_2)) ) ) ).
fof(fact_4550_gcd__nat__induct,axiom,
! [Ma,Na,P_1] :
( ! [M_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),zero_zero_nat))
=> ( ! [M_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),hAPP_nat_nat(div_mod_nat(M_1),N_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,M_1),N_1)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Ma),Na)) ) ) ).
fof(fact_4551_divides__fact,axiom,
! [N,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))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(dvd_dvd_nat,P_3),hAPP_nat_nat(fact,N))) ) ) ).
fof(fact_4552_mod__code__int__def,axiom,
! [N,M] : hAPP_Q1762011733de_int(div_mo231679042de_int(N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(div_mod_int(quickcheck_int_of(N)),quickcheck_int_of(M))) ).
fof(fact_4553_number__of__code__int__def,axiom,
number1226105091de_int = quickcheck_of_int ).
fof(fact_4554_int__of__of__int,axiom,
! [N] :
( is_int(N)
=> quickcheck_int_of(hAPP_i1732201573de_int(quickcheck_of_int,N)) = N ) ).
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__inverse,axiom,
! [X] : hAPP_i1732201573de_int(quickcheck_of_int,quickcheck_int_of(X)) = X ).
fof(fact_4557_Quickcheck__Narrowing_Oof__int__inject,axiom,
! [Na,Ma] :
( ( is_int(Na)
& is_int(Ma) )
=> ( hAPP_i1732201573de_int(quickcheck_of_int,Na) = hAPP_i1732201573de_int(quickcheck_of_int,Ma)
<=> Na = Ma ) ) ).
fof(fact_4558_fact__mono,axiom,
! [M,N] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(fact,M)),hAPP_nat_nat(fact,N))) ) ).
fof(fact_4559_times__code__int__def,axiom,
! [N,M] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(times_123202395de_int,N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,quickcheck_int_of(N)),quickcheck_int_of(M))) ).
fof(fact_4560_plus__code__int__def,axiom,
! [N,M] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(plus_p1446045655de_int,N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,quickcheck_int_of(N)),quickcheck_int_of(M))) ).
fof(fact_4561_minus__code__int__def,axiom,
! [N,M] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(minus_534354567de_int,N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,quickcheck_int_of(N)),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] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(fact,N))) ).
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] : hAPP_nat_nat(fact,hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(suc,N)),hAPP_nat_nat(fact,N)) ).
fof(fact_4566_fact__le,axiom,
! [N] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,one_one_nat),hAPP_nat_nat(fact,N))) ).
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,M] : hAPP_Q1762011733de_int(hAPP_Q1729056540de_int(div_di1430059507de_int,N),M) = hAPP_i1732201573de_int(quickcheck_of_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,quickcheck_int_of(N)),quickcheck_int_of(M))) ).
fof(fact_4569_euclid__bound,axiom,
! [N] :
? [P_4] :
( hBOOL(hAPP_nat_bool(prime,P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),P_4))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,P_4),hAPP_nat_nat(suc,hAPP_nat_nat(fact,N)))) ) ).
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_1] : hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_2),K_1)) = 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_number__of__real__def,axiom,
! [X] : hAPP_int_real(number267125858f_real,X) = hAPP_int_real(ring_1_of_int_real,X) ).
fof(fact_4576_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_4577_complex__number__of__def,axiom,
! [W] : hAPP_int_complex(number528085621omplex,W) = hAPP_int_complex(ring_11397209091omplex,W) ).
fof(fact_4578_of__int__int__eq,axiom,
! [N] : hAPP_int_int(ring_1_of_int_int,hAPP_nat_int(semiri1621563631at_int,N)) = hAPP_nat_int(semiri1621563631at_int,N) ).
fof(fact_4579_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_4580_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_4581_complex__Im__of__int,axiom,
! [Z_1] : hAPP_complex_real(im,hAPP_int_complex(ring_11397209091omplex,Z_1)) = zero_zero_real ).
fof(fact_4582_real__eq__of__int,axiom,
real_int = ring_1_of_int_real ).
fof(fact_4583_real__of__int__def,axiom,
real_int = ring_1_of_int_real ).
fof(fact_4584_phi__def,axiom,
! [Ma] : phi(Ma) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma)) ).
fof(fact_4585_norR__mem__unique,axiom,
! [A_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(A_1),Ma) = one_one_int
=> ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),X_1),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(norRRset,Ma)))
& ! [Y_1] :
( is_int(Y_1)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),Y_1),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Y_1),hAPP_i1948725293t_bool(norRRset,Ma))) )
=> Y_1 = X_1 ) ) ) ) ) ).
fof(fact_4586_card__nor__eq__noX,axiom,
! [Xa,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(Xa),Ma) = one_one_int
=> hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(noXRRset(Ma),Xa)) = hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(norRRset,Ma)) ) ) ).
fof(fact_4587_noX__is__RRset,axiom,
! [Xa,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(Xa),Ma) = one_one_int
=> hBOOL(hAPP_int_bool(is_RRset(hAPP_i1948725293t_bool(noXRRset(Ma),Xa)),Ma)) ) ) ).
fof(fact_4588_RRset__gcd,axiom,
! [A_1,A,Ma] :
( hBOOL(hAPP_int_bool(is_RRset(A),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A))
=> hAPP_int_int(legacy_zgcd(A_1),Ma) = one_one_int ) ) ).
fof(fact_4589_RRset__zcong__eq,axiom,
! [A_1,B_1,A,Ma] :
( ( is_int(A_1)
& is_int(B_1) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A),Ma))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),B_1),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,B_1),A))
=> A_1 = B_1 ) ) ) ) ) ) ).
fof(fact_4590_RRset2norRR__correct,axiom,
! [A_1,A,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),hAPP_int_int(rRset2norRR(A,Ma),A_1)),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A,Ma),A_1)),hAPP_i1948725293t_bool(norRRset,Ma))) ) ) ) ) ).
fof(fact_4591_RRset2norRR__correct2,axiom,
! [A_1,A,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A))
=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_int_int(rRset2norRR(A,Ma),A_1)),hAPP_i1948725293t_bool(norRRset,Ma))) ) ) ) ).
fof(fact_4592_RRset2norRR__correct1,axiom,
! [A_1,A,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),hAPP_int_int(rRset2norRR(A,Ma),A_1)),Ma)) ) ) ) ).
fof(fact_4593_is__RRset__def,axiom,
! [A,Ma] :
( hBOOL(hAPP_int_bool(is_RRset(A),Ma))
<=> ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A),rsetR(Ma)))
& hAPP_f957591787ol_nat(finite_card_int,A) = phi(Ma) ) ) ).
fof(fact_4594_RRset2norRR__eq__norR,axiom,
! [A,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A),Ma))
=> image_int_int(rRset2norRR(A,Ma),A) = hAPP_i1948725293t_bool(norRRset,Ma) ) ) ).
fof(fact_4595_RsetR__fin,axiom,
! [A,Ma] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A),rsetR(Ma)))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,A)) ) ).
fof(fact_4596_Bnor__in__RsetR,axiom,
! [A_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),Ma))
=> hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,hAPP_i1948725293t_bool(bnorRset(A_1),Ma)),rsetR(Ma))) ) ).
fof(fact_4597_ResSet__image,axiom,
! [F,A,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(resSet(Ma),A))
=> ( ! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),A))
=> ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(hAPP_int_int(F,X_1)),hAPP_int_int(F,Xa_1)),Ma))
=> X_1 = Xa_1 ) ) ) ) )
=> hBOOL(hAPP_f448129468l_bool(resSet(Ma),image_int_int(F,A))) ) ) ) ).
fof(fact_4598_transfer__nat__int__set__relations_I1_J,axiom,
! [A] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))
<=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,image_nat_int(semiri1621563631at_int,A))) ) ).
fof(fact_4599_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
! [A] : hAPP_f22106695ol_nat(finite_card_nat,A) = hAPP_f957591787ol_nat(finite_card_int,image_nat_int(semiri1621563631at_int,A)) ).
fof(fact_4600_transfer__nat__int__set__relations_I3_J,axiom,
! [A,B] :
( A = B
<=> image_nat_int(semiri1621563631at_int,A) = image_nat_int(semiri1621563631at_int,B) ) ).
fof(fact_4601_transfer__nat__int__set__relations_I2_J,axiom,
! [Xa,A] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),A))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,hAPP_nat_int(semiri1621563631at_int,Xa)),image_nat_int(semiri1621563631at_int,A))) ) ).
fof(fact_4602_transfer__int__nat__set__return__embed,axiom,
! [A] : image_int_nat(nat,image_nat_int(semiri1621563631at_int,A)) = A ).
fof(fact_4603_transfer__nat__int__set__relations_I4_J,axiom,
! [A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A),B))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).
fof(fact_4604_transfer__nat__int__set__relations_I5_J,axiom,
! [A,B] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
<=> hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,image_nat_int(semiri1621563631at_int,A)),image_nat_int(semiri1621563631at_int,B))) ) ).
fof(fact_4605_SetS__def,axiom,
! [A_1,P_5] : setS(A_1,P_5) = image_275383677t_bool(multInvPair(A_1,P_5),hAPP_i1948725293t_bool(sRStar,P_5)) ).
fof(fact_4606_infinite__nat__iff__unbounded__le,axiom,
! [S] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
<=> ! [M_1] :
? [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
& hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_1),S)) ) ) ).
fof(fact_4607_infinite__nat__iff__unbounded,axiom,
! [S] :
( ~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
<=> ! [M_1] :
? [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
& hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_1),S)) ) ) ).
fof(fact_4608_transfer__int__nat__set__relations_I4_J,axiom,
! [B,A] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le1912455174t_bool,A),B))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,image_int_nat(nat,A)),image_int_nat(nat,B))) ) ) ) ).
fof(fact_4609_finite__nat__set__iff__bounded__le,axiom,
! [N_3] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
<=> ? [M_1] :
! [X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),M_1)) ) ) ).
fof(fact_4610_nat__set__def,axiom,
! [S] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,S))
<=> ! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),S))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),X_1)) ) ) ) ).
fof(fact_4611_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J,axiom,
! [Xa,A] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) ) ) ).
fof(fact_4612_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_4613_transfer__int__nat__set__relations_I3_J,axiom,
! [B,A] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B))
=> ( A = B
<=> image_int_nat(nat,A) = image_int_nat(nat,B) ) ) ) ).
fof(fact_4614_transfer__nat__int__set__return__embed,axiom,
! [A] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> image_nat_int(semiri1621563631at_int,image_int_nat(nat,A)) = A ) ).
fof(fact_4615_transfer__int__nat__set__relations_I1_J,axiom,
! [A] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> ( hBOOL(hAPP_f448129468l_bool(finite_finite_int,A))
<=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_int_nat(nat,A))) ) ) ).
fof(fact_4616_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J,axiom,
! [A] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> hAPP_f957591787ol_nat(finite_card_int,A) = hAPP_f22106695ol_nat(finite_card_nat,image_int_nat(nat,A)) ) ).
fof(fact_4617_transfer__int__nat__set__relations_I5_J,axiom,
! [B,A] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,B))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_f284875647l_bool(ord_le951220754t_bool,A),B))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,image_int_nat(nat,A)),image_int_nat(nat,B))) ) ) ) ).
fof(fact_4618_finite__nat__set__iff__bounded,axiom,
! [N_3] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3))
<=> ? [M_1] :
! [X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),M_1)) ) ) ).
fof(fact_4619_bounded__nat__set__is__finite,axiom,
! [Na,N_3] :
( ! [X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),N_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Na)) )
=> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N_3)) ) ).
fof(fact_4620_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_4621_finite__greaterThanLessThan__int,axiom,
! [L,U] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_gr1297742076an_int(L),U))) ).
fof(fact_4622_Rats__abs__nat__div__natE,axiom,
! [Xa] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),field_1210416355s_real))
=> ~ ! [M_1,N_1] :
( N_1 != zero_zero_nat
=> ( hAPP_real_real(abs_abs_real,Xa) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_nat_real(real_nat,M_1)),hAPP_nat_real(real_nat,N_1))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,M_1),N_1) != one_one_nat ) ) ) ).
fof(fact_4623_Nitpick_Oint__lcm__def,axiom,
! [X,Y] : hAPP_int_int(int_lcm(X),Y) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_lcm(hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y)))) ).
fof(fact_4624_finite__greaterThanLessThan,axiom,
! [L,U] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_gr660468384an_nat(L),U))) ).
fof(fact_4625_Rats__real__nat,axiom,
! [Na] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,Na)),field_1210416355s_real)) ).
fof(fact_4626_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_4627_Rats__dense__in__nn__real,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
=> ? [X_1] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),field_1210416355s_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Ya)) ) ) ) ).
fof(fact_4628_DERIV__isconst3,axiom,
! [F,Ya,Xa,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1))
=> ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),hAPP_r1134773055l_bool(ord_gr788844697n_real(A_1),B_1)))
=> ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Ya),hAPP_r1134773055l_bool(ord_gr788844697n_real(A_1),B_1)))
=> ( ! [X_1] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),hAPP_r1134773055l_bool(ord_gr788844697n_real(A_1),B_1)))
=> hBOOL(hAPP_real_bool(deriv_real(F,X_1),zero_zero_real)) )
=> hAPP_real_real(F,Xa) = hAPP_real_real(F,Ya) ) ) ) ) ).
fof(fact_4629_iszero__rat,axiom,
! [K] :
( hBOOL(hAPP_rat_bool(iszero_rat,hAPP_int_rat(number_number_of_rat,K)))
<=> hBOOL(hAPP_int_bool(iszero_int,hAPP_int_int(number_number_of_int,K))) ) ).
fof(fact_4630_Rats__dense__in__real,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),Ya))
=> ? [X_1] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),field_1210416355s_real))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),Ya)) ) ) ).
fof(fact_4631_Sup__greaterThanLessThan,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa))
=> comple124823625p_real(hAPP_r1134773055l_bool(ord_gr788844697n_real(Ya),Xa)) = Xa ) ).
fof(fact_4632_Sup__upper__EX,axiom,
! [Xa,X_2] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),X_2))
=> ( ? [Z] :
! [X_1] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Z)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),comple124823625p_real(X_2))) ) ) ).
fof(fact_4633_SupInf_OSup__upper,axiom,
! [Z_2,Xa,X_2] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),X_2))
=> ( ! [X_1] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Z_2)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),comple124823625p_real(X_2))) ) ) ).
fof(fact_4634_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_4635_sgn__rat__def,axiom,
! [Q_1] :
( ( Q_1 = zero_zero_rat
=> hAPP_rat_rat(sgn_sgn_rat,Q_1) = zero_zero_rat )
& ( Q_1 != zero_zero_rat
=> ( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_1))
=> hAPP_rat_rat(sgn_sgn_rat,Q_1) = one_one_rat )
& ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Q_1))
=> hAPP_rat_rat(sgn_sgn_rat,Q_1) = hAPP_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ) ).
fof(fact_4636_divide__rat__def,axiom,
! [Q_1,R_2] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,Q_1),R_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,Q_1),hAPP_rat_rat(inverse_inverse_rat,R_2)) ).
fof(fact_4637_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_4638_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_4639_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_4640_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_4641_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_4642_Sup__eq__maximum,axiom,
! [Z_2,X_2] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Z_2),X_2))
=> ( ! [X_1] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Z_2)) )
=> comple124823625p_real(X_2) = Z_2 ) ) ).
fof(fact_4643_SupInf_OSup__upper2,axiom,
! [Z_2,Ya,Xa,X_2] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,Xa),X_2))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa))
=> ( ! [X_1] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),X_2))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),Z_2)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),comple124823625p_real(X_2))) ) ) ) ).
fof(fact_4644_diff__rat__def,axiom,
! [Q_1,R_2] : hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,Q_1),R_2) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,Q_1),hAPP_rat_rat(uminus_uminus_rat,R_2)) ).
fof(fact_4645_abs__rat__def,axiom,
! [Q_1] :
( ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_1),zero_zero_rat))
=> hAPP_rat_rat(abs_abs_rat,Q_1) = hAPP_rat_rat(uminus_uminus_rat,Q_1) )
& ( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Q_1),zero_zero_rat))
=> hAPP_rat_rat(abs_abs_rat,Q_1) = Q_1 ) ) ).
fof(fact_4646_less__rat__def,axiom,
! [Z_2,Wa] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Z_2),Wa))
<=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Z_2),Wa))
& Z_2 != Wa ) ) ).
fof(fact_4647_obtain__pos__sum,axiom,
! [R_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_2))
=> ~ ! [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_2 != hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,S_2),T) ) ) ) ).
fof(fact_4648_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_4649_quotient__of__inject__eq,axiom,
! [A_1,B_1] :
( quotient_of(A_1) = quotient_of(B_1)
<=> A_1 = B_1 ) ).
fof(fact_4650_quotient__of__inject,axiom,
! [A_3,B_2] :
( quotient_of(A_3) = quotient_of(B_2)
=> A_3 = B_2 ) ).
fof(fact_4651_quotient__of__denom__pos,axiom,
! [R_2,P_3,Q_1] :
( quotient_of(R_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_1)) ) ).
fof(fact_4652_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_4653_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_4654_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_4655_Nitpick_Onat__lcm__def,axiom,
! [X,Y] : hAPP_nat_nat(nat_lcm(X),Y) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(div_div_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,X),Y)),hAPP_nat_nat(nat_gcd(X),Y)) ).
fof(fact_4656_normalize__negative,axiom,
! [P_3,Q_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Q_1),zero_zero_int))
=> normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)) = 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_1))) ) ).
fof(fact_4657_gcd__eq__nitpick__gcd,axiom,
! [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y) = hAPP_nat_nat(nat_gcd(X),Y) ).
fof(fact_4658_nat__gcd_Osimps,axiom,
! [X,Y] :
( ( Y = zero_zero_nat
=> hAPP_nat_nat(nat_gcd(X),Y) = X )
& ( Y != zero_zero_nat
=> hAPP_nat_nat(nat_gcd(X),Y) = hAPP_nat_nat(nat_gcd(Y),hAPP_nat_nat(div_mod_nat(X),Y)) ) ) ).
fof(fact_4659_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_4660_normalize__denom__pos,axiom,
! [R_2,P_3,Q_1] :
( normalize(R_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_1)) ) ).
fof(fact_4661_normalize__crossproduct,axiom,
! [P_3,R_2,S_1,Q_1] :
( ( is_int(S_1)
& is_int(Q_1) )
=> ( Q_1 != zero_zero_int
=> ( S_1 != zero_zero_int
=> ( normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)) = normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,R_2),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_2),Q_1) ) ) ) ) ).
fof(fact_4662_nat__gcd_Opsimps,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,Xa),Ya)))
=> ( ( Ya = zero_zero_nat
=> hAPP_nat_nat(nat_gcd(Xa),Ya) = Xa )
& ( Ya != zero_zero_nat
=> hAPP_nat_nat(nat_gcd(Xa),Ya) = hAPP_nat_nat(nat_gcd(Ya),hAPP_nat_nat(div_mod_nat(Xa),Ya)) ) ) ) ).
fof(fact_4663_Nitpick_Oint__gcd__def,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(int_gcd,X),Y) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(nat_gcd(hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,X))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y)))) ).
fof(fact_4664_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_1,Y_1] :
( hBOOL(hAPP_P1555980039t_bool(accp_P490777396at_nat(nat_gcd_rel),hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,X_1),Y_1)))
=> ( ( Y_1 != zero_zero_nat
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,Y_1),hAPP_nat_nat(div_mod_nat(X_1),Y_1))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,X_1),Y_1)) ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,A0),A1)) ) ) ).
fof(fact_4665_ratrel__iff,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_f526364711l_bool(hAPP_P1062890741l_bool(member2143287562nt_int,hAPP_P230595783nt_int(hAPP_P941956607nt_int(produc883642259nt_int,Xa),Ya)),ratrel))
<=> ( hAPP_P1175774780nt_int(product_snd_int_int,Xa) != zero_zero_int
& hAPP_P1175774780nt_int(product_snd_int_int,Ya) != zero_zero_int
& hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,Xa)),hAPP_P1175774780nt_int(product_snd_int_int,Ya)) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_P1175774780nt_int(product_fst_int_int,Ya)),hAPP_P1175774780nt_int(product_snd_int_int,Xa)) ) ) ).
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_2,S_1] : ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,hAPP_int_rat(number_number_of_rat,R_2)),hAPP_int_rat(number_number_of_rat,S_1))) = hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,hAPP_int_real(number267125858f_real,R_2)),hAPP_int_real(number267125858f_real,S_1)) ).
fof(fact_4668_quotient__of__eq,axiom,
! [A_3,B_2,P_3,Q_1] :
( 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_1)
=> hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_1) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ) ).
fof(fact_4669_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_4670_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_4671_eq__rat_I1_J,axiom,
! [A_1,C,D_1,B_1] :
( ( is_int(D_1)
& is_int(B_1) )
=> ( B_1 != zero_zero_int
=> ( D_1 != zero_zero_int
=> ( hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D_1)
<=> hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),D_1) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1) ) ) ) ) ).
fof(fact_4672_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_4673_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_4674_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_4675_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_4676_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_4677_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_4678_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_4679_real__floor__code,axiom,
! [X] : archim1246769320r_real(ratreal(X)) = archim791455193or_rat(X) ).
fof(fact_4680_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_4681_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_4682_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_4683_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_4684_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_4685_one__rat,axiom,
one_one_rat = hAPP_int_rat(hAPP_int_fun_int_rat(fract,one_one_int),one_one_int) ).
fof(fact_4686_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_4687_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_4688_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_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_1] :
( 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_1)
=> hAPP_int_rat(hAPP_int_fun_int_rat(fract,P_3),Q_1) = hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_3),B_2) ) ).
fof(fact_4692_real__less__code,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,ratreal(Xa)),ratreal(Ya)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,Xa),Ya)) ) ).
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_one__real__code,axiom,
one_one_real = ratreal(one_one_rat) ).
fof(fact_4698_Ratreal__number__collapse_I2_J,axiom,
ratreal(one_one_rat) = one_one_real ).
fof(fact_4699_real__less__eq__code,axiom,
! [Xa,Ya] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,ratreal(Xa)),ratreal(Ya)))
<=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,Xa),Ya)) ) ).
fof(fact_4700_real__plus__code,axiom,
! [X,Y] : hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,X),Y)) ).
fof(fact_4701_real__minus__code,axiom,
! [X,Y] : hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(minus_minus_rat,X),Y)) ).
fof(fact_4702_real__uminus__code,axiom,
! [X] : hAPP_real_real(uminus_uminus_real,ratreal(X)) = ratreal(hAPP_rat_rat(uminus_uminus_rat,X)) ).
fof(fact_4703_real__divide__code,axiom,
! [X,Y] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(inverse_divide_rat,X),Y)) ).
fof(fact_4704_real__times__code,axiom,
! [X,Y] : hAPP_real_real(hAPP_r1250527377l_real(times_times_real,ratreal(X)),ratreal(Y)) = ratreal(hAPP_rat_rat(hAPP_rat_fun_rat_rat(times_times_rat,X),Y)) ).
fof(fact_4705_real__inverse__code,axiom,
! [X] : hAPP_real_real(inverse_inverse_real,ratreal(X)) = ratreal(hAPP_rat_rat(inverse_inverse_rat,X)) ).
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_1,C,D_1,B_1] :
( ( is_int(D_1)
& is_int(B_1) )
=> ( B_1 != zero_zero_int
=> ( D_1 != zero_zero_int
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D_1)))
<=> 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_1),D_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1)))) ) ) ) ) ).
fof(fact_4713_Ratreal__number__of__quotient,axiom,
! [R_2,S_1] : hAPP_real_real(hAPP_r1250527377l_real(inverse_divide_real,ratreal(hAPP_int_rat(number_number_of_rat,R_2))),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_2)),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_zero__less__Fract__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),A_1)) ) ) ).
fof(fact_4716_Fract__less__zero__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),zero_zero_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),zero_zero_int)) ) ) ).
fof(fact_4717_le__rat,axiom,
! [A_1,C,D_1,B_1] :
( ( is_int(D_1)
& is_int(B_1) )
=> ( B_1 != zero_zero_int
=> ( D_1 != zero_zero_int
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),hAPP_int_rat(hAPP_int_fun_int_rat(fract,C),D_1)))
<=> 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_1),D_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1))),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,C),B_1)),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,B_1),D_1)))) ) ) ) ) ).
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_one__less__Fract__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),A_1)) ) ) ).
fof(fact_4721_Fract__less__one__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),one_one_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),B_1)) ) ) ).
fof(fact_4722_Fract__add__one,axiom,
! [M,N] :
( is_int(N)
=> ( N != 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)),N) = hAPP_rat_rat(hAPP_rat_fun_rat_rat(plus_plus_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,M),N)),one_one_rat) ) ) ).
fof(fact_4723_zero__le__Fract__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,zero_zero_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),A_1)) ) ) ).
fof(fact_4724_Fract__le__zero__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),zero_zero_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),zero_zero_int)) ) ) ).
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_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,one_one_rat),hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,B_1),A_1)) ) ) ).
fof(fact_4727_Fract__le__one__iff,axiom,
! [A_1,B_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),B_1))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_eq_rat,hAPP_int_rat(hAPP_int_fun_int_rat(fract,A_1),B_1)),one_one_rat))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,A_1),B_1)) ) ) ).
fof(fact_4728_Rat__induct__pos,axiom,
! [Q_2,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_2)) ) ).
fof(fact_4729_Ints__real__of__nat,axiom,
! [Na] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_nat_real(real_nat,Na)),ring_1_Ints_real)) ).
fof(fact_4730_Ints__real__of__int,axiom,
! [Xa] : hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,hAPP_int_real(real_int,Xa)),ring_1_Ints_real)) ).
fof(fact_4731_adjust__eq,axiom,
! [B_1,Q_2,Ra] : hAPP_P1975530577nt_int(adjust(B_1),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,Q_2),Ra)) = 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,Ra),B_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,hAPP_int_int(number_number_of_int,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))),Q_2)),one_one_int)),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Ra),B_1))),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_2)),Ra)) ).
fof(fact_4732_lemma__DERIV__sin__cos__minus,axiom,
! [X_1] : 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_1),zero_zero_real)) ).
fof(fact_4733_finite__Collect__less__nat,axiom,
! [K] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),K)))) ).
fof(fact_4734_finite__Collect__le__nat,axiom,
! [K] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),K)))) ).
fof(fact_4735_Bseq__realpow,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),one_one_real))
=> hBOOL(bseq_real(hAPP_r474017924t_real(power_power_real,Xa))) ) ) ).
fof(fact_4736_DERIV__cos__add,axiom,
! [K,Xa] : 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)),Xa),hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),K))))) ).
fof(fact_4737_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_4738_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_4739_bdd__int__set__l__le__finite,axiom,
! [Na] : 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),Na))))) ).
fof(fact_4740_bdd__int__set__l__finite,axiom,
! [Na] : 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),Na))))) ).
fof(fact_4741_bdd__int__set__le__finite,axiom,
! [Na] : 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),Na))))) ).
fof(fact_4742_card__bdd__int__set__l,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na))
=> 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),Na)))) = hAPP_int_nat(nat,Na) ) ).
fof(fact_4743_card__bdd__int__set__l__le,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na))
=> 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),Na)))) = hAPP_int_nat(nat,Na) ) ).
fof(fact_4744_less__eq__Suc__le__raw,axiom,
! [X_1] : hAPP_n1699378549t_bool(ord_less_nat,X_1) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X_1)) ).
fof(fact_4745_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_4746_DERIV__sin__sin__mult,axiom,
! [Xa] : 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),Xa),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Xa)),hAPP_real_real(sin,Xa))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Xa)),hAPP_real_real(sin,Xa))))) ).
fof(fact_4747_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_4748_DERIV__sin__add,axiom,
! [K,Xa] : 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)),Xa),hAPP_real_real(cos,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,Xa),K)))) ).
fof(fact_4749_card__Collect__le__nat,axiom,
! [Na] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),Na))) = hAPP_nat_nat(suc,Na) ).
fof(fact_4750_card__bdd__nat__set__le,axiom,
! [Xa] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),Xa))) = hAPP_nat_nat(suc,Xa) ).
fof(fact_4751_bdd__nat__set__le__finite,axiom,
! [Xa] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_eq_nat),Xa)))) ).
fof(fact_4752_primes__infinite,axiom,
~ hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(prime))) ).
fof(fact_4753_Re_OCauchy,axiom,
! [X_2] :
( hBOOL(cauchy_complex(X_2))
=> hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_2))) ) ).
fof(fact_4754_Im_OCauchy,axiom,
! [X_2] :
( hBOOL(cauchy_complex(X_2))
=> hBOOL(cauchy_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_2))) ) ).
fof(fact_4755_cnj_OCauchy,axiom,
! [X_2] :
( hBOOL(cauchy_complex(X_2))
=> hBOOL(cauchy_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_2))) ) ).
fof(fact_4756_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_4757_DERIV__fun__sin,axiom,
! [G,Xa,Ma] :
( hBOOL(hAPP_real_bool(deriv_real(G,Xa),Ma))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,sin),G),Xa),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,hAPP_real_real(G,Xa))),Ma))) ) ).
fof(fact_4758_card__bdd__nat__set__l,axiom,
! [Xa] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),Xa))) = Xa ).
fof(fact_4759_card__Collect__less__nat,axiom,
! [Na] : hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),Na))) = Na ).
fof(fact_4760_bdd__int__set__l__l__finite,axiom,
! [Na] : 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),Na))))) ).
fof(fact_4761_bdd__nat__set__l__finite,axiom,
! [Xa] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(hAPP_f229349961t_bool(cOMBC_nat_nat_bool,ord_less_nat),Xa)))) ).
fof(fact_4762_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_4763_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_4764_DERIV__fun__cos,axiom,
! [G,Xa,Ma] :
( hBOOL(hAPP_real_bool(deriv_real(G,Xa),Ma))
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,cos),G),Xa),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,hAPP_real_real(G,Xa)))),Ma))) ) ).
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_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_4767_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_4768_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_4769_transfer__morphism__int__nat,axiom,
hBOOL(hAPP_f54304608l_bool(nat_tr160667106at_int(semiri1621563631at_int),cOMBK_bool_nat(fTrue))) ).
fof(fact_4770_int__card__bdd__int__set__l__le,axiom,
! [Na] :
( is_int(Na)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na))
=> 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),Na))))) = Na ) ) ).
fof(fact_4771_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_4772_noXRRset__def,axiom,
! [Ma,Xa] : hAPP_i1948725293t_bool(noXRRset(Ma),Xa) = image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),Xa),hAPP_i1948725293t_bool(norRRset,Ma)) ).
fof(fact_4773_DERIV__fun__pow,axiom,
! [Na,G,Xa,Ma] :
( hBOOL(hAPP_real_bool(deriv_real(G,Xa),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)),Na),Xa),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(G,Xa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)))),Ma))) ) ).
fof(fact_4774_int__card__bdd__int__set__l__l,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Na))
=> 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),Na))))) = hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Na),one_one_int) ) ).
fof(fact_4775_card__bdd__int__set__le,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na))
=> 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),Na)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_int_nat(nat,Na)),one_one_nat) ) ).
fof(fact_4776_card__bdd__int__set__l__l,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Na))
=> 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),Na)))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(nat,Na)),one_one_nat) ) ).
fof(fact_4777_DERIV__cos__cos__mult,axiom,
! [Xa] : 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),Xa),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,Xa))),hAPP_real_real(cos,Xa))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,Xa))),hAPP_real_real(cos,Xa))))) ).
fof(fact_4778_eq__number__of__Suc,axiom,
! [Va,Na] :
( hAPP_int_nat(number_number_of_nat,Va) = hAPP_nat_nat(suc,Na)
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
=> hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))) = Na ) ) ) ).
fof(fact_4779_Suc__eq__number__of,axiom,
! [Na,Va] :
( hAPP_nat_nat(suc,Na) = hAPP_int_nat(number_number_of_nat,Va)
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
=> hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))) = Na ) ) ) ).
fof(fact_4780_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_4781_DERIV__pow,axiom,
! [Na,Xa] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(hAPP_f248847589l_real(cOMBC_real_nat_real,power_power_real),Na),Xa),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),hAPP_nat_nat(suc,zero_zero_nat)))))) ).
fof(fact_4782_DERIV__cos__cos__mult2,axiom,
! [Xa] : 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),Xa),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,Xa))),hAPP_real_real(sin,Xa)))) ).
fof(fact_4783_less__Suc__number__of,axiom,
! [Na,Va] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,Na)),hAPP_int_nat(number_number_of_nat,Va)))
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Na),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))) ) ) ) ).
fof(fact_4784_less__number__of__Suc,axiom,
! [Va,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_nat_nat(suc,Na)))
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
=> 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,Va)))),Na)) ) ) ).
fof(fact_4785_le__Suc__number__of,axiom,
! [Na,Va] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),hAPP_int_nat(number_number_of_nat,Va)))
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
& ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))) ) ) ) ).
fof(fact_4786_le__number__of__Suc,axiom,
! [Va,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_nat_nat(suc,Na)))
<=> ( ~ hBOOL(hAPP_int_bool(nat_neg,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va))))
=> 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,Va)))),Na)) ) ) ).
fof(fact_4787_SR__pos,axiom,
! [X_2,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_2)),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_4788_RsetR__zmult__mono,axiom,
! [Xa,A,Ma] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,A),rsetR(Ma)))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Ma))
=> ( hAPP_int_int(legacy_zgcd(Xa),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),Xa),A)),rsetR(Ma))) ) ) ) ).
fof(fact_4789_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_4790_DERIV__sin__sin__mult2,axiom,
! [Xa] : 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),Xa),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,Xa))),hAPP_real_real(sin,Xa)))) ).
fof(fact_4791_DERIV__log,axiom,
! [B_1,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> hBOOL(hAPP_real_bool(deriv_real(log(B_1),Xa),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_1)),Xa)))) ) ).
fof(fact_4792_DERIV__cos__cos__mult3,axiom,
! [Xa] : 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),Xa),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,Xa))),hAPP_real_real(sin,Xa))))) ).
fof(fact_4793_DERIV__sin__circle__all__zero,axiom,
! [X_1] : 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_1),zero_zero_real)) ).
fof(fact_4794_DERIV__sin__realpow2a,axiom,
! [Xa] : 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)))),Xa),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,Xa))),hAPP_real_real(sin,Xa)))) ).
fof(fact_4795_diff__nat__number__of,axiom,
! [Va,V] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_int_nat(number_number_of_nat,V)) = hAPP_int_nat(number_number_of_nat,Va) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,V),pls))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_int_nat(number_number_of_nat,Va)),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(hAPP_int_fun_int_int(plus_plus_int,Va),hAPP_int_int(uminus_uminus_int,V))))),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,Va),hAPP_int_int(uminus_uminus_int,V))))) ) ) ).
fof(fact_4796_DERIV__sin__realpow2,axiom,
! [Xa] : 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)))),Xa),hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Xa)),hAPP_real_real(sin,Xa))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(cos,Xa)),hAPP_real_real(sin,Xa))))) ).
fof(fact_4797_DERIV__cos__realpow2a,axiom,
! [Xa] : 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)))),Xa),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,Xa))),hAPP_real_real(sin,Xa)))) ).
fof(fact_4798_DERIV__cos__realpow2b,axiom,
! [Xa] : 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)))),Xa),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,Xa))),hAPP_real_real(sin,Xa))))) ).
fof(fact_4799_DERIV__cos__realpow2,axiom,
! [Xa] : 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)))),Xa),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,Xa))),hAPP_real_real(cos,Xa))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,Xa))),hAPP_real_real(cos,Xa))))) ).
fof(fact_4800_lemma__DERIV__tan,axiom,
! [Xa] :
( hAPP_real_real(cos,Xa) != 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),Xa),hAPP_real_real(inverse_inverse_real,hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(cos,Xa)),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))))) ) ).
fof(fact_4801_lemma__DERIV__sin__cos__add,axiom,
! [Ya,X_1] : 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),Ya)))),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,Ya)))),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,Ya)))))),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),Ya)))),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,Ya)))),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,Ya)))))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls))))),X_1),zero_zero_real)) ).
fof(fact_4802_DERIV__sin__circle__all,axiom,
! [X_1] : 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_1),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_1))),hAPP_real_real(sin,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(cos,X_1))),hAPP_real_real(sin,X_1))))) ).
fof(fact_4803_norm__frac_Opsimps,axiom,
! [A_1,B_1] :
( ( is_int(A_1)
& is_int(B_1) )
=> ( hBOOL(hAPP_P603027463t_bool(accp_P2006205492nt_int(norm_frac_rel),hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,A_1),B_1)))
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
=> norm_frac(A_1,B_1) = norm_frac(hAPP_int_int(uminus_uminus_int,A_1),hAPP_int_int(uminus_uminus_int,B_1)) )
& ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,B_1),zero_zero_int))
=> ( ( ( A_1 = zero_zero_int
| B_1 = zero_zero_int )
=> norm_frac(A_1,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),one_one_int) )
& ( ~ ( A_1 = zero_zero_int
| B_1 = zero_zero_int )
=> norm_frac(A_1,B_1) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_1),B_1))),hAPP_int_int(hAPP_int_fun_int_int(div_div_int,B_1),hAPP_int_int(hAPP_int_fun_int_int(int_gcd,A_1),B_1))) ) ) ) ) ) ) ).
fof(fact_4804_monoseq__arctan__series,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Xa)),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,Xa)),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,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),one_one_real))
=> hBOOL(monoseq_real(hAPP_r474017924t_real(power_power_real,Xa))) ) ) ).
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_1,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_1),zero_zero_int),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A_1,P_5)))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,A_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_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_1] : big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(d22set,A_1)) = hAPP_int_int(zfact,A_1) ).
fof(fact_4810_Bnor__prod__zgcd,axiom,
! [A_1,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),Ma))
=> hAPP_int_int(legacy_zgcd(big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A_1),Ma))),Ma) = one_one_int ) ).
fof(fact_4811_Bnor__prod__power,axiom,
! [A_1,Ma,Xa] :
( is_int(Xa)
=> ( Xa != zero_zero_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,A_1),Ma))
=> big_co1548731110nt_int(cOMBI_int,image_int_int(hAPP_int_fun_int_int(hAPP_f2135977429nt_int(cOMBC_int_int_int,times_times_int),Xa),hAPP_i1948725293t_bool(bnorRset(A_1),Ma))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(bnorRset(A_1),Ma))),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,Xa),hAPP_f957591787ol_nat(finite_card_int,hAPP_i1948725293t_bool(bnorRset(A_1),Ma)))) ) ) ) ).
fof(fact_4812_SetS__setprod__prop,axiom,
! [Xa,A_1,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_1),zero_zero_int),P_5))
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(quadRes,P_5),A_1))
=> ( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,Xa),setS(A_1,P_5)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,Xa)),A_1),P_5)) ) ) ) ) ) ).
fof(fact_4813_wset__zcong__prod__1,axiom,
! [A_1,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_1),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_1),P_5))),one_one_int),P_5)) ) ) ) ).
fof(fact_4814_Union__SetS__setprod__prop2,axiom,
! [A_1,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_1),zero_zero_int),P_5))
=> big_co1548731110nt_int(cOMBI_int,comple219730294t_bool(setS(A_1,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] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> ( ! [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_eq_int,zero_zero_int),hAPP_int_int(F,X_1))) ) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),big_co1548731110nt_int(F,A))) ) ) ).
fof(fact_4816_arctan__series,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Xa)),one_one_real))
=> hAPP_real_real(arctan,Xa) = 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,Xa)),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,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(abs_abs_real,Xa)),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))))),Xa),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,Xa)),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,
! [Xa] : hAPP_real_real(sin,Xa) = 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,Xa))) ).
fof(fact_4819_lemma__sin__ext,axiom,
! [X_1] : hAPP_real_real(sin,X_1) = 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_1))) ).
fof(fact_4820_cos__def,axiom,
! [Xa] : hAPP_real_real(cos,Xa) = 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,Xa))) ).
fof(fact_4821_lemma__cos__ext,axiom,
! [X_1] : hAPP_real_real(cos,X_1) = 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_1))) ).
fof(fact_4822_lemma__sin__minus,axiom,
! [Xa] : hAPP_real_real(uminus_uminus_real,hAPP_real_real(sin,Xa)) = 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,Xa)))) ).
fof(fact_4823_exp__first__two__terms,axiom,
! [Xa] : hAPP_real_real(exp_real,Xa) = hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,hAPP_real_real(hAPP_r1250527377l_real(plus_plus_real,one_one_real),Xa)),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,Xa)),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,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),hAPP_int_real(number267125858f_real,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))
=> hAPP_real_real(ln,Xa) = 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,Xa),one_one_real))),suc))) ) ) ).
fof(fact_4826_summable__arctan__series,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Xa)),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,Xa)),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_2] :
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),N_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_1))),hAPP_nat_real(G,N_1))) )
=> ( 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,
! [Xa] : 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,Xa)))) ).
fof(fact_4831_summable__cos,axiom,
! [Xa] : 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,Xa)))) ).
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,
! [Xa] : 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,Xa)))) ).
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_1,B_1] : divmod_int_rel(A_1,B_1) = 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_1))),hAPP_f1760145644nt_int(hAPP_f1629352853nt_int(cOMBB_1496585939nt_int,plus_plus_int),hAPP_int_fun_int_int(times_times_int,B_1)))))),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_1))),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_1))))),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_1)))),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_1))),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,
! [Xa] : 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,Xa)),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_1] : adjust(B_1) = 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_1))))),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_1))))),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_3] :
( ! [X_1] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_3)),R_3)))
=> 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_1)))) )
=> ( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X0),hAPP_r1134773055l_bool(ord_gr788844697n_real(hAPP_real_real(uminus_uminus_real,R_3)),R_3)))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),R_3))
=> 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_2] :
( hBOOL(summable_complex(X_2))
=> hAPP_complex_complex(cnj,hAPP_f213450978omplex(suminf_complex,X_2)) = hAPP_f213450978omplex(suminf_complex,hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_2)) ) ).
fof(fact_4844_cnj_Osummable,axiom,
! [X_2] :
( hBOOL(summable_complex(X_2))
=> hBOOL(summable_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_2))) ) ).
fof(fact_4845_Re_Osummable,axiom,
! [X_2] :
( hBOOL(summable_complex(X_2))
=> hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_2))) ) ).
fof(fact_4846_Im_Osummable,axiom,
! [X_2] :
( hBOOL(summable_complex(X_2))
=> hBOOL(summable_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_2))) ) ).
fof(fact_4847_Re_Osuminf,axiom,
! [X_2] :
( hBOOL(summable_complex(X_2))
=> hAPP_complex_real(re,hAPP_f213450978omplex(suminf_complex,X_2)) = hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_2)) ) ).
fof(fact_4848_Im_Osuminf,axiom,
! [X_2] :
( hBOOL(summable_complex(X_2))
=> hAPP_complex_real(im,hAPP_f213450978omplex(suminf_complex,X_2)) = hAPP_f352196356l_real(suminf_real,hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_2)) ) ).
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,Na] :
( ( ( Na = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) )
=> divmod_nat(Ma,Na) = hAPP_n1289843868at_nat(hAPP_n1865633855at_nat(product_Pair_nat_nat,zero_zero_nat),Ma) )
& ( ~ ( Na = zero_zero_nat
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Ma),Na)) )
=> divmod_nat(Ma,Na) = 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),Na),Na)) ) ) ).
fof(fact_4851_Bolzano__bisect_Osimps_I2_J,axiom,
! [P_1,A_1,B_1,Na] : hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),hAPP_nat_nat(suc,Na)) = 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_1,B_1),Na)) ).
fof(fact_4852_suminf__exist__split,axiom,
! [A_1,Ra] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Ra))
=> ( hBOOL(summable_real(A_1))
=> ? [N_2] :
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),N_1))
=> 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_1),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),N_1))))),Ra)) ) ) ) ).
fof(fact_4853_divmod__int__pdivmod,axiom,
! [L,K] :
( ( is_int(L)
& is_int(K) )
=> ( ( K = zero_zero_int
=> divmod_int(K,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) )
& ( K != zero_zero_int
=> ( ( L = zero_zero_int
=> divmod_int(K,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K) )
& ( L != zero_zero_int
=> divmod_int(K,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)),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),zero_zero_int)))),pdivmod(K,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,L)))) ) ) ) ) ) ).
fof(fact_4854_negateSnd__def,axiom,
negateSnd = produc713050258nt_int(uminus_uminus_int) ).
fof(fact_4855_divmod__int__code,axiom,
! [L,K] :
( ( is_int(L)
& is_int(K) )
=> ( ( K = zero_zero_int
=> divmod_int(K,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),zero_zero_int) )
& ( K != zero_zero_int
=> ( ( L = zero_zero_int
=> divmod_int(K,L) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,zero_zero_int),K) )
& ( L != zero_zero_int
=> divmod_int(K,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)),hAPP_int_int(sgn_sgn_int,L))),pdivmod(K,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,L)))) ) ) ) ) ) ).
fof(fact_4856_summable__le2,axiom,
! [F,G] :
( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,hAPP_nat_real(F,N_1))),hAPP_nat_real(G,N_1)))
=> ( 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_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(F,N_1)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_f352196356l_real(suminf_real,F))) ) ) ).
fof(fact_4858_suminf__0__le,axiom,
! [F] :
( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_1)))
=> ( 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_4859_suminf__ge__zero,axiom,
! [F] :
( hBOOL(summable_real(F))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_1)))
=> 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_1] : int_ge_less_than2(D_1) = 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_1)))),ord_less_int))) ).
fof(fact_4861_int__ge__less__than__def,axiom,
! [D_1] : int_ge_less_than(D_1) = 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_1)))),ord_less_int))) ).
fof(fact_4862_sin__paired,axiom,
! [Xa] : 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,Xa)),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,Xa))) ).
fof(fact_4863_sin__converges,axiom,
! [Xa] : 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,Xa))),hAPP_real_real(sin,Xa))) ).
fof(fact_4864_cos__converges,axiom,
! [Xa] : 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,Xa))),hAPP_real_real(cos,Xa))) ).
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,
! [Xa] : 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,Xa),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,Xa),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))))) ).
fof(fact_4867_sums__if_H,axiom,
! [G,Xa] :
( hBOOL(hAPP_real_bool(sums_real(G),Xa))
=> 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))))))),Xa)) ) ).
fof(fact_4868_sums__if,axiom,
! [F,Ya,G,Xa] :
( hBOOL(hAPP_real_bool(sums_real(G),Xa))
=> ( hBOOL(hAPP_real_bool(sums_real(F),Ya))
=> 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,Xa),Ya))) ) ) ).
fof(fact_4869_cos__paired,axiom,
! [Xa] : 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,Xa)),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,Xa))) ).
fof(fact_4870_cnj_Osums,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_complex_bool(sums_complex(X_2),A_1))
=> hBOOL(hAPP_complex_bool(sums_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_2)),hAPP_complex_complex(cnj,A_1))) ) ).
fof(fact_4871_Re_Osums,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_complex_bool(sums_complex(X_2),A_1))
=> hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_2)),hAPP_complex_real(re,A_1))) ) ).
fof(fact_4872_Im_Osums,axiom,
! [X_2,A_1] :
( hBOOL(hAPP_complex_bool(sums_complex(X_2),A_1))
=> hBOOL(hAPP_real_bool(sums_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_2)),hAPP_complex_real(im,A_1))) ) ).
fof(fact_4873_sin__fdiffs2,axiom,
! [Na] : hAPP_nat_real(diffs_real(sin_coeff),Na) = hAPP_nat_real(cos_coeff,Na) ).
fof(fact_4874_sin__fdiffs,axiom,
diffs_real(sin_coeff) = cos_coeff ).
fof(fact_4875_exp__fdiffs,axiom,
! [X_1] : 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_1) = hAPP_real_real(inverse_inverse_real,hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,X_1))) ).
fof(fact_4876_less__eq__nat_Osimps_I2_J,axiom,
! [Ma,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Ma)),Na))
<=> hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,Ma)),Na)) ) ).
fof(fact_4877_cos__fdiffs,axiom,
! [X_1] : hAPP_nat_real(diffs_real(cos_coeff),X_1) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,X_1)) ).
fof(fact_4878_cos__fdiffs2,axiom,
! [Na] : hAPP_nat_real(diffs_real(cos_coeff),Na) = hAPP_real_real(uminus_uminus_real,hAPP_nat_real(sin_coeff,Na)) ).
fof(fact_4879_diff__Suc,axiom,
! [Ma,Na] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),hAPP_nat_nat(suc,Na)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Ma),Na)) ).
fof(fact_4880_summable__le,axiom,
! [F,G] :
( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_1)),hAPP_nat_real(G,N_1)))
=> ( 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,
! [Ya] : hAPP_real_real(arctan,Ya) = 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)),Ya)))) ).
fof(fact_4882_arcsin__def,axiom,
! [Ya] : hAPP_real_real(arcsin,Ya) = 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)),Ya)))) ).
fof(fact_4883_ln__def,axiom,
! [Xa] : hAPP_real_real(ln,Xa) = 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)),Xa)) ).
fof(fact_4884_root__def,axiom,
! [Na,Xa] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) = 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),Na))),Xa))) )
& ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> ( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real))
=> hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) = 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),Na))),hAPP_real_real(uminus_uminus_real,Xa))))) )
& ( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real))
=> hAPP_real_real(hAPP_n546711566l_real(root,Na),Xa) = zero_zero_real ) ) ) ) ).
fof(fact_4885_arccos__def,axiom,
! [Ya] : hAPP_real_real(arccos,Ya) = 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)),Ya)))) ).
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,Na] : divmod_nat(Ma,Na) = the_Pr588456374at_nat(divmod_nat_rel(Ma,Na)) ).
fof(fact_4889_floor__real__def,axiom,
! [Xa] : archim1246769320r_real(Xa) = 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)),Xa))),hAPP_f429384717t_bool(hAPP_f1173902867t_bool(cOMBB_real_bool_int,hAPP_r1134773055l_bool(ord_less_real,Xa)),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,
! [Xa] : archim791455193or_rat(Xa) = 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)),Xa))),hAPP_f1241350704t_bool(hAPP_f1131424617t_bool(cOMBB_rat_bool_int,hAPP_r2115560837t_bool(ord_less_rat,Xa)),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_1,A,Ma] :
( ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
& hBOOL(hAPP_int_bool(is_RRset(A),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A)) )
=> hAPP_int_int(rRset2norRR(A,Ma),A_1) = 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_1)),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),Ma))
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A)) )
=> hAPP_int_int(rRset2norRR(A,Ma),A_1) = zero_zero_int ) ) ).
fof(fact_4892_aux__some,axiom,
! [A_1,A,Ma] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,one_one_int),Ma))
=> ( hBOOL(hAPP_int_bool(is_RRset(A),Ma))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,A_1),A))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(A_1),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_1)),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_1)),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,
! [Xa,Na] : 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,Xa)),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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))))),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,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,hAPP_real_real(abs_abs_real,Xa)),Na)))) ).
fof(fact_4894_finite__less__ub,axiom,
! [U,F] :
( ! [N_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),hAPP_nat_nat(F,N_1)))
=> 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__offset4,axiom,
! [K,N_1,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_1),K))) = 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))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),N_1))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K))) ).
fof(fact_4897_sumr__offset2,axiom,
! [K,Na,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))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na)) = 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,Na),K)))),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,F_2),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),K))) ).
fof(fact_4898_sumr__diff__mult__const,axiom,
! [F,Na,Ra] : 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),Na))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),Ra)) = 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)),Ra)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na)) ).
fof(fact_4899_sumr__one__lb__realpow__zero,axiom,
! [F,Na] : 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)),Na)) = zero_zero_real ).
fof(fact_4900_ex__nat__less__eq,axiom,
! [P_1,Na] :
( ? [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
& hBOOL(hAPP_nat_bool(P_1,M_1)) )
<=> ? [X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na)))
& hBOOL(hAPP_nat_bool(P_1,X_1)) ) ) ).
fof(fact_4901_all__nat__less__eq,axiom,
! [P_1,Na] :
( ! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
=> hBOOL(hAPP_nat_bool(P_1,M_1)) )
<=> ! [X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na)))
=> hBOOL(hAPP_nat_bool(P_1,X_1)) ) ) ).
fof(fact_4902_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_4903_sumr__geometric,axiom,
! [Na,Xa] :
( Xa != one_one_real
=> hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(power_power_real,Xa)),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na)) = 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,Xa),Na)),one_one_real)),hAPP_real_real(hAPP_r1250527377l_real(minus_minus_real,Xa),one_one_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,I_1,J] : image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K),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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K)) ).
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,Na,Xa] :
( Xa = zero_zero_real
=> ( Na != 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na)) = hAPP_real_real(hAPP_n546711566l_real(Diff,zero_zero_nat),zero_zero_real) ) ) ).
fof(fact_4908_subset__card__intvl__is__intvl,axiom,
! [A,K] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),hAPP_n1699378549t_bool(ord_at4362885an_nat(K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),hAPP_f22106695ol_nat(finite_card_nat,A)))))
=> A = hAPP_n1699378549t_bool(ord_at4362885an_nat(K),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),hAPP_f22106695ol_nat(finite_card_nat,A))) ) ).
fof(fact_4909_sum__split__even__odd,axiom,
! [F,G,Na] : 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)))),Na))) = 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),Na))),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),Na))) ).
fof(fact_4910_sumr__minus__one__realpow__zero,axiom,
! [Na] : 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)))),Na))) = zero_zero_real ).
fof(fact_4911_lemma__STAR__cos2,axiom,
! [Na] : 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),Na)) = zero_zero_real ).
fof(fact_4912_sumr__cos__zero__one,axiom,
! [Na] : 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,Na))) = one_one_real ).
fof(fact_4913_Maclaurin__sin__expansion2,axiom,
! [Na,Xa] :
? [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,Xa)))
& hAPP_real_real(sin,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) ) ).
fof(fact_4914_Maclaurin__cos__expansion,axiom,
! [Na,Xa] :
? [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,Xa)))
& hAPP_real_real(cos,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) ) ).
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_finite__atLeastZeroLessThan__int,axiom,
! [U] : hBOOL(hAPP_f448129468l_bool(finite_finite_int,hAPP_i1948725293t_bool(ord_at641636577an_int(zero_zero_int),U))) ).
fof(fact_4917_Sup__atLeastLessThan,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Ya),Xa))
=> comple124823625p_real(hAPP_r1134773055l_bool(ord_at1496968948n_real(Ya),Xa)) = Xa ) ).
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,
! [Xa,Na] :
? [T] : hAPP_real_real(sin,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na))),pi)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) ).
fof(fact_4923_Maclaurin__lemma2,axiom,
! [B,K,Diff,H,Na] :
( ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
& 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))) )
=> ( Na = hAPP_nat_nat(suc,K)
=> ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
& 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,Na),M_1))))),hAPP_f1950183573l_real(hAPP_f1992866895l_real(cOMBB_real_real_real,hAPP_r1250527377l_real(times_times_real,B)),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,Na),M_1)))),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),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,Na),hAPP_nat_nat(suc,M_1))))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,B),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,Na),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,Na),hAPP_nat_nat(suc,M_1)))))))))) ) ) ) ).
fof(fact_4924_Maclaurin__exp__lt,axiom,
! [Na,Xa] :
( Xa != zero_zero_real
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ? [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,Xa)))
& hAPP_real_real(exp_real,Xa) = 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,Xa))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) ) ) ) ).
fof(fact_4925_Maclaurin__exp__le,axiom,
! [Na,Xa] :
? [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,Xa)))
& hAPP_real_real(exp_real,Xa) = 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,Xa))),hAPP_f1676298106t_real(hAPP_f1566856189t_real(cOMBB_nat_real_nat,real_nat),fact_fact_nat))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) ) ).
fof(fact_4926_sumr__pos__lt__pair,axiom,
! [K,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),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),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))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).
fof(fact_4927_Maclaurin__lemma,axiom,
! [F,J,Na,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),Na))),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),Na)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na))))) ) ).
fof(fact_4928_series__pos__less,axiom,
! [Na,F] :
( hBOOL(summable_real(F))
=> ( ! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),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),Na))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).
fof(fact_4929_series__pos__le,axiom,
! [Na,F] :
( hBOOL(summable_real(F))
=> ( ! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),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),Na))),hAPP_f352196356l_real(suminf_real,F))) ) ) ).
fof(fact_4930_Maclaurin__all__lt__objl,axiom,
! [Na,Xa,Diff,F] :
( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
& ! [M_1,X_1] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_1)))
& Xa != zero_zero_real
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na)) )
=> ? [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,Xa)))
& hAPP_real_real(F,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) ) ) ).
fof(fact_4931_Maclaurin__all__lt,axiom,
! [Xa,Na,Diff,F] :
( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( Xa != zero_zero_real
=> ( ! [M_1,X_1] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_1)))
=> ? [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,Xa)))
& hAPP_real_real(F,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) ) ) ) ) ) ).
fof(fact_4932_Maclaurin__bi__le,axiom,
! [Xa,Na,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),Na))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,T)),hAPP_real_real(abs_abs_real,Xa))) )
=> 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,Xa)))
& hAPP_real_real(F,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) ) ) ) ).
fof(fact_4933_Maclaurin__objl,axiom,
! [Diff,F,Na,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),Na))
& hAPP_n546711566l_real(Diff,zero_zero_nat) = F
& ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
& 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),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),Na))) ) ) ).
fof(fact_4934_Maclaurin2__objl,axiom,
! [Na,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),Na))
& 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),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),Na))) ) ) ).
fof(fact_4935_Maclaurin__minus,axiom,
! [Diff,F,Na,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),Na))
=> ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
=> ( ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
& 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),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),Na))) ) ) ) ) ) ).
fof(fact_4936_Maclaurin2,axiom,
! [Na,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),Na))
& 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),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),Na))) ) ) ) ) ).
fof(fact_4937_Maclaurin__minus__objl,axiom,
! [Diff,F,Na,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),Na))
& hAPP_n546711566l_real(Diff,zero_zero_nat) = F
& ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
& 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),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),Na))) ) ) ).
fof(fact_4938_Maclaurin,axiom,
! [Diff,F,Na,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),Na))
=> ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
=> ( ! [M_1,T] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),Na))
& 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),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,H),Na))) ) ) ) ) ) ).
fof(fact_4939_Maclaurin__all__le,axiom,
! [Na,Xa,Diff,F] :
( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
=> ( ! [M_1,X_1] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_1)))
=> ? [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,Xa)))
& hAPP_real_real(F,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) ) ) ) ).
fof(fact_4940_Maclaurin__all__le__objl,axiom,
! [Na,Xa,Diff,F] :
( ( hAPP_n546711566l_real(Diff,zero_zero_nat) = F
& ! [M_1,X_1] : hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(Diff,M_1),X_1),hAPP_real_real(hAPP_n546711566l_real(Diff,hAPP_nat_nat(suc,M_1)),X_1))) )
=> ? [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,Xa)))
& hAPP_real_real(F,Xa) = 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,Xa))),hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na))),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,Na),T)),hAPP_nat_real(real_nat,hAPP_nat_nat(fact_fact_nat,Na)))),hAPP_nat_real(hAPP_r474017924t_real(power_power_real,Xa),Na))) ) ) ).
fof(fact_4941_real__setsum__nat__ivl__bounded,axiom,
! [F,K_3,Na] :
( ! [P_4] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,P_4),Na))
=> 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),Na))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,hAPP_nat_real(real_nat,Na)),K_3))) ) ).
fof(fact_4942_DERIV__sumr,axiom,
! [F,Xa,F_1,Na,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),Na))) )
=> hBOOL(hAPP_real_bool(deriv_real(hAPP_n546711566l_real(F,R_1),Xa),hAPP_real_real(hAPP_n546711566l_real(F_1,R_1),Xa))) )
=> 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),Na)),Xa),hAPP_f1406902706l_real(hAPP_f1509287679l_real(big_co604158596t_real,hAPP_r474017924t_real(hAPP_f42270917t_real(cOMBC_nat_real_real,F_1),Xa)),hAPP_n1699378549t_bool(ord_at4362885an_nat(Ma),Na)))) ) ).
fof(fact_4943_suminf__le,axiom,
! [Xa,F] :
( hBOOL(summable_real(F))
=> ( ! [N_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_1))),Xa))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_f352196356l_real(suminf_real,F)),Xa)) ) ) ).
fof(fact_4944_transfer__nat__int__sum__prod__closure_I1_J,axiom,
! [F,A] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> ( ! [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_eq_int,zero_zero_int),hAPP_int_int(F,X_1))) ) )
=> 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))) ) ) ).
fof(fact_4945_pos__summable,axiom,
! [Xa,F] :
( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(F,N_1)))
=> ( ! [N_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_1))),Xa))
=> hBOOL(summable_real(F)) ) ) ).
fof(fact_4946_arith__series__int,axiom,
! [A_1,D_1,Na] : 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_1)),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_1))),hAPP_n1699378549t_bool(ord_lessThan_nat,Na))) = hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_nat_int(semiri1621563631at_int,Na)),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,A_1),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,Na),one_one_nat))),D_1)))) ).
fof(fact_4947_BseqI2,axiom,
! [K_3,K,F] :
( ! [N_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,K),hAPP_nat_real(F,N_1)))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_1)),K_3)) )
=> hBOOL(bseq_real(F)) ) ).
fof(fact_4948_finite__lessThan,axiom,
! [K] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,K))) ).
fof(fact_4949_card__lessThan,axiom,
! [U] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_lessThan_nat,U)) = U ).
fof(fact_4950_finite__nat__iff__bounded,axiom,
! [S] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
<=> ? [K_1] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K_1))) ) ).
fof(fact_4951_atLeast0LessThan,axiom,
! [Na] : hAPP_n1699378549t_bool(ord_at4362885an_nat(zero_zero_nat),Na) = hAPP_n1699378549t_bool(ord_lessThan_nat,Na) ).
fof(fact_4952_arith__series__nat,axiom,
! [A_1,D_1,Na] : 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_1)),hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,times_times_nat),D_1))),hAPP_n1699378549t_bool(ord_lessThan_nat,Na))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,Na),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(times_times_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),D_1)))) ).
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_1] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_lessThan_nat,K_1))) ) ).
fof(fact_4955_sum__diff__distrib,axiom,
! [Na,Q_3,P_1] :
( ! [X_1] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(Q_3,X_1)),hAPP_nat_nat(P_1,X_1)))
=> 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,Na))),hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,Q_3),hAPP_n1699378549t_bool(ord_lessThan_nat,Na))) = 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,Na)) ) ).
fof(fact_4956_Sup__lessThan,axiom,
! [Xa] : comple124823625p_real(hAPP_r1134773055l_bool(ord_lessThan_real,Xa)) = Xa ).
fof(fact_4957_transfer__nat__int__set__cong,axiom,
! [P_2,P_1] :
( ! [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(P_1,X_1))
<=> hBOOL(hAPP_int_bool(P_2,X_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)) = 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__dvd2__int,axiom,
! [X,Y] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)),Y)) ).
fof(fact_4960_gcd__dvd1__int,axiom,
! [X,Y] : hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)),X)) ).
fof(fact_4961_coprime__exp2__int,axiom,
! [N,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)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),M)) = one_one_int ) ).
fof(fact_4962_divides__mult__int,axiom,
! [N,M,R_2] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,M),R_2))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,N),R_2))
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) = 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)),R_2)) ) ) ) ).
fof(fact_4963_coprime__dvd__mult__iff__int,axiom,
! [Ma,K,Na] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K),Na) = one_one_int
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,Ma),Na)))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Ma)) ) ) ).
fof(fact_4964_coprime__dvd__mult__int,axiom,
! [M,K_2,N] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N) = 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)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K_2),M)) ) ) ).
fof(fact_4965_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_4966_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_4967_gcd__0__int,axiom,
! [X] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),zero_zero_int) = hAPP_int_int(abs_abs_int,X) ).
fof(fact_4968_gcd__0__left__int,axiom,
! [X] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,zero_zero_int),X) = hAPP_int_int(abs_abs_int,X) ).
fof(fact_4969_gcd__add__mult__int,axiom,
! [M,K_2,N] : 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)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) ).
fof(fact_4970_gcd__mult__distrib__int,axiom,
! [K_2,M,N] : 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)) = 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)) ).
fof(fact_4971_gcd__add1__int,axiom,
! [M,N] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,M),N)),N) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) ).
fof(fact_4972_gcd__add2__int,axiom,
! [M,N] : 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)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) ).
fof(fact_4973_gcd__proj1__if__dvd__int,axiom,
! [X,Y] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),Y))
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(abs_abs_int,X) ) ).
fof(fact_4974_gcd__proj2__if__dvd__int,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,Y),X))
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(abs_abs_int,Y) ) ).
fof(fact_4975_abs__gcd__int,axiom,
! [X,Y] : hAPP_int_int(abs_abs_int,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).
fof(fact_4976_gcd__idem__int,axiom,
! [X] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),X) = hAPP_int_int(abs_abs_int,X) ).
fof(fact_4977_gcd__abs__int,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X)),hAPP_int_int(abs_abs_int,Y)) ).
fof(fact_4978_gcd__abs2__int,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),hAPP_int_int(abs_abs_int,Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).
fof(fact_4979_gcd__abs1__int,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(abs_abs_int,X)),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).
fof(fact_4980_coprime__plus__one__int,axiom,
! [N] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,N),one_one_int)),N) = one_one_int ).
fof(fact_4981_coprime__minus__one__int,axiom,
! [N] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,N),one_one_int)),N) = one_one_int ).
fof(fact_4982_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_4983_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_4984_coprime__exp__int,axiom,
! [N,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)) = one_one_int ) ).
fof(fact_4985_gcd__red__int,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y),hAPP_int_int(div_mod_int(X),Y)) ).
fof(fact_4986_gcd__exp__int,axiom,
! [A_3,N,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)),hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,B_2),N)) = 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) ).
fof(fact_4987_dvd__gcd__D2__int,axiom,
! [I,M,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),N)) ) ).
fof(fact_4988_dvd__gcd__D1__int,axiom,
! [I,M,N] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,I),M)) ) ).
fof(fact_4989_gcd__greatest__int,axiom,
! [N,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))
=> 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))) ) ) ).
fof(fact_4990_gcd__greatest__iff__int,axiom,
! [K,Ma,Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),Na)))
<=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Ma))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,K),Na)) ) ) ).
fof(fact_4991_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_4992_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_4993_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_4994_gcd__neg2__int,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),hAPP_int_int(uminus_uminus_int,Y)) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).
fof(fact_4995_gcd__neg1__int,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,X)),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) ).
fof(fact_4996_gcd__zero__int,axiom,
! [Ma,Na] :
( ( is_int(Ma)
& is_int(Na) )
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Ma),Na) = zero_zero_int
<=> ( Ma = zero_zero_int
& Na = zero_zero_int ) ) ) ).
fof(fact_4997_gcd__pos__int,axiom,
! [Ma,Na] :
( ( is_int(Ma)
& is_int(Na) )
=> ( 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),Na)))
<=> ( Ma != zero_zero_int
| Na != zero_zero_int ) ) ) ).
fof(fact_4998_coprime__mul__eq__int,axiom,
! [D_1,A_1,B_1] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,A_1),B_1)) = one_one_int
<=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_1),A_1) = one_one_int
& hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,D_1),B_1) = one_one_int ) ) ).
fof(fact_4999_gcd__mult__cancel__int,axiom,
! [M,K_2,N] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K_2),N) = 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) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,M),N) ) ).
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_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_5002_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_5003_transfer__int__nat__gcd_I1_J,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_nat_int(semiri1621563631at_int,X)),hAPP_nat_int(semiri1621563631at_int,Y)) = hAPP_nat_int(semiri1621563631at_int,hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,X),Y)) ).
fof(fact_5004_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_5005_gcd__non__0__int,axiom,
! [X,Y] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Y))
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Y),hAPP_int_int(div_mod_int(X),Y)) ) ).
fof(fact_5006_invertible__coprime__int,axiom,
! [X,Y,M] :
( hAPP_int_int(div_mod_int(hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y)),M) = one_one_int
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),M) = one_one_int ) ).
fof(fact_5007_coprime__crossproduct__int,axiom,
! [B_1,C,A_1,D_1] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_1),D_1) = one_one_int
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,B_1),C) = one_one_int
=> ( hAPP_int_int(hAPP_int_fun_int_int(times_times_int,hAPP_int_int(abs_abs_int,A_1)),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_1)),hAPP_int_int(abs_abs_int,D_1))
<=> ( hAPP_int_int(abs_abs_int,A_1) = hAPP_int_int(abs_abs_int,B_1)
& hAPP_int_int(abs_abs_int,C) = hAPP_int_int(abs_abs_int,D_1) ) ) ) ) ).
fof(fact_5008_gcd__unique__int,axiom,
! [B_1,A_1,D_1] :
( is_int(D_1)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),D_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),A_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),B_1))
& ! [E] :
( is_int(E)
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),A_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),B_1)) )
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,E),D_1)) ) ) )
<=> D_1 = hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_1),B_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__ge__0__int,axiom,
! [X,Y] : 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),Y))) ).
fof(fact_5012_transfer__nat__int__gcd__closures_I1_J,axiom,
! [Y,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),Y))
=> 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),Y))) ) ) ).
fof(fact_5013_gcd__cases__int,axiom,
! [P_1,Ya,Xa] :
( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Xa),Ya))) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),zero_zero_int))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Xa),hAPP_int_int(uminus_uminus_int,Ya)))) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Ya))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_int_int(uminus_uminus_int,Xa)),Ya))) ) )
=> ( ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Xa),zero_zero_int))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ya),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,Xa)),hAPP_int_int(uminus_uminus_int,Ya)))) ) )
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,Xa),Ya))) ) ) ) ) ).
fof(fact_5014_gcd__code__int,axiom,
! [K,L] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,K),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),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)),hAPP_int_int(abs_abs_int,L))))) ).
fof(fact_5015_coprime__common__divisor__int,axiom,
! [X,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),A_3))
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,X),B_2))
=> hAPP_int_int(abs_abs_int,X) = one_one_int ) ) ) ).
fof(fact_5016_quotient__of__coprime,axiom,
! [R_2,P_3,Q_1] :
( quotient_of(R_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_1) = one_one_int ) ).
fof(fact_5017_normalize__coprime,axiom,
! [R_2,P_3,Q_1] :
( normalize(R_2) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)
=> hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_1) = one_one_int ) ).
fof(fact_5018_coprime__int,axiom,
! [A_1,B_1] :
( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,A_1),B_1) = 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_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_2),B_1)) )
<=> D_2 = one_one_int ) ) ) ).
fof(fact_5019_transfer__nat__int__gcd_I1_J,axiom,
! [Y,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),Y))
=> hAPP_nat_nat(hAPP_nat_fun_nat_nat(gcd_gcd_nat,hAPP_int_nat(nat,X)),hAPP_int_nat(nat,Y)) = hAPP_int_nat(nat,hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y)) ) ) ).
fof(fact_5020_gcd__int__def,axiom,
! [X,Y] : hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,X),Y) = 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))),hAPP_int_nat(nat,hAPP_int_int(abs_abs_int,Y)))) ).
fof(fact_5021_normalize__stable,axiom,
! [P_3,Q_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),Q_1))
=> ( hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,P_3),Q_1) = one_one_int
=> normalize(hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1)) = hAPP_i1524277240nt_int(hAPP_i1584592887nt_int(product_Pair_int_int,P_3),Q_1) ) ) ).
fof(fact_5022_quotient__of__def,axiom,
! [Xa] : quotient_of(Xa) = 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,Xa)),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_2] :
? [X_1] :
( R_2 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,X_1)),hAPP_P1175774780nt_int(product_snd_int_int,X_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,X_1)))
& hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,X_1)),hAPP_P1175774780nt_int(product_snd_int_int,X_1)) = one_one_int
& ! [Y_1] :
( ( R_2 = hAPP_int_rat(hAPP_int_fun_int_rat(fract,hAPP_P1175774780nt_int(product_fst_int_int,Y_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_1))
& hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)))
& hAPP_int_int(hAPP_int_fun_int_int(gcd_gcd_int,hAPP_P1175774780nt_int(product_fst_int_int,Y_1)),hAPP_P1175774780nt_int(product_snd_int_int,Y_1)) = one_one_int )
=> Y_1 = X_1 ) ) ).
fof(fact_5024_Rat__cases__nonzero,axiom,
! [Q_1] :
( ! [A_2,B_4] :
( ( is_int(A_2)
& is_int(B_4) )
=> ( Q_1 = 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_1 = zero_zero_rat ) ).
fof(fact_5025_Rat__induct,axiom,
! [Q_2,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_2)) ) ).
fof(fact_5026_Rat__cases,axiom,
! [Q_1] :
~ ! [A_2,B_4] :
( ( is_int(A_2)
& is_int(B_4) )
=> ( Q_1 = 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_5027_card__setsum__aux,axiom,
! [Na,S] :
( hBOOL(hAPP_f215623910l_bool(finite1395289673t_bool,S))
=> ( ! [X_1] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),S))
=> hBOOL(hAPP_f448129468l_bool(finite_finite_int,X_1)) )
=> ( ! [X_1] :
( hBOOL(hAPP_f215623910l_bool(hAPP_f628503027l_bool(member_fun_int_bool,X_1),S))
=> hAPP_f957591787ol_nat(finite_card_int,X_1) = Na )
=> 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(Na)),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_1] :
( hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially))
=> ( hBOOL(monoseq_real(A_1))
=> 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_1))),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_1)),sequentially)) ) ) ).
fof(fact_5032_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [Ra] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,Ra)),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)))),Ra,sequentially)) ).
fof(fact_5033_LIMSEQ__inverse__realpow__zero,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Xa))
=> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),hAPP_r474017924t_real(power_power_real,Xa)),zero_zero_real,sequentially)) ) ).
fof(fact_5034_LIMSEQ__divide__realpow__zero,axiom,
! [A_1,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,one_one_real),Xa))
=> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(inverse_divide_real,A_1)),hAPP_r474017924t_real(power_power_real,Xa)),zero_zero_real,sequentially)) ) ).
fof(fact_5035_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_5036_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_5037_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_5038_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_5039_LIMSEQ__inverse__real__of__nat__add,axiom,
! [Ra] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(plus_plus_real,Ra)),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))),Ra,sequentially)) ).
fof(fact_5040_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_5041_LIMSEQ__neg__powr,axiom,
! [Sb] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Sb))
=> 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,Sb)),zero_zero_real,sequentially)) ) ).
fof(fact_5042_monoseq__le,axiom,
! [Xa,A_1] :
( hBOOL(monoseq_real(A_1))
=> ( hBOOL(tendsto_nat_real(A_1,Xa,sequentially))
=> ( ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,N_1)),Xa))
& ! [M_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,M_1)),hAPP_nat_real(A_1,N_1))) ) )
| ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),hAPP_nat_real(A_1,N_1)))
& ! [M_1,N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,N_1)),hAPP_nat_real(A_1,M_1))) ) ) ) ) ) ).
fof(fact_5043_LIMSEQ__le__const,axiom,
! [A_1,X_2,Xa] :
( hBOOL(tendsto_nat_real(X_2,Xa,sequentially))
=> ( ? [N_2] :
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),N_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),hAPP_nat_real(X_2,N_1))) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),Xa)) ) ) ).
fof(fact_5044_LIMSEQ__le__const2,axiom,
! [A_1,X_2,Xa] :
( hBOOL(tendsto_nat_real(X_2,Xa,sequentially))
=> ( ? [N_2] :
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),N_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_2,N_1)),A_1)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),A_1)) ) ) ).
fof(fact_5045_LIMSEQ__le,axiom,
! [Y_2,Ya,X_2,Xa] :
( hBOOL(tendsto_nat_real(X_2,Xa,sequentially))
=> ( hBOOL(tendsto_nat_real(Y_2,Ya,sequentially))
=> ( ? [N_2] :
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),N_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(X_2,N_1)),hAPP_nat_real(Y_2,N_1))) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),Ya)) ) ) ) ).
fof(fact_5046_LIMSEQ__realpow__zero,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real))
=> hBOOL(tendsto_nat_real(hAPP_r474017924t_real(power_power_real,Xa),zero_zero_real,sequentially)) ) ) ).
fof(fact_5047_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [Ra] : hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,hAPP_r1250527377l_real(times_times_real,Ra)),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))))),Ra,sequentially)) ).
fof(fact_5048_summable__Leibniz_I1_J,axiom,
! [A_1] :
( hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially))
=> ( hBOOL(monoseq_real(A_1))
=> 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_1))) ) ) ).
fof(fact_5049_zeroseq__arctan__series,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_real_real(abs_abs_real,Xa)),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,Xa)),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_1] :
( hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially))
=> ( hBOOL(monoseq_real(A_1))
=> 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_1))),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_1)),sequentially)) ) ) ).
fof(fact_5052_summable__Leibniz_H_I4_J,axiom,
! [Na,A_1] :
( hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A_1,N_1)))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,hAPP_nat_nat(suc,N_1))),hAPP_nat_real(A_1,N_1)))
=> 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_1))),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_1)),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)))),Na)),one_one_nat))))) ) ) ) ).
fof(fact_5053_summable__Leibniz_H_I5_J,axiom,
! [A_1] :
( hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A_1,N_1)))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,hAPP_nat_nat(suc,N_1))),hAPP_nat_real(A_1,N_1)))
=> 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_1))),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_1)),sequentially)) ) ) ) ).
fof(fact_5054_cnj_OLIMSEQ,axiom,
! [X_2,A_1] :
( hBOOL(tendsto_nat_complex(X_2,A_1,sequentially))
=> hBOOL(tendsto_nat_complex(hAPP_f1646117269omplex(hAPP_f1457396693omplex(cOMBB_117013006ex_nat,cnj),X_2),hAPP_complex_complex(cnj,A_1),sequentially)) ) ).
fof(fact_5055_Re_OLIMSEQ,axiom,
! [X_2,A_1] :
( hBOOL(tendsto_nat_complex(X_2,A_1,sequentially))
=> hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,re),X_2),hAPP_complex_real(re,A_1),sequentially)) ) ).
fof(fact_5056_Im_OLIMSEQ,axiom,
! [X_2,A_1] :
( hBOOL(tendsto_nat_complex(X_2,A_1,sequentially))
=> hBOOL(tendsto_nat_real(hAPP_f1290670424t_real(hAPP_f327026105t_real(cOMBB_722610049al_nat,im),X_2),hAPP_complex_real(im,A_1),sequentially)) ) ).
fof(fact_5057_LIMSEQ__Complex,axiom,
! [Y_2,B_1,X_2,A_1] :
( hBOOL(tendsto_nat_real(X_2,A_1,sequentially))
=> ( hBOOL(tendsto_nat_real(Y_2,B_1,sequentially))
=> hBOOL(tendsto_nat_complex(hAPP_f219720582omplex(hAPP_f1052840010omplex(cOMBS_696648398omplex,hAPP_f23851874omplex(hAPP_f406530075omplex(cOMBB_1714920583ex_nat,complex),X_2)),Y_2),hAPP_real_complex(hAPP_r265291036omplex(complex,A_1),B_1),sequentially)) ) ) ).
fof(fact_5058_Bolzano__nest__unique,axiom,
! [P_1,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),B_1))
=> ( 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_1,B_1)))),hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A_1,B_1))),zero_zero_real,sequentially))
=> ? [L_1] :
( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_P731461727l_real(produc1935615926l_real,hAPP_n332733730l_real(bolzano_bisect(P_1,A_1,B_1),N_1))),L_1))
& hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc1935615926l_real),bolzano_bisect(P_1,A_1,B_1)),L_1,sequentially))
& ! [N_1] : 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_1,B_1),N_1))))
& hBOOL(tendsto_nat_real(hAPP_f73570213t_real(hAPP_f918606171t_real(cOMBB_64450052al_nat,produc556554744l_real),bolzano_bisect(P_1,A_1,B_1)),L_1,sequentially)) ) ) ) ) ) ).
fof(fact_5059_summable__Leibniz_H_I3_J,axiom,
! [A_1] :
( hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A_1,N_1)))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,hAPP_nat_nat(suc,N_1))),hAPP_nat_real(A_1,N_1)))
=> 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_1))),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_1)),sequentially)) ) ) ) ).
fof(fact_5060_summable__Leibniz_H_I2_J,axiom,
! [Na,A_1] :
( hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A_1,N_1)))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,hAPP_nat_nat(suc,N_1))),hAPP_nat_real(A_1,N_1)))
=> 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_1)),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)))),Na)))),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_1)))) ) ) ) ).
fof(fact_5061_summable__Leibniz_I2_J,axiom,
! [A_1] :
( hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially))
=> ( hBOOL(monoseq_real(A_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_nat_real(A_1,zero_zero_nat)))
=> ! [N_1] : 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_1))),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_1)),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_1)))),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_1)),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_1)),one_one_nat)))))) ) ) ) ).
fof(fact_5062_summable__Leibniz_I3_J,axiom,
! [A_1] :
( hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially))
=> ( hBOOL(monoseq_real(A_1))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_nat_real(A_1,zero_zero_nat)),zero_zero_real))
=> ! [N_1] : 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_1))),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_1)),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_1)),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_1)),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_1)))))) ) ) ) ).
fof(fact_5063_Sup__atLeastAtMost,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Ya),Xa))
=> comple124823625p_real(hAPP_r1134773055l_bool(ord_at1589558736t_real(Ya),Xa)) = Xa ) ).
fof(fact_5064_summable,axiom,
! [A_1] :
( hBOOL(tendsto_nat_real(A_1,zero_zero_real,sequentially))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_nat_real(A_1,N_1)))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(A_1,hAPP_nat_nat(suc,N_1))),hAPP_nat_real(A_1,N_1)))
=> 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_1))) ) ) ) ).
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_ex__nat__less,axiom,
! [P_1,Na] :
( ? [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na))
& hBOOL(hAPP_nat_bool(P_1,M_1)) )
<=> ? [X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),Na)))
& hBOOL(hAPP_nat_bool(P_1,X_1)) ) ) ).
fof(fact_5068_all__nat__less,axiom,
! [P_1,Na] :
( ! [M_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),Na))
=> hBOOL(hAPP_nat_bool(P_1,M_1)) )
<=> ! [X_1] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),Na)))
=> hBOOL(hAPP_nat_bool(P_1,X_1)) ) ) ).
fof(fact_5069_SetInterval_Otransfer__nat__int__set__function__closures,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa))
=> hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(Xa),Ya))) ) ).
fof(fact_5070_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_5071_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_5072_fact__altdef__nat,axiom,
! [Na] : hAPP_nat_nat(fact_fact_nat,Na) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(one_one_nat),Na)) ).
fof(fact_5073_image__add__atLeastAtMost,axiom,
! [K,I_1,J] : image_nat_nat(hAPP_nat_fun_nat_nat(hAPP_f416620757at_nat(cOMBC_nat_nat_nat,plus_plus_nat),K),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)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K)) ).
fof(fact_5074_SetInterval_Otransfer__nat__int__set__functions_I2_J,axiom,
! [Ma,Na] : hAPP_n1699378549t_bool(ord_at238088361st_nat(Ma),Na) = image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(hAPP_nat_int(semiri1621563631at_int,Ma)),hAPP_nat_int(semiri1621563631at_int,Na))) ).
fof(fact_5075_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_5076_Re_Ocont,axiom,
! [A_1] : hBOOL(tendsto_complex_real(re,hAPP_complex_real(re,A_1),at_complex(A_1))) ).
fof(fact_5077_Im_Ocont,axiom,
! [A_1] : hBOOL(tendsto_complex_real(im,hAPP_complex_real(im,A_1),at_complex(A_1))) ).
fof(fact_5078_cnj_Ocont,axiom,
! [A_1] : hBOOL(tendst1507391555omplex(cnj,hAPP_complex_complex(cnj,A_1),at_complex(A_1))) ).
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,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)) = hAPP_f22106695ol_nat(hAPP_f782000547ol_nat(big_co387207925at_nat,F),hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),K)) ) ).
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,
! [Na,Ma] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),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,Na)) = big_co1705425894at_nat(cOMBI_nat,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),Ma)) ) ).
fof(fact_5083_fact__altdef__int,axiom,
! [Na] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Na))
=> hAPP_int_int(fact_fact_int,Na) = big_co1548731110nt_int(cOMBI_int,hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),Na)) ) ).
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_1] :
( ( X_1 != 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_1))),R_1)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,X_1)),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_1] :
( ( X_1 != 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_1))),R_1)) )
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),hAPP_real_real(F,X_1))) ) ) ) ) ).
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_1] :
( ( X_1 != 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_1))),R_1)) )
=> hAPP_real_real(F,X_1) != zero_zero_real ) ) ) ) ).
fof(fact_5088_aset_I8_J,axiom,
! [Ta,A,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D))) ) ) ) ) ).
fof(fact_5089_aset_I6_J,axiom,
! [Ta,A,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( 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))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Ta))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D)),Ta)) ) ) ) ) ) ).
fof(fact_5090_bset_I8_J,axiom,
! [Ta,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( 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))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D))) ) ) ) ) ) ).
fof(fact_5091_bset_I6_J,axiom,
! [Ta,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,X_1),Ta))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D)),Ta)) ) ) ) ) ).
fof(fact_5092_bset_I3_J,axiom,
! [Ta,B,D] :
( is_int(Ta)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( 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))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) ) ) ) )
=> ( X_1 = Ta
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D) = Ta ) ) ) ) ) ) ).
fof(fact_5093_aset_I3_J,axiom,
! [Ta,A,D] :
( is_int(Ta)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( 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))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) ) ) ) )
=> ( X_1 = Ta
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D) = Ta ) ) ) ) ) ) ).
fof(fact_5094_aset_I7_J,axiom,
! [Ta,A,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D))) ) ) ) ) ).
fof(fact_5095_bset_I5_J,axiom,
! [Ta,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Ta))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D)),Ta)) ) ) ) ) ).
fof(fact_5096_aset_I5_J,axiom,
! [Ta,A,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Ta))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D)),Ta)) ) ) ) ) ) ).
fof(fact_5097_bset_I7_J,axiom,
! [Ta,B,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),X_1))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Ta),hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D))) ) ) ) ) ) ).
fof(fact_5098_aset_I4_J,axiom,
! [Ta,A,D] :
( is_int(Ta)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),A))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) ) ) ) )
=> ( X_1 != Ta
=> hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D) != Ta ) ) ) ) ) ) ).
fof(fact_5099_bset_I4_J,axiom,
! [Ta,B,D] :
( is_int(Ta)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Ta),B))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) ) ) ) )
=> ( X_1 != Ta
=> hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D) != Ta ) ) ) ) ) ) ).
fof(fact_5100_periodic__finite__ex,axiom,
! [P_1,D_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D_1))
=> ( ! [X_1,K_1] :
( ( is_int(X_1)
& is_int(K_1) )
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
<=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D_1)))) ) )
=> ( ? [X1] :
( is_int(X1)
& hBOOL(hAPP_int_bool(P_1,X1)) )
<=> ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D_1)))
& hBOOL(hAPP_int_bool(P_1,X_1)) ) ) ) ) ).
fof(fact_5101_aset_I10_J,axiom,
! [Ta,A,D_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),D))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) ) ) ) )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta)))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D)),Ta))) ) ) ) ) ).
fof(fact_5102_bset_I10_J,axiom,
! [Ta,B,D_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),D))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) ) ) ) )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta)))
=> ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D)),Ta))) ) ) ) ) ).
fof(fact_5103_bset_I9_J,axiom,
! [Ta,B,D_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),D))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D)),Ta))) ) ) ) ) ).
fof(fact_5104_aset_I9_J,axiom,
! [Ta,A,D_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),D))
=> ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( is_int(Xa_1)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( is_int(Xb)
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) ) ) ) )
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),Ta)))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(dvd_dvd_int,D_1),hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D)),Ta))) ) ) ) ) ).
fof(fact_5105_LIMSEQ__inverse__zero,axiom,
! [X_2] :
( ! [R_1] :
? [N_2] :
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_2),N_1))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,R_1),hAPP_nat_real(X_2,N_1))) )
=> hBOOL(tendsto_nat_real(hAPP_f1888375463t_real(hAPP_f1690286043t_real(cOMBB_real_real_nat,inverse_inverse_real),X_2),zero_zero_real,sequentially)) ) ).
fof(fact_5106_Re_OscaleR,axiom,
! [R_2,X] : hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_2),X)) = hAPP_real_real(scaleR_scaleR_real(R_2),hAPP_complex_real(re,X)) ).
fof(fact_5107_Im_OscaleR,axiom,
! [R_2,X] : hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_2),X)) = hAPP_real_real(scaleR_scaleR_real(R_2),hAPP_complex_real(im,X)) ).
fof(fact_5108_cnj_OscaleR,axiom,
! [R_2,X] : hAPP_complex_complex(cnj,hAPP_complex_complex(scaleR1652505878omplex(R_2),X)) = hAPP_complex_complex(scaleR1652505878omplex(R_2),hAPP_complex_complex(cnj,X)) ).
fof(fact_5109_real__scaleR__def,axiom,
! [A_3,X] : hAPP_real_real(scaleR_scaleR_real(A_3),X) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,A_3),X) ).
fof(fact_5110_complex__scaleR,axiom,
! [R_2,A_3,B_2] : hAPP_complex_complex(scaleR1652505878omplex(R_2),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_2),A_3)),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),B_2)) ).
fof(fact_5111_complex__Im__scaleR,axiom,
! [R_2,X] : hAPP_complex_real(im,hAPP_complex_complex(scaleR1652505878omplex(R_2),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_complex_real(im,X)) ).
fof(fact_5112_complex__Re__scaleR,axiom,
! [R_2,X] : hAPP_complex_real(re,hAPP_complex_complex(scaleR1652505878omplex(R_2),X)) = hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_complex_real(re,X)) ).
fof(fact_5113_complex__sgn__def,axiom,
! [X] : hAPP_complex_complex(sgn_sgn_complex,X) = hAPP_complex_complex(scaleR1652505878omplex(hAPP_real_real(inverse_inverse_real,hAPP_complex_real(norm_norm_complex,X))),X) ).
fof(fact_5114_complex__scaleR__def,axiom,
! [R_2,X] : hAPP_complex_complex(scaleR1652505878omplex(R_2),X) = hAPP_real_complex(hAPP_r265291036omplex(complex,hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_complex_real(re,X))),hAPP_real_real(hAPP_r1250527377l_real(times_times_real,R_2),hAPP_complex_real(im,X))) ).
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,
! [Na,Va] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,Na)),hAPP_int_nat(number_number_of_nat,Va)) = 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,Va)))),zero_zero_nat),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Na),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))))) ).
fof(fact_5119_min__Suc__number__of,axiom,
! [Va,Na] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_nat_nat(suc,Na)) = 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,Va)))),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,Va)))),Na))) ).
fof(fact_5120_min__0R,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).
fof(fact_5121_min__0L,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).
fof(fact_5122_min__Suc__Suc,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N)) ).
fof(fact_5123_min__diff,axiom,
! [M,I,N] : 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),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)),I) ).
fof(fact_5124_min__Suc2,axiom,
! [Ma,Na] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Ma),hAPP_nat_nat(suc,Na)) = 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),Na))),Ma) ).
fof(fact_5125_min__Suc1,axiom,
! [Na,Ma] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,Na)),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,Na))),Ma) ).
fof(fact_5126_vanishes__mult__bounded,axiom,
! [Y_2,X_2] :
( ? [A_2] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),A_2))
& ! [N_1] : hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_2,N_1))),A_2)) )
=> ( hBOOL(vanishes(Y_2))
=> 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_2)),Y_2))) ) ) ).
fof(fact_5127_SetInterval_Otransfer__int__nat__set__functions,axiom,
! [Na,Ma] :
( hBOOL(hAPP_int_bool(nat_is_nat,Ma))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Na))
=> hAPP_i1948725293t_bool(ord_at875362053st_int(Ma),Na) = image_nat_int(semiri1621563631at_int,hAPP_n1699378549t_bool(ord_at238088361st_nat(hAPP_int_nat(nat,Ma)),hAPP_int_nat(nat,Na))) ) ) ).
fof(fact_5128_is__nat__def,axiom,
! [Xa] :
( hBOOL(hAPP_int_bool(nat_is_nat,Xa))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),Xa)) ) ).
fof(fact_5129_Nat__Transfer_Otransfer__int__nat__function__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(nat_is_nat,X))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X),Y))) ) ) ).
fof(fact_5130_Nat__Transfer_Otransfer__int__nat__function__closures_I6_J,axiom,
hBOOL(hAPP_int_bool(nat_is_nat,one_one_int)) ).
fof(fact_5131_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J,axiom,
! [Xa,A] :
( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A))
=> hBOOL(hAPP_int_bool(nat_is_nat,Xa)) ) ) ).
fof(fact_5132_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J,axiom,
! [N,X] :
( hBOOL(hAPP_int_bool(nat_is_nat,X))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,X),N))) ) ).
fof(fact_5133_Divides_Otransfer__int__nat__function__closures_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(nat_is_nat,X))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(div_mod_int(X),Y))) ) ) ).
fof(fact_5134_Divides_Otransfer__int__nat__function__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(nat_is_nat,X))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(div_div_int,X),Y))) ) ) ).
fof(fact_5135_Nat__Transfer_Otransfer__int__nat__function__closures_I3_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(nat_is_nat,X))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(nat_tsub(X),Y))) ) ) ).
fof(fact_5136_Nat__Transfer_Otransfer__int__nat__function__closures_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP_int_bool(nat_is_nat,X))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
=> hBOOL(hAPP_int_bool(nat_is_nat,hAPP_int_int(hAPP_int_fun_int_int(times_times_int,X),Y))) ) ) ).
fof(fact_5137_Nat__Transfer_Otransfer__int__nat__function__closures_I5_J,axiom,
hBOOL(hAPP_int_bool(nat_is_nat,zero_zero_int)) ).
fof(fact_5138_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_5139_vanishes__minus,axiom,
! [X_2] :
( hBOOL(vanishes(X_2))
=> hBOOL(vanishes(hAPP_f2062659861at_rat(hAPP_f1082758869at_rat(cOMBB_rat_rat_nat,uminus_uminus_rat),X_2))) ) ).
fof(fact_5140_vanishes__const,axiom,
! [C] :
( hBOOL(vanishes(cOMBK_rat_nat(C)))
<=> C = zero_zero_rat ) ).
fof(fact_5141_vanishes__diff,axiom,
! [Y_2,X_2] :
( hBOOL(vanishes(X_2))
=> ( hBOOL(vanishes(Y_2))
=> 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_2)),Y_2))) ) ) ).
fof(fact_5142_vanishes__add,axiom,
! [Y_2,X_2] :
( hBOOL(vanishes(X_2))
=> ( hBOOL(vanishes(Y_2))
=> 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_2)),Y_2))) ) ) ).
fof(fact_5143_SetInterval_Otransfer__int__nat__set__function__closures,axiom,
! [Ya,Xa] :
( hBOOL(hAPP_int_bool(nat_is_nat,Xa))
=> hBOOL(hAPP_f448129468l_bool(nat_nat_set,hAPP_i1948725293t_bool(ord_at875362053st_int(Xa),Ya))) ) ).
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,X] :
( hBOOL(hAPP_int_bool(nat_is_nat,X))
=> ( hBOOL(hAPP_int_bool(nat_is_nat,Y))
=> 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),Y))) ) ) ).
fof(fact_5146_transfer__int__nat__factorial__closure,axiom,
! [X] :
( hBOOL(hAPP_int_bool(nat_is_nat,X))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_eq_int,zero_zero_int),hAPP_int_int(fact_fact_int,X))) ) ).
fof(fact_5147_transfer__int__nat__set__relations_I2_J,axiom,
! [A,Xa] :
( hBOOL(hAPP_int_bool(nat_is_nat,Xa))
=> ( hBOOL(hAPP_f448129468l_bool(nat_nat_set,A))
=> ( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa),A))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_int_nat(nat,Xa)),image_int_nat(nat,A))) ) ) ) ).
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_vanishesD,axiom,
! [Ra,X_2] :
( hBOOL(vanishes(X_2))
=> ( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),Ra))
=> ? [K_1] :
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_2,N_1))),Ra)) ) ) ) ).
fof(fact_5150_vanishes__def,axiom,
! [X_2] :
( hBOOL(vanishes(X_2))
<=> ! [R_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1))
=> ? [K_1] :
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_2,N_1))),R_1)) ) ) ) ).
fof(fact_5151_vanishesI,axiom,
! [X_2] :
( ! [R_1] :
( hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,zero_zero_rat),R_1))
=> ? [K_1] :
! [N_1] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1))
=> hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(ord_less_rat,hAPP_rat_rat(abs_abs_rat,hAPP_nat_rat(X_2,N_1))),R_1)) ) )
=> hBOOL(vanishes(X_2)) ) ).
fof(fact_5152_zcongm__def,axiom,
! [Ma,X_1,Xa_1] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcongm(Ma),X_1),Xa_1))
<=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(X_1),Xa_1),Ma)) ) ).
fof(fact_5153_increasing__LIMSEQ,axiom,
! [L,F] :
( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_1)),hAPP_nat_real(F,hAPP_nat_nat(suc,N_1))))
=> ( ! [N_1] : hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,hAPP_nat_real(F,N_1)),L))
=> ( ! [E] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),E))
=> ? [N_1] : 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_1)),E))) )
=> hBOOL(tendsto_nat_real(F,L,sequentially)) ) ) ) ).
fof(fact_5154_bijzcong__zcong__prod,axiom,
! [A,B,Ma] :
( hBOOL(hAPP_f1469637905l_bool(hAPP_P1681069257l_bool(member308914164t_bool,hAPP_f538843494t_bool(hAPP_f1418982953t_bool(produc713279741t_bool,A),B)),bijR_int_int(zcongm(Ma))))
=> hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(zcong(big_co1548731110nt_int(cOMBI_int,A)),big_co1548731110nt_int(cOMBI_int,B)),Ma)) ) ).
fof(fact_5155_isCont__real__root,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,Na)),Xa)) ) ).
fof(fact_5156_isCont__ln,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> hBOOL(hAPP_real_bool(isCont_real_real(ln),Xa)) ) ).
fof(fact_5157_isCont__tan,axiom,
! [Xa] :
( hAPP_real_real(cos,Xa) != zero_zero_real
=> hBOOL(hAPP_real_bool(isCont_real_real(tan),Xa)) ) ).
fof(fact_5158_isCont__abs,axiom,
! [A_1] : hBOOL(hAPP_real_bool(isCont_real_real(abs_abs_real),A_1)) ).
fof(fact_5159_isCont__sin,axiom,
! [Xa] : hBOOL(hAPP_real_bool(isCont_real_real(sin),Xa)) ).
fof(fact_5160_isCont__cos,axiom,
! [Xa] : hBOOL(hAPP_real_bool(isCont_real_real(cos),Xa)) ).
fof(fact_5161_isCont__real__sqrt,axiom,
! [Xa] : hBOOL(hAPP_real_bool(isCont_real_real(sqrt),Xa)) ).
fof(fact_5162_isCont__arctan,axiom,
! [Xa] : hBOOL(hAPP_real_bool(isCont_real_real(arctan),Xa)) ).
fof(fact_5163_cnj_OisCont,axiom,
! [A_1] : hBOOL(hAPP_complex_bool(isCont156215680omplex(cnj),A_1)) ).
fof(fact_5164_Im_OisCont,axiom,
! [A_1] : hBOOL(hAPP_complex_bool(isCont_complex_real(im),A_1)) ).
fof(fact_5165_Re_OisCont,axiom,
! [A_1] : hBOOL(hAPP_complex_bool(isCont_complex_real(re),A_1)) ).
fof(fact_5166_isCont__root__zero,axiom,
! [Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,Na)),zero_zero_real)) ) ).
fof(fact_5167_isCont__root__pos,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),Xa))
=> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,Na)),Xa)) ) ) ).
fof(fact_5168_isCont__root__neg,axiom,
! [Xa,Na] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),Na))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),zero_zero_real))
=> hBOOL(hAPP_real_bool(isCont_real_real(hAPP_n546711566l_real(root,Na)),Xa)) ) ) ).
fof(fact_5169_isCont__arccos,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real))
=> hBOOL(hAPP_real_bool(isCont_real_real(arccos),Xa)) ) ) ).
fof(fact_5170_isCont__arcsin,axiom,
! [Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_int_real(number267125858f_real,min)),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),one_one_real))
=> hBOOL(hAPP_real_bool(isCont_real_real(arcsin),Xa)) ) ) ).
fof(fact_5171_LIM__less__bound,axiom,
! [F,B_1,Xa] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,B_1),Xa))
=> ( ! [X_1] :
( hBOOL(hAPP_f1501066425l_bool(hAPP_r1130899993l_bool(member_real,X_1),hAPP_r1134773055l_bool(ord_gr788844697n_real(B_1),Xa)))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,X_1))) )
=> ( hBOOL(hAPP_real_bool(isCont_real_real(F),Xa))
=> hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,zero_zero_real),hAPP_real_real(F,Xa))) ) ) ) ).
fof(fact_5172_DERIV__inverse__function,axiom,
! [B_1,A_1,F,G,Xa,D] :
( hBOOL(hAPP_real_bool(deriv_real(F,hAPP_real_real(G,Xa)),D))
=> ( D != zero_zero_real
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Xa),B_1))
=> ( ! [Y_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),Y_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,Y_1),B_1)) )
=> hAPP_real_real(F,hAPP_real_real(G,Y_1)) = Y_1 )
=> ( hBOOL(hAPP_real_bool(isCont_real_real(G),Xa))
=> hBOOL(hAPP_real_bool(deriv_real(G,Xa),hAPP_real_real(inverse_inverse_real,D))) ) ) ) ) ) ) ).
fof(fact_5173_isCont__inv__fun,axiom,
! [G,F,Xa,D_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1))
=> ( ! [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),Xa))),D_1))
=> 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),Xa))),D_1))
=> hBOOL(hAPP_real_bool(isCont_real_real(F),Z)) )
=> hBOOL(hAPP_real_bool(isCont_real_real(G),hAPP_real_real(F,Xa))) ) ) ) ).
fof(fact_5174_DERIV__isconst1,axiom,
! [F,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1))
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) )
=> hBOOL(hAPP_real_bool(isCont_real_real(F),X_1)) )
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B_1)) )
=> hBOOL(hAPP_real_bool(deriv_real(F,X_1),zero_zero_real)) )
=> ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) )
=> hAPP_real_real(F,X_1) = hAPP_real_real(F,A_1) ) ) ) ) ).
fof(fact_5175_DERIV__isconst2,axiom,
! [Xa,F,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1))
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) )
=> hBOOL(hAPP_real_bool(isCont_real_real(F),X_1)) )
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B_1)) )
=> hBOOL(hAPP_real_bool(deriv_real(F,X_1),zero_zero_real)) )
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),Xa))
=> ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,Xa),B_1))
=> hAPP_real_real(F,Xa) = hAPP_real_real(F,A_1) ) ) ) ) ) ).
fof(fact_5176_DERIV__isconst__end,axiom,
! [F,A_1,B_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),B_1))
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_eq_real,X_1),B_1)) )
=> hBOOL(hAPP_real_bool(isCont_real_real(F),X_1)) )
=> ( ! [X_1] :
( ( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,A_1),X_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,X_1),B_1)) )
=> hBOOL(hAPP_real_bool(deriv_real(F,X_1),zero_zero_real)) )
=> hAPP_real_real(F,B_1) = hAPP_real_real(F,A_1) ) ) ) ).
fof(fact_5177_isCont__inv__fun__inv,axiom,
! [G,F,Xa,D_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1))
=> ( ! [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),Xa))),D_1))
=> 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),Xa))),D_1))
=> 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_1] :
( ( 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_1),hAPP_real_real(F,Xa)))))
& 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_1),hAPP_real_real(F,Xa)))),E)) )
=> hAPP_real_real(F,hAPP_real_real(G,Y_1)) = Y_1 ) ) ) ) ) ).
fof(fact_5178_lemma__isCont__inj,axiom,
! [G,F,Xa,D_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1))
=> ( ! [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),Xa))),D_1))
=> 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),Xa))),D_1))
=> 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),Xa))),D_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Xa)),hAPP_real_real(F,Z))) ) ) ) ) ).
fof(fact_5179_lemma__isCont__inj2,axiom,
! [G,F,Xa,D_1] :
( hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,zero_zero_real),D_1))
=> ( ! [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),Xa))),D_1))
=> 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),Xa))),D_1))
=> 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),Xa))),D_1))
& hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(ord_less_real,hAPP_real_real(F,Z)),hAPP_real_real(F,Xa))) ) ) ) ) ).
fof(fact_5180_cpmi,axiom,
! [B,P_2,P_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ? [Z] :
! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,X_1),Z))
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
<=> hBOOL(hAPP_int_bool(P_2,X_1)) ) ) )
=> ( ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),B))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xb),Xa_1) ) )
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),D))) ) ) )
=> ( ! [X_1,K_1] :
( ( is_int(X_1)
& is_int(K_1) )
=> ( hBOOL(hAPP_int_bool(P_2,X_1))
<=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))) ) )
=> ( ? [X1] :
( is_int(X1)
& hBOOL(hAPP_int_bool(P_1,X1)) )
<=> ( ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
& hBOOL(hAPP_int_bool(P_2,X_1)) )
| ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
& ? [Xa_1] :
( is_int(Xa_1)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),B))
& hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,Xa_1),X_1))) ) ) ) ) ) ) ) ) ).
fof(fact_5181_cppi,axiom,
! [A,P_2,P_1,D] :
( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,zero_zero_int),D))
=> ( ? [Z] :
! [X_1] :
( is_int(X_1)
=> ( hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(ord_less_int,Z),X_1))
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
<=> hBOOL(hAPP_int_bool(P_2,X_1)) ) ) )
=> ( ! [X_1] :
( is_int(X_1)
=> ( ! [Xa_1] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
=> ! [Xb] :
( hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xb),A))
=> X_1 != hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xb),Xa_1) ) )
=> ( hBOOL(hAPP_int_bool(P_1,X_1))
=> hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,X_1),D))) ) ) )
=> ( ! [X_1,K_1] :
( ( is_int(X_1)
& is_int(K_1) )
=> ( hBOOL(hAPP_int_bool(P_2,X_1))
<=> hBOOL(hAPP_int_bool(P_2,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,X_1),hAPP_int_int(hAPP_int_fun_int_int(times_times_int,K_1),D)))) ) )
=> ( ? [X1] :
( is_int(X1)
& hBOOL(hAPP_int_bool(P_1,X1)) )
<=> ( ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
& hBOOL(hAPP_int_bool(P_2,X_1)) )
| ? [X_1] :
( is_int(X_1)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,X_1),hAPP_i1948725293t_bool(ord_at875362053st_int(one_one_int),D)))
& ? [Xa_1] :
( is_int(Xa_1)
& hBOOL(hAPP_f448129468l_bool(hAPP_i2112223885l_bool(member_int,Xa_1),A))
& hBOOL(hAPP_int_bool(P_1,hAPP_int_int(hAPP_int_fun_int_int(minus_minus_int,Xa_1),X_1))) ) ) ) ) ) ) ) ) ).
fof(fact_5182_SetInterval_Otransfer__nat__int__set__functions_I1_J,axiom,
! [Na] : hAPP_n1699378549t_bool(ord_atMost_nat,Na) = image_int_nat(nat,hAPP_i1948725293t_bool(ord_at875362053st_int(zero_zero_int),hAPP_nat_int(semiri1621563631at_int,Na))) ).
fof(fact_5183_finite__atMost,axiom,
! [K] : hBOOL(hAPP_f54304608l_bool(finite_finite_nat,hAPP_n1699378549t_bool(ord_atMost_nat,K))) ).
fof(fact_5184_finite__nat__iff__bounded__le,axiom,
! [S] :
( hBOOL(hAPP_f54304608l_bool(finite_finite_nat,S))
<=> ? [K_1] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,S),hAPP_n1699378549t_bool(ord_atMost_nat,K_1))) ) ).
fof(fact_5185_lessThan__Suc__atMost,axiom,
! [K] : hAPP_n1699378549t_bool(ord_lessThan_nat,hAPP_nat_nat(suc,K)) = hAPP_n1699378549t_bool(ord_atMost_nat,K) ).
fof(fact_5186_card__atMost,axiom,
! [U] : hAPP_f22106695ol_nat(finite_card_nat,hAPP_n1699378549t_bool(ord_atMost_nat,U)) = hAPP_nat_nat(suc,U) ).
fof(fact_5187_atLeast0AtMost,axiom,
! [Na] : hAPP_n1699378549t_bool(ord_at238088361st_nat(zero_zero_nat),Na) = hAPP_n1699378549t_bool(ord_atMost_nat,Na) ).
fof(fact_5188_max__number__of__Suc,axiom,
! [Na,Va] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,Na)),hAPP_int_nat(number_number_of_nat,Va)) = 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,Va)))),hAPP_nat_nat(suc,Na)),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Na),hAPP_int_nat(nat,hAPP_int_int(number_number_of_int,hAPP_int_int(pred,Va)))))) ).
fof(fact_5189_max__Suc__number__of,axiom,
! [Va,Na] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_int_nat(number_number_of_nat,Va)),hAPP_nat_nat(suc,Na)) = 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,Va)))),hAPP_nat_nat(suc,Na)),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,Va)))),Na))) ).
fof(fact_5190_Sup__atMost,axiom,
! [Xa] : comple124823625p_real(hAPP_r1134773055l_bool(ord_atMost_real,Xa)) = Xa ).
fof(fact_5191_max__Suc2,axiom,
! [Ma,Na] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Ma),hAPP_nat_nat(suc,Na)) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,Na),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),Na))),Ma) ).
fof(fact_5192_max__Suc1,axiom,
! [Na,Ma] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,Na)),Ma) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,Na),hAPP_f1914919701at_nat(hAPP_f1585078997at_nat(cOMBB_nat_nat_nat,suc),hAPP_nat_fun_nat_nat(ord_max_nat,Na))),Ma) ).
fof(fact_5193_max__0R,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).
fof(fact_5194_max__0L,axiom,
! [N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).
fof(fact_5195_max__Suc__Suc,axiom,
! [M,N] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M),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,Y] :
( is_int(X)
=> hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fTrue),X),Y) = X ) ).
fof(help_If_2_1_If_000tc__Int__Oint_T,axiom,
! [X,Y] :
( is_int(Y)
=> hAPP_int_int(hAPP_int_fun_int_int(hAPP_b1463609396nt_int(if_int,fFalse),X),Y) = Y ) ).
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,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fTrue),X),Y) = X ).
fof(help_If_2_1_If_000tc__Nat__Onat_T,axiom,
! [X,Y] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(hAPP_b992065680at_nat(if_nat,fFalse),X),Y) = Y ).
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] :
( ~ hBOOL(hAPP_int_bool(P,X))
| hBOOL(hAPP_f448129468l_bool(fEx_int,P)) ) ).
fof(help_fEx_1_1_fEx_000tc__Nat__Onat_U,axiom,
! [P,X] :
( ~ hBOOL(hAPP_nat_bool(P,X))
| 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,Y] :
( ( is_int(X)
& is_int(Y) )
=> ( ~ hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y))
| X = Y ) ) ).
fof(help_fequal_2_1_fequal_000tc__Int__Oint_T,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_int_bool(hAPP_i1948725293t_bool(fequal_int,X),Y)) ) ).
fof(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
| X = Y ) ).
fof(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).
fof(help_fequal_1_1_fequal_000tc__Rat__Orat_T,axiom,
! [X,Y] :
( ~ hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y))
| X = Y ) ).
fof(help_fequal_2_1_fequal_000tc__Rat__Orat_T,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_rat_bool(hAPP_r2115560837t_bool(fequal_rat,X),Y)) ) ).
fof(help_If_1_1_If_000tc__RealDef__Oreal_T,axiom,
! [X,Y] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fTrue),X),Y) = X ).
fof(help_If_2_1_If_000tc__RealDef__Oreal_T,axiom,
! [X,Y] : hAPP_real_real(hAPP_r1250527377l_real(hAPP_b646336293l_real(if_real,fFalse),X),Y) = Y ).
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,Y] :
( ~ hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y))
| X = Y ) ).
fof(help_fequal_2_1_fequal_000tc__RealDef__Oreal_T,axiom,
! [X,Y] :
( X != Y
| hBOOL(hAPP_real_bool(hAPP_r1134773055l_bool(fequal_real,X),Y)) ) ).
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,Y] : hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fTrue),X),Y) = X ).
fof(help_If_2_1_If_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_T,axiom,
! [X,Y] : hAPP_P1975530577nt_int(hAPP_P1145851913nt_int(hAPP_b40753821nt_int(if_Pro1731782967nt_int,fFalse),X),Y) = Y ).
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,Y] : hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fTrue),X),Y) = X ).
fof(help_If_2_1_If_000tc__prod_Itc__RealDef__Oreal_Mtc__RealDef__Oreal_J_T,axiom,
! [X,Y] : hAPP_P1860904029l_real(hAPP_P1982799375l_real(hAPP_b1482221219l_real(if_Pro313124157l_real,fFalse),X),Y) = Y ).
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,
hAPP_nat_int(hAPP_int_fun_nat_int(power_power_int,hAPP_int_int(hAPP_int_fun_int_int(plus_plus_int,one_one_int),hAPP_nat_int(semiri1621563631at_int,n))),hAPP_int_nat(number_number_of_nat,hAPP_int_int(bit0,hAPP_int_int(bit1,pls)))) != zero_zero_int ).
%------------------------------------------------------------------------------