TPTP Problem File: NUM923+8.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM923+8 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 23, 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_fofpt_l23 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 5741 (1867 unt; 0 def)
% Number of atoms : 14404 (4117 equ)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 9672 (1009 ~; 188 |;1123 &)
% ( 909 <=>;6443 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 5 avg)
% Maximal term depth : 25 ( 2 avg)
% Number of predicates : 102 ( 101 usr; 0 prp; 1-2 aty)
% Number of functors : 259 ( 259 usr; 140 con; 0-4 aty)
% Number of variables : 13639 (13295 !; 344 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:49:10
% : Encoded with polymorphic tags.
%------------------------------------------------------------------------------
%----Explicit typings (275)
fof(tsy_c_Archimedean__Field_Oceiling_res,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ti(fun(X_a,int),archimedean_ceiling(X_a)) = archimedean_ceiling(X_a) ) ).
fof(tsy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_res,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ti(fun(X_a,int),archim2085082626_floor(X_a)) = archim2085082626_floor(X_a) ) ).
fof(tsy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum_res,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ti(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),big_co1399186613setsum(X_a,X_b)) = big_co1399186613setsum(X_a,X_b) ) ).
fof(tsy_c_Big__Operators_Ocomm__monoid__mult__class_Osetprod_res,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ti(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),big_co302546470etprod(X_a,X_b)) = big_co302546470etprod(X_a,X_b) ) ).
fof(tsy_c_BijectionRel_ObijR_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,fun(X_b,bool)),fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool)),bijR(X_a,X_b)) = bijR(X_a,X_b) ).
fof(tsy_c_COMBB_res,axiom,
! [X_b,X_c,X_a] : ti(fun(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c))),combb(X_b,X_c,X_a)) = combb(X_b,X_c,X_a) ).
fof(tsy_c_COMBC_res,axiom,
! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c))),combc(X_a,X_b,X_c)) = combc(X_a,X_b,X_c) ).
fof(tsy_c_COMBI_res,axiom,
! [X_a] : ti(fun(X_a,X_a),combi(X_a)) = combi(X_a) ).
fof(tsy_c_COMBK_res,axiom,
! [X_a,X_b] : ti(fun(X_a,fun(X_b,X_a)),combk(X_a,X_b)) = combk(X_a,X_b) ).
fof(tsy_c_COMBS_res,axiom,
! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c))),combs(X_a,X_b,X_c)) = combs(X_a,X_b,X_c) ).
fof(tsy_c_Code__Numeral_OSuc__code__numeral_res,axiom,
ti(fun(code_code_numeral,code_code_numeral),code_S1047413653umeral) = code_S1047413653umeral ).
fof(tsy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__case_res,axiom,
! [X_a] : ti(fun(X_a,fun(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a))),code_c600343653l_case(X_a)) = code_c600343653l_case(X_a) ).
fof(tsy_c_Code__Numeral_Ocode__numeral_Ocode__numeral__size_res,axiom,
ti(fun(code_code_numeral,nat),code_c271388182l_size) = code_c271388182l_size ).
fof(tsy_c_Code__Numeral_Odiv__mod__code__numeral_res,axiom,
ti(fun(code_code_numeral,fun(code_code_numeral,product_prod(code_code_numeral,code_code_numeral))),code_d418564891umeral) = code_d418564891umeral ).
fof(tsy_c_Code__Numeral_Oint__of_res,axiom,
ti(fun(code_code_numeral,int),code_int_of) = code_int_of ).
fof(tsy_c_Code__Numeral_Onat__of__aux_res,axiom,
ti(fun(code_code_numeral,fun(nat,nat)),code_nat_of_aux) = code_nat_of_aux ).
fof(tsy_c_Complete__Lattice_OSup__class_OSup_0_res,axiom,
ti(fun(fun(real,bool),real),complete_Sup_Sup(real)) = complete_Sup_Sup(real) ).
fof(tsy_c_Complete__Lattice_OSup__class_OSup_1_res,axiom,
! [X_a] :
( comple44059174attice(X_a)
=> ti(fun(fun(X_a,bool),X_a),complete_Sup_Sup(X_a)) = complete_Sup_Sup(X_a) ) ).
fof(tsy_c_Complex_OIm_res,axiom,
ti(fun(complex,real),im) = im ).
fof(tsy_c_Complex_ORe_res,axiom,
ti(fun(complex,real),re) = re ).
fof(tsy_c_Complex_Oarg_res,axiom,
ti(fun(complex,real),arg) = arg ).
fof(tsy_c_Complex_Ocis_res,axiom,
ti(fun(real,complex),cis) = cis ).
fof(tsy_c_Complex_Ocnj_res,axiom,
ti(fun(complex,complex),cnj) = cnj ).
fof(tsy_c_Complex_Ocomplex_OComplex_res,axiom,
ti(fun(real,fun(real,complex)),complex_1) = complex_1 ).
fof(tsy_c_Complex_Ocomplex_Ocomplex__case_res,axiom,
! [X_a] : ti(fun(fun(real,fun(real,X_a)),fun(complex,X_a)),complex_case(X_a)) = complex_case(X_a) ).
fof(tsy_c_Complex_Ocomplex_Ocomplex__rec_res,axiom,
! [X_a] : ti(fun(fun(real,fun(real,X_a)),fun(complex,X_a)),complex_rec(X_a)) = complex_rec(X_a) ).
fof(tsy_c_Complex_Ocomplex_Ocomplex__size_res,axiom,
ti(fun(complex,nat),complex_size) = complex_size ).
fof(tsy_c_Complex_Oexpi_res,axiom,
ti(fun(complex,complex),expi) = expi ).
fof(tsy_c_Complex_Oii_res,axiom,
ti(complex,ii) = ii ).
fof(tsy_c_Complex_Orcis_res,axiom,
ti(fun(real,fun(real,complex)),rcis) = rcis ).
fof(tsy_c_Deriv_OBolzano__bisect_res,axiom,
ti(fun(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real))))),bolzano_bisect) = bolzano_bisect ).
fof(tsy_c_Deriv_Oderiv_res,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ti(fun(fun(X_a,X_a),fun(X_a,fun(X_a,bool))),deriv(X_a)) = deriv(X_a) ) ).
fof(tsy_c_Divides_Oadjust_res,axiom,
ti(fun(int,fun(product_prod(int,int),product_prod(int,int))),adjust) = adjust ).
fof(tsy_c_Divides_Odiv__class_Odiv_res,axiom,
! [X_a] :
( semiring_div(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),div_div(X_a)) = div_div(X_a) ) ).
fof(tsy_c_Divides_Odiv__class_Omod_res,axiom,
! [X_a] :
( semiring_div(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),div_mod(X_a)) = div_mod(X_a) ) ).
fof(tsy_c_Divides_Odivmod__int_res,axiom,
ti(fun(int,fun(int,product_prod(int,int))),divmod_int) = divmod_int ).
fof(tsy_c_Divides_Odivmod__int__rel_res,axiom,
ti(fun(int,fun(int,fun(product_prod(int,int),bool))),divmod_int_rel) = divmod_int_rel ).
fof(tsy_c_Divides_Odivmod__nat_res,axiom,
ti(fun(nat,fun(nat,product_prod(nat,nat))),divmod_nat) = divmod_nat ).
fof(tsy_c_Divides_Odivmod__nat__rel_res,axiom,
ti(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),divmod_nat_rel) = divmod_nat_rel ).
fof(tsy_c_Divides_OnegDivAlg_res,axiom,
ti(fun(int,fun(int,product_prod(int,int))),negDivAlg) = negDivAlg ).
fof(tsy_c_Divides_OnegDivAlg__rel_res,axiom,
ti(fun(product_prod(int,int),fun(product_prod(int,int),bool)),negDivAlg_rel) = negDivAlg_rel ).
fof(tsy_c_Divides_OnegateSnd_res,axiom,
ti(fun(product_prod(int,int),product_prod(int,int)),negateSnd) = negateSnd ).
fof(tsy_c_Divides_Opdivmod_res,axiom,
ti(fun(int,fun(int,product_prod(int,int))),pdivmod) = pdivmod ).
fof(tsy_c_Divides_OposDivAlg_res,axiom,
ti(fun(int,fun(int,product_prod(int,int))),posDivAlg) = posDivAlg ).
fof(tsy_c_Divides_OposDivAlg__rel_res,axiom,
ti(fun(product_prod(int,int),fun(product_prod(int,int),bool)),posDivAlg_rel) = posDivAlg_rel ).
fof(tsy_c_EulerFermat_OBnorRset_res,axiom,
ti(fun(int,fun(int,fun(int,bool))),bnorRset) = bnorRset ).
fof(tsy_c_EulerFermat_ORRset2norRR_res,axiom,
ti(fun(fun(int,bool),fun(int,fun(int,int))),rRset2norRR) = rRset2norRR ).
fof(tsy_c_EulerFermat_ORsetR_res,axiom,
ti(fun(int,fun(fun(int,bool),bool)),rsetR) = rsetR ).
fof(tsy_c_EulerFermat_Ois__RRset_res,axiom,
ti(fun(fun(int,bool),fun(int,bool)),is_RRset) = is_RRset ).
fof(tsy_c_EulerFermat_OnoXRRset_res,axiom,
ti(fun(int,fun(int,fun(int,bool))),noXRRset) = noXRRset ).
fof(tsy_c_EulerFermat_OnorRRset_res,axiom,
ti(fun(int,fun(int,bool)),norRRset) = norRRset ).
fof(tsy_c_EulerFermat_Ophi_res,axiom,
ti(fun(int,nat),phi) = phi ).
fof(tsy_c_EulerFermat_Ozcongm_res,axiom,
ti(fun(int,fun(int,fun(int,bool))),zcongm) = zcongm ).
fof(tsy_c_Euler_OMultInvPair_res,axiom,
ti(fun(int,fun(int,fun(int,fun(int,bool)))),multInvPair) = multInvPair ).
fof(tsy_c_Euler_OSetS_res,axiom,
ti(fun(int,fun(int,fun(fun(int,bool),bool))),setS) = setS ).
fof(tsy_c_EvenOdd_OzEven_res,axiom,
ti(fun(int,bool),zEven) = zEven ).
fof(tsy_c_EvenOdd_OzOdd_res,axiom,
ti(fun(int,bool),zOdd) = zOdd ).
fof(tsy_c_Fact_Ofact__class_Ofact_0_res,axiom,
ti(fun(int,int),fact_fact(int)) = fact_fact(int) ).
fof(tsy_c_Fact_Ofact__class_Ofact_1_res,axiom,
ti(fun(nat,nat),fact_fact(nat)) = fact_fact(nat) ).
fof(tsy_c_Fields_Oinverse__class_Odivide_0_res,axiom,
! [X_a] :
( ( inverse(X_a)
& real_normed_vector(X_a) )
=> ti(fun(X_a,fun(X_a,X_a)),inverse_divide(X_a)) = inverse_divide(X_a) ) ).
fof(tsy_c_Fields_Oinverse__class_Odivide_1_res,axiom,
! [X_a] :
( division_ring(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),inverse_divide(X_a)) = inverse_divide(X_a) ) ).
fof(tsy_c_Fields_Oinverse__class_Oinverse_res,axiom,
! [X_a] :
( division_ring(X_a)
=> ti(fun(X_a,X_a),inverse_inverse(X_a)) = inverse_inverse(X_a) ) ).
fof(tsy_c_Finite__Set_Ocard_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),nat),finite_card(X_a)) = finite_card(X_a) ).
fof(tsy_c_Finite__Set_Ofinite_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),bool),finite_finite_1(X_a)) = finite_finite_1(X_a) ).
fof(tsy_c_FunDef_Oin__rel_res,axiom,
! [X_a,X_b] : ti(fun(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool))),in_rel(X_a,X_b)) = in_rel(X_a,X_b) ).
fof(tsy_c_FunDef_Opair__leq_res,axiom,
ti(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),pair_leq) = pair_leq ).
fof(tsy_c_FunDef_Opair__less_res,axiom,
ti(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),pair_less) = pair_less ).
fof(tsy_c_GCD_Obezw_res,axiom,
ti(fun(nat,fun(nat,product_prod(int,int))),bezw) = bezw ).
fof(tsy_c_GCD_Ogcd__class_Ogcd_0_res,axiom,
ti(fun(int,fun(int,int)),gcd_gcd(int)) = gcd_gcd(int) ).
fof(tsy_c_GCD_Ogcd__class_Ogcd_1_res,axiom,
ti(fun(nat,fun(nat,nat)),gcd_gcd(nat)) = gcd_gcd(nat) ).
fof(tsy_c_Groups_Oabs__class_Oabs_0_res,axiom,
! [X_a] :
( abs_if(X_a)
=> ti(fun(X_a,X_a),abs_abs(X_a)) = abs_abs(X_a) ) ).
fof(tsy_c_Groups_Oabs__class_Oabs_1_res,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ti(fun(X_a,X_a),abs_abs(X_a)) = abs_abs(X_a) ) ).
fof(tsy_c_Groups_Ominus__class_Ominus_res,axiom,
! [X_a] :
( minus(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),minus_minus(X_a)) = minus_minus(X_a) ) ).
fof(tsy_c_Groups_Oone__class_Oone_res,axiom,
! [X_a] :
( one(X_a)
=> ti(X_a,one_one(X_a)) = one_one(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_0_res,axiom,
! [X_a] :
( ( plus(X_a)
& linorder(X_a)
& dvd(X_a) )
=> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_1_res,axiom,
! [X_a] :
( monoid_add(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_2_res,axiom,
! [X_a] :
( ab_semigroup_add(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).
fof(tsy_c_Groups_Oplus__class_Oplus_3_res,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),plus_plus(X_a)) = plus_plus(X_a) ) ).
fof(tsy_c_Groups_Osgn__class_Osgn_0_res,axiom,
! [X_a] :
( sgn_div_norm(X_a)
=> ti(fun(X_a,X_a),sgn_sgn(X_a)) = sgn_sgn(X_a) ) ).
fof(tsy_c_Groups_Osgn__class_Osgn_1_res,axiom,
! [X_a] :
( sgn_if(X_a)
=> ti(fun(X_a,X_a),sgn_sgn(X_a)) = sgn_sgn(X_a) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_0_res,axiom,
! [X_a] :
( dvd(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_1_res,axiom,
! [X_a] :
( power(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_2_res,axiom,
! [X_a] :
( mult_zero(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_3_res,axiom,
! [X_a] :
( no_zero_divisors(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_4_res,axiom,
! [X_a] :
( semiring(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_Groups_Otimes__class_Otimes_5_res,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),times_times(X_a)) = times_times(X_a) ) ).
fof(tsy_c_Groups_Ouminus__class_Ouminus_res,axiom,
! [X_a] :
( uminus(X_a)
=> ti(fun(X_a,X_a),uminus_uminus(X_a)) = uminus_uminus(X_a) ) ).
fof(tsy_c_Groups_Ozero__class_Ozero_res,axiom,
! [X_a] :
( zero(X_a)
=> ti(X_a,zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(tsy_c_HOL_OThe_0_res,axiom,
! [X_a] :
( metric_space(X_a)
=> ti(fun(fun(X_a,bool),X_a),the(X_a)) = the(X_a) ) ).
fof(tsy_c_HOL_OThe_1_res,axiom,
ti(fun(fun(int,bool),int),the(int)) = the(int) ).
fof(tsy_c_HOL_OThe_2_res,axiom,
! [X_a,X_b] : ti(fun(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b)),the(product_prod(X_a,X_b))) = the(product_prod(X_a,X_b)) ).
fof(tsy_c_HOL_OThe_3_res,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> ti(fun(fun(X_a,bool),X_a),the(X_a)) = the(X_a) ) ).
fof(tsy_c_HOL_Oundefined_res,axiom,
! [X_a] : ti(X_a,undefined(X_a)) = undefined(X_a) ).
fof(tsy_c_Hilbert__Choice_OEps_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),X_a),hilbert_Eps(X_a)) = hilbert_Eps(X_a) ).
fof(tsy_c_If_res,axiom,
! [X_a] : ti(fun(bool,fun(X_a,fun(X_a,X_a))),if(X_a)) = if(X_a) ).
fof(tsy_c_Int2_OMultInv_res,axiom,
ti(fun(int,fun(int,int)),multInv) = multInv ).
fof(tsy_c_IntFact_Od22set_res,axiom,
ti(fun(int,fun(int,bool)),d22set) = d22set ).
fof(tsy_c_IntFact_Ozfact_res,axiom,
ti(fun(int,int),zfact) = zfact ).
fof(tsy_c_IntPrimes_Oxzgcd_res,axiom,
ti(fun(int,fun(int,product_prod(int,product_prod(int,int)))),xzgcd) = xzgcd ).
fof(tsy_c_IntPrimes_Oxzgcda_res,axiom,
ti(fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))))),xzgcda) = xzgcda ).
fof(tsy_c_IntPrimes_Ozcong_res,axiom,
ti(fun(int,fun(int,fun(int,bool))),zcong) = zcong ).
fof(tsy_c_IntPrimes_Ozprime_res,axiom,
ti(fun(int,bool),zprime) = zprime ).
fof(tsy_c_Int_OBit0_res,axiom,
ti(fun(int,int),bit0) = bit0 ).
fof(tsy_c_Int_OBit1_res,axiom,
ti(fun(int,int),bit1) = bit1 ).
fof(tsy_c_Int_OMin_res,axiom,
ti(int,min) = min ).
fof(tsy_c_Int_OPls_res,axiom,
ti(int,pls) = pls ).
fof(tsy_c_Int_Oint__ge__less__than_res,axiom,
ti(fun(int,fun(product_prod(int,int),bool)),int_ge_less_than) = int_ge_less_than ).
fof(tsy_c_Int_Oint__ge__less__than2_res,axiom,
ti(fun(int,fun(product_prod(int,int),bool)),int_ge_less_than2) = int_ge_less_than2 ).
fof(tsy_c_Int_Oiszero_res,axiom,
! [X_a] :
( semiring_1(X_a)
=> ti(fun(X_a,bool),iszero(X_a)) = iszero(X_a) ) ).
fof(tsy_c_Int_Onat_res,axiom,
ti(fun(int,nat),nat_1) = nat_1 ).
fof(tsy_c_Int_Onat__aux_res,axiom,
ti(fun(int,fun(nat,nat)),nat_aux) = nat_aux ).
fof(tsy_c_Int_Onumber__class_Onumber__of_res,axiom,
! [X_a] :
( number(X_a)
=> ti(fun(int,X_a),number_number_of(X_a)) = number_number_of(X_a) ) ).
fof(tsy_c_Int_Opred_res,axiom,
ti(fun(int,int),pred) = pred ).
fof(tsy_c_Int_Oring__1__class_OInts_res,axiom,
! [X_a] :
( ring_1(X_a)
=> ti(fun(X_a,bool),ring_1_Ints(X_a)) = ring_1_Ints(X_a) ) ).
fof(tsy_c_Int_Oring__1__class_Oof__int_res,axiom,
! [X_a] :
( ring_1(X_a)
=> ti(fun(int,X_a),ring_1_of_int(X_a)) = ring_1_of_int(X_a) ) ).
fof(tsy_c_Int_Osucc_res,axiom,
ti(fun(int,int),succ) = succ ).
fof(tsy_c_Lazy__Sequence_Osmall__lazy_H__rel_res,axiom,
ti(fun(product_prod(int,int),fun(product_prod(int,int),bool)),lazy_small_lazy_rel) = lazy_small_lazy_rel ).
fof(tsy_c_Legacy__GCD_Ozgcd_res,axiom,
ti(fun(int,fun(int,int)),legacy_zgcd) = legacy_zgcd ).
fof(tsy_c_Lim_OisCont_res,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& topological_space(X_b) )
=> ti(fun(fun(X_a,X_b),fun(X_a,bool)),isCont(X_a,X_b)) = isCont(X_a,X_b) ) ).
fof(tsy_c_Limits_Oat_res,axiom,
! [X_a] :
( topological_space(X_a)
=> ti(fun(X_a,filter(X_a)),at(X_a)) = at(X_a) ) ).
fof(tsy_c_Limits_Osequentially_res,axiom,
ti(filter(nat),sequentially) = sequentially ).
fof(tsy_c_Limits_Otendsto_res,axiom,
! [X_a,X_b] :
( topological_space(X_b)
=> ti(fun(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool))),tendsto(X_a,X_b)) = tendsto(X_a,X_b) ) ).
fof(tsy_c_Limits_Otrivial__limit_res,axiom,
! [X_a] : ti(fun(filter(X_a),bool),trivial_limit(X_a)) = trivial_limit(X_a) ).
fof(tsy_c_List_Oupto__rel_res,axiom,
ti(fun(product_prod(int,int),fun(product_prod(int,int),bool)),upto_rel) = upto_rel ).
fof(tsy_c_Log_Olog_res,axiom,
ti(fun(real,fun(real,real)),log) = log ).
fof(tsy_c_Log_Opowr_res,axiom,
ti(fun(real,fun(real,real)),powr) = powr ).
fof(tsy_c_Nat_OSuc_res,axiom,
ti(fun(nat,nat),suc) = suc ).
fof(tsy_c_Nat_Onat_Onat__case_res,axiom,
! [X_a] : ti(fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),nat_case(X_a)) = nat_case(X_a) ).
fof(tsy_c_Nat_Onat_Onat__rec_res,axiom,
! [X_a] : ti(fun(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a))),nat_rec(X_a)) = nat_rec(X_a) ).
fof(tsy_c_Nat_Onat_Onat__size_res,axiom,
ti(fun(nat,nat),nat_size) = nat_size ).
fof(tsy_c_Nat_Osemiring__1__class_Oof__nat_res,axiom,
! [X_a] :
( semiring_1(X_a)
=> ti(fun(nat,X_a),semiring_1_of_nat(X_a)) = semiring_1_of_nat(X_a) ) ).
fof(tsy_c_Nat_Osemiring__1__class_Oof__nat__aux_res,axiom,
! [X_a] :
( semiring_1(X_a)
=> ti(fun(fun(X_a,X_a),fun(nat,fun(X_a,X_a))),semiri532925092at_aux(X_a)) = semiri532925092at_aux(X_a) ) ).
fof(tsy_c_Nat_Osize__class_Osize_0_res,axiom,
! [X_a] : ti(fun(list(X_a),nat),size_size(list(X_a))) = size_size(list(X_a)) ).
fof(tsy_c_Nat_Osize__class_Osize_1_res,axiom,
ti(fun(code_code_numeral,nat),size_size(code_code_numeral)) = size_size(code_code_numeral) ).
fof(tsy_c_Nat_Osize__class_Osize_2_res,axiom,
ti(fun(nat,nat),size_size(nat)) = size_size(nat) ).
fof(tsy_c_Nat_Osize__class_Osize_3_res,axiom,
ti(fun(complex,nat),size_size(complex)) = size_size(complex) ).
fof(tsy_c_Nat__Numeral_Oneg_res,axiom,
ti(fun(int,bool),nat_neg) = nat_neg ).
fof(tsy_c_Nat__Transfer_Ois__nat_res,axiom,
ti(fun(int,bool),nat_is_nat) = nat_is_nat ).
fof(tsy_c_Nat__Transfer_Onat__set_res,axiom,
ti(fun(fun(int,bool),bool),nat_nat_set) = nat_nat_set ).
fof(tsy_c_Nat__Transfer_Otransfer__morphism_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,X_b),fun(fun(X_a,bool),bool)),nat_tr1645093318rphism(X_a,X_b)) = nat_tr1645093318rphism(X_a,X_b) ).
fof(tsy_c_Nat__Transfer_Otsub_res,axiom,
ti(fun(int,fun(int,int)),nat_tsub) = nat_tsub ).
fof(tsy_c_Nitpick_OAbs__Frac_res,axiom,
! [X_a] : ti(fun(product_prod(int,int),X_a),abs_Frac(X_a)) = abs_Frac(X_a) ).
fof(tsy_c_Nitpick_OFrac_res,axiom,
ti(fun(product_prod(int,int),bool),frac_1) = frac_1 ).
fof(tsy_c_Nitpick_Ofrac_res,axiom,
! [X_a] : ti(fun(int,fun(int,X_a)),frac(X_a)) = frac(X_a) ).
fof(tsy_c_Nitpick_Oint__gcd_res,axiom,
ti(fun(int,fun(int,int)),int_gcd) = int_gcd ).
fof(tsy_c_Nitpick_Oint__lcm_res,axiom,
ti(fun(int,fun(int,int)),int_lcm) = int_lcm ).
fof(tsy_c_Nitpick_Onat__gcd_res,axiom,
ti(fun(nat,fun(nat,nat)),nat_gcd) = nat_gcd ).
fof(tsy_c_Nitpick_Onat__gcd__rel_res,axiom,
ti(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),nat_gcd_rel) = nat_gcd_rel ).
fof(tsy_c_Nitpick_Onat__lcm_res,axiom,
ti(fun(nat,fun(nat,nat)),nat_lcm) = nat_lcm ).
fof(tsy_c_Nitpick_Onorm__frac_res,axiom,
ti(fun(int,fun(int,product_prod(int,int))),norm_frac) = norm_frac ).
fof(tsy_c_Nitpick_Onorm__frac__rel_res,axiom,
ti(fun(product_prod(int,int),fun(product_prod(int,int),bool)),norm_frac_rel) = norm_frac_rel ).
fof(tsy_c_Nitpick_Onumber__of__frac_res,axiom,
! [X_a] : ti(fun(int,X_a),number_of_frac(X_a)) = number_of_frac(X_a) ).
fof(tsy_c_Nitpick_Oone__frac_res,axiom,
! [X_a] : ti(X_a,one_frac(X_a)) = one_frac(X_a) ).
fof(tsy_c_Nitpick_Oprod_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,bool),fun(fun(X_b,bool),fun(product_prod(X_a,X_b),bool))),prod(X_a,X_b)) = prod(X_a,X_b) ).
fof(tsy_c_Nitpick_Orefl_H_res,axiom,
! [X_a] : ti(fun(fun(product_prod(X_a,X_a),bool),bool),refl(X_a)) = refl(X_a) ).
fof(tsy_c_Nitpick_Ozero__frac_res,axiom,
! [X_a] : ti(X_a,zero_frac(X_a)) = zero_frac(X_a) ).
fof(tsy_c_NthRoot_Oroot_res,axiom,
ti(fun(nat,fun(real,real)),root) = root ).
fof(tsy_c_NthRoot_Osqrt_res,axiom,
ti(fun(real,real),sqrt) = sqrt ).
fof(tsy_c_Orderings_Oord_Omax_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a))),max(X_a)) = max(X_a) ) ).
fof(tsy_c_Orderings_Oord_Omin_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a))),min_1(X_a)) = min_1(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Oless_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,bool)),ord_less(X_a)) = ord_less(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Oless__eq_res,axiom,
! [X_b] :
( ord(X_b)
=> ti(fun(X_b,fun(X_b,bool)),ord_less_eq(X_b)) = ord_less_eq(X_b) ) ).
fof(tsy_c_Orderings_Oord__class_Omax_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),ord_max(X_a)) = ord_max(X_a) ) ).
fof(tsy_c_Orderings_Oord__class_Omin_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,X_a)),ord_min(X_a)) = ord_min(X_a) ) ).
fof(tsy_c_Orderings_Oorder_Omono_res,axiom,
! [X_a] :
( order(X_a)
=> ti(fun(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool)),mono(nat,X_a)) = mono(nat,X_a) ) ).
fof(tsy_c_Orderings_Oorder_Ostrict__mono_res,axiom,
! [X_a] :
( order(X_a)
=> ti(fun(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool)),strict_mono(nat,X_a)) = strict_mono(nat,X_a) ) ).
fof(tsy_c_Parity_Oeven__odd__class_Oeven_0_res,axiom,
ti(fun(int,bool),even_odd_even(int)) = even_odd_even(int) ).
fof(tsy_c_Parity_Oeven__odd__class_Oeven_1_res,axiom,
ti(fun(nat,bool),even_odd_even(nat)) = even_odd_even(nat) ).
fof(tsy_c_Power_Opower_Opower_res,axiom,
! [X_a] : ti(fun(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)))),power_1(X_a)) = power_1(X_a) ).
fof(tsy_c_Power_Opower__class_Opower_res,axiom,
! [X_a] :
( power(X_a)
=> ti(fun(X_a,fun(nat,X_a)),power_power(X_a)) = power_power(X_a) ) ).
fof(tsy_c_Predicate_OPowp_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),powp(X_a)) = powp(X_a) ).
fof(tsy_c_Primes_Ocoprime_res,axiom,
ti(fun(nat,fun(nat,bool)),coprime) = coprime ).
fof(tsy_c_Primes_Ofact_res,axiom,
ti(fun(nat,nat),fact) = fact ).
fof(tsy_c_Primes_Oprime_res,axiom,
ti(fun(nat,bool),prime) = prime ).
fof(tsy_c_Product__Type_OPair_res,axiom,
! [X_a,X_b] : ti(fun(X_a,fun(X_b,product_prod(X_a,X_b))),product_Pair(X_a,X_b)) = product_Pair(X_a,X_b) ).
fof(tsy_c_Product__Type_Oapsnd_res,axiom,
! [X_c,X_b,X_a] : ti(fun(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b))),product_apsnd(X_c,X_b,X_a)) = product_apsnd(X_c,X_b,X_a) ).
fof(tsy_c_Product__Type_Ocurry_res,axiom,
! [X_b,X_c,X_a] : ti(fun(fun(product_prod(X_b,X_c),X_a),fun(X_b,fun(X_c,X_a))),product_curry(X_b,X_c,X_a)) = product_curry(X_b,X_c,X_a) ).
fof(tsy_c_Product__Type_Ofst_res,axiom,
! [X_a,X_b] : ti(fun(product_prod(X_a,X_b),X_a),product_fst(X_a,X_b)) = product_fst(X_a,X_b) ).
fof(tsy_c_Product__Type_Ointernal__split_res,axiom,
! [X_b,X_c,X_a] : ti(fun(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a)),produc1605651328_split(X_b,X_c,X_a)) = produc1605651328_split(X_b,X_c,X_a) ).
fof(tsy_c_Product__Type_Oprod_Oprod__case_res,axiom,
! [X_a,X_b,X_c] : ti(fun(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c)),product_prod_case(X_a,X_b,X_c)) = product_prod_case(X_a,X_b,X_c) ).
fof(tsy_c_Product__Type_Oprod_Oprod__rec_res,axiom,
! [X_b,X_c,X_a] : ti(fun(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a)),product_prod_rec(X_b,X_c,X_a)) = product_prod_rec(X_b,X_c,X_a) ).
fof(tsy_c_Product__Type_Oprod_Oprod__size_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,nat),fun(fun(X_b,nat),fun(product_prod(X_a,X_b),nat))),product_prod_size(X_a,X_b)) = product_prod_size(X_a,X_b) ).
fof(tsy_c_Product__Type_Osnd_res,axiom,
! [X_a,X_b] : ti(fun(product_prod(X_a,X_b),X_b),product_snd(X_a,X_b)) = product_snd(X_a,X_b) ).
fof(tsy_c_Quickcheck__Narrowing_Oaround__zero_res,axiom,
ti(fun(int,list(int)),quickc666637781d_zero) = quickc666637781d_zero ).
fof(tsy_c_Quickcheck__Narrowing_Oaround__zero__rel_res,axiom,
ti(fun(int,fun(int,bool)),quickc1265749348ro_rel) = quickc1265749348ro_rel ).
fof(tsy_c_Quickcheck__Narrowing_Odiv__mod__code__int_res,axiom,
ti(fun(quickcheck_code_int,fun(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int))),quickc495462417de_int) = quickc495462417de_int ).
fof(tsy_c_Quickcheck__Narrowing_Oint__of_res,axiom,
ti(fun(quickcheck_code_int,int),quickcheck_int_of) = quickcheck_int_of ).
fof(tsy_c_Quickcheck__Narrowing_Onat__of_res,axiom,
ti(fun(quickcheck_code_int,nat),quickcheck_nat_of) = quickcheck_nat_of ).
fof(tsy_c_Quickcheck__Narrowing_Oof__int_res,axiom,
ti(fun(int,quickcheck_code_int),quickcheck_of_int) = quickcheck_of_int ).
fof(tsy_c_RComplete_Onatceiling_res,axiom,
ti(fun(real,nat),natceiling) = natceiling ).
fof(tsy_c_RComplete_Onatfloor_res,axiom,
ti(fun(real,nat),natfloor) = natfloor ).
fof(tsy_c_Rat_OFract_res,axiom,
ti(fun(int,fun(int,rat)),fract) = fract ).
fof(tsy_c_Rat_OFrct_res,axiom,
ti(fun(product_prod(int,int),rat),frct) = frct ).
fof(tsy_c_Rat_Ofield__char__0__class_ORats_res,axiom,
! [X_a] :
( field_char_0(X_a)
=> ti(fun(X_a,bool),field_char_0_Rats(X_a)) = field_char_0_Rats(X_a) ) ).
fof(tsy_c_Rat_Onormalize_res,axiom,
ti(fun(product_prod(int,int),product_prod(int,int)),normalize) = normalize ).
fof(tsy_c_Rat_Oquotient__of_res,axiom,
ti(fun(rat,product_prod(int,int)),quotient_of) = quotient_of ).
fof(tsy_c_Rat_Oratrel_res,axiom,
ti(fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),ratrel) = ratrel ).
fof(tsy_c_RealDef_ORatreal_res,axiom,
ti(fun(rat,real),ratreal) = ratreal ).
fof(tsy_c_RealDef_Oreal_0_res,axiom,
ti(fun(int,real),real_1(int)) = real_1(int) ).
fof(tsy_c_RealDef_Oreal_1_res,axiom,
ti(fun(nat,real),real_1(nat)) = real_1(nat) ).
fof(tsy_c_RealDef_Ovanishes_res,axiom,
ti(fun(fun(nat,rat),bool),vanishes) = vanishes ).
fof(tsy_c_RealVector_Obounded__bilinear_res,axiom,
! [X_c,X_b,X_a] :
( ( real_normed_vector(X_a)
& real_normed_vector(X_b)
& real_normed_vector(X_c) )
=> ti(fun(fun(X_a,fun(X_b,X_c)),bool),bounded_bilinear(X_a,X_b,X_c)) = bounded_bilinear(X_a,X_b,X_c) ) ).
fof(tsy_c_RealVector_Odist__class_Odist_0_res,axiom,
! [X_a] :
( dist_norm(X_a)
=> ti(fun(X_a,fun(X_a,real)),dist_dist(X_a)) = dist_dist(X_a) ) ).
fof(tsy_c_RealVector_Odist__class_Odist_1_res,axiom,
! [X_a] :
( metric_space(X_a)
=> ti(fun(X_a,fun(X_a,real)),dist_dist(X_a)) = dist_dist(X_a) ) ).
fof(tsy_c_RealVector_Onorm__class_Onorm_0_res,axiom,
! [X_a] :
( sgn_div_norm(X_a)
=> ti(fun(X_a,real),norm_norm(X_a)) = norm_norm(X_a) ) ).
fof(tsy_c_RealVector_Onorm__class_Onorm_1_res,axiom,
! [X_a] :
( dist_norm(X_a)
=> ti(fun(X_a,real),norm_norm(X_a)) = norm_norm(X_a) ) ).
fof(tsy_c_RealVector_Oof__real_res,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ti(fun(real,X_a),of_real(X_a)) = of_real(X_a) ) ).
fof(tsy_c_RealVector_OscaleR__class_OscaleR_0_res,axiom,
! [X_a] :
( sgn_div_norm(X_a)
=> ti(fun(real,fun(X_a,X_a)),scaleR_scaleR(X_a)) = scaleR_scaleR(X_a) ) ).
fof(tsy_c_RealVector_OscaleR__class_OscaleR_1_res,axiom,
! [X_a] :
( real_vector(X_a)
=> ti(fun(real,fun(X_a,X_a)),scaleR_scaleR(X_a)) = scaleR_scaleR(X_a) ) ).
fof(tsy_c_Residues_OLegendre_res,axiom,
ti(fun(int,fun(int,int)),legendre) = legendre ).
fof(tsy_c_Residues_OQuadRes_res,axiom,
ti(fun(int,fun(int,bool)),quadRes) = quadRes ).
fof(tsy_c_Residues_OResSet_res,axiom,
ti(fun(int,fun(fun(int,bool),bool)),resSet) = resSet ).
fof(tsy_c_Residues_OSR_res,axiom,
ti(fun(int,fun(int,bool)),sr) = sr ).
fof(tsy_c_Residues_OSRStar_res,axiom,
ti(fun(int,fun(int,bool)),sRStar) = sRStar ).
fof(tsy_c_Residues_OStandardRes_res,axiom,
ti(fun(int,fun(int,int)),standardRes) = standardRes ).
fof(tsy_c_Rings_Odvd__class_Odvd_res,axiom,
! [X_a] :
( dvd(X_a)
=> ti(fun(X_a,fun(X_a,bool)),dvd_dvd(X_a)) = dvd_dvd(X_a) ) ).
fof(tsy_c_SEQ_OBseq_res,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ti(fun(fun(nat,X_a),bool),bseq(X_a)) = bseq(X_a) ) ).
fof(tsy_c_SEQ_OCauchy_res,axiom,
! [X_a] :
( metric_space(X_a)
=> ti(fun(fun(nat,X_a),bool),cauchy(X_a)) = cauchy(X_a) ) ).
fof(tsy_c_SEQ_Olim_res,axiom,
! [X_a] :
( metric_space(X_a)
=> ti(fun(fun(nat,X_a),X_a),lim(X_a)) = lim(X_a) ) ).
fof(tsy_c_SEQ_Omonoseq_res,axiom,
! [X_a] :
( order(X_a)
=> ti(fun(fun(nat,X_a),bool),monoseq(X_a)) = monoseq(X_a) ) ).
fof(tsy_c_SMT_Oz3div_res,axiom,
ti(fun(int,fun(int,int)),z3div) = z3div ).
fof(tsy_c_SMT_Oz3mod_res,axiom,
ti(fun(int,fun(int,int)),z3mod) = z3mod ).
fof(tsy_c_Series_Osuminf_res,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> ti(fun(fun(nat,X_a),X_a),suminf(X_a)) = suminf(X_a) ) ).
fof(tsy_c_Series_Osummable_res,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> ti(fun(fun(nat,X_a),bool),summable(X_a)) = summable(X_a) ) ).
fof(tsy_c_Series_Osums_res,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> ti(fun(fun(nat,X_a),fun(X_a,bool)),sums(X_a)) = sums(X_a) ) ).
fof(tsy_c_SetInterval_Oord__class_OatLeastAtMost_res,axiom,
! [X_a] :
( order(X_a)
=> ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_atLeastAtMost(X_a)) = ord_atLeastAtMost(X_a) ) ).
fof(tsy_c_SetInterval_Oord__class_OatLeastLessThan_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_atLeastLessThan(X_a)) = ord_atLeastLessThan(X_a) ) ).
fof(tsy_c_SetInterval_Oord__class_OatMost_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,bool)),ord_atMost(X_a)) = ord_atMost(X_a) ) ).
fof(tsy_c_SetInterval_Oord__class_OgreaterThanLessThan_0_res,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_gr676655987ssThan(X_a)) = ord_gr676655987ssThan(X_a) ) ).
fof(tsy_c_SetInterval_Oord__class_OgreaterThanLessThan_1_res,axiom,
ti(fun(nat,fun(nat,fun(nat,bool))),ord_gr676655987ssThan(nat)) = ord_gr676655987ssThan(nat) ).
fof(tsy_c_SetInterval_Oord__class_OgreaterThanLessThan_2_res,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ti(fun(X_a,fun(X_a,fun(X_a,bool))),ord_gr676655987ssThan(X_a)) = ord_gr676655987ssThan(X_a) ) ).
fof(tsy_c_SetInterval_Oord__class_OlessThan_res,axiom,
! [X_a] :
( ord(X_a)
=> ti(fun(X_a,fun(X_a,bool)),ord_lessThan(X_a)) = ord_lessThan(X_a) ) ).
fof(tsy_c_Set_OCollect_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),fun(X_a,bool)),collect(X_a)) = collect(X_a) ).
fof(tsy_c_Set_OPow_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),pow(X_a)) = pow(X_a) ).
fof(tsy_c_Set_Oimage_res,axiom,
! [X_b,X_a] : ti(fun(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool))),image(X_b,X_a)) = image(X_b,X_a) ).
fof(tsy_c_Sum__Type_OPlus_res,axiom,
! [X_a,X_b] : ti(fun(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool))),sum_Plus(X_a,X_b)) = sum_Plus(X_a,X_b) ).
fof(tsy_c_Transcendental_Oarccos_res,axiom,
ti(fun(real,real),arccos) = arccos ).
fof(tsy_c_Transcendental_Oarcsin_res,axiom,
ti(fun(real,real),arcsin) = arcsin ).
fof(tsy_c_Transcendental_Oarctan_res,axiom,
ti(fun(real,real),arctan) = arctan ).
fof(tsy_c_Transcendental_Ocos_res,axiom,
ti(fun(real,real),cos) = cos ).
fof(tsy_c_Transcendental_Ocos__coeff_res,axiom,
ti(fun(nat,real),cos_coeff) = cos_coeff ).
fof(tsy_c_Transcendental_Odiffs_res,axiom,
! [X_a] :
( ring_1(X_a)
=> ti(fun(fun(nat,X_a),fun(nat,X_a)),diffs(X_a)) = diffs(X_a) ) ).
fof(tsy_c_Transcendental_Oexp_res,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ti(fun(X_a,X_a),exp(X_a)) = exp(X_a) ) ).
fof(tsy_c_Transcendental_Oln_res,axiom,
ti(fun(real,real),ln) = ln ).
fof(tsy_c_Transcendental_Opi_res,axiom,
ti(real,pi) = pi ).
fof(tsy_c_Transcendental_Osin_res,axiom,
ti(fun(real,real),sin) = sin ).
fof(tsy_c_Transcendental_Osin__coeff_res,axiom,
ti(fun(nat,real),sin_coeff) = sin_coeff ).
fof(tsy_c_Transcendental_Otan_res,axiom,
ti(fun(real,real),tan) = tan ).
fof(tsy_c_TwoSquares__Mirabelle__gltvpfochi_Ois__sum2sq_res,axiom,
ti(fun(int,bool),twoSqu1091635170sum2sq) = twoSqu1091635170sum2sq ).
fof(tsy_c_TwoSquares__Mirabelle__gltvpfochi_Osum2sq_res,hypothesis,
ti(fun(product_prod(int,int),int),twoSqu1094008024sum2sq) = twoSqu1094008024sum2sq ).
fof(tsy_c_Wellfounded_Oaccp_res,axiom,
! [X_a] : ti(fun(fun(X_a,fun(X_a,bool)),fun(X_a,bool)),accp(X_a)) = accp(X_a) ).
fof(tsy_c_Wellfounded_Ofinite__psubset_res,axiom,
! [X_a] : ti(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),finite_psubset(X_a)) = finite_psubset(X_a) ).
fof(tsy_c_Wellfounded_Olex__prod_res,axiom,
! [X_a,X_b] : ti(fun(fun(product_prod(X_a,X_a),bool),fun(fun(product_prod(X_b,X_b),bool),fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool))),lex_prod(X_a,X_b)) = lex_prod(X_a,X_b) ).
fof(tsy_c_Wellfounded_Omeasure_res,axiom,
! [X_a] : ti(fun(fun(X_a,nat),fun(product_prod(X_a,X_a),bool)),measure(X_a)) = measure(X_a) ).
fof(tsy_c_Wellfounded_Omlex__prod_res,axiom,
! [X_a] : ti(fun(fun(X_a,nat),fun(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool))),mlex_prod(X_a)) = mlex_prod(X_a) ).
fof(tsy_c_Wellfounded_Opred__nat_res,axiom,
ti(fun(product_prod(nat,nat),bool),pred_nat) = pred_nat ).
fof(tsy_c_WilsonRuss_Oinv_res,axiom,
ti(fun(int,fun(int,int)),inv) = inv ).
fof(tsy_c_WilsonRuss_Owset_res,axiom,
ti(fun(int,fun(int,fun(int,bool))),wset) = wset ).
fof(tsy_c_fEx_res,axiom,
! [X_a] : ti(fun(fun(X_a,bool),bool),fEx(X_a)) = fEx(X_a) ).
fof(tsy_c_fFalse_res,axiom,
ti(bool,fFalse) = fFalse ).
fof(tsy_c_fNot_res,axiom,
ti(fun(bool,bool),fNot) = fNot ).
fof(tsy_c_fTrue_res,axiom,
ti(bool,fTrue) = fTrue ).
fof(tsy_c_fconj_res,axiom,
ti(fun(bool,fun(bool,bool)),fconj) = fconj ).
fof(tsy_c_fdisj_res,axiom,
ti(fun(bool,fun(bool,bool)),fdisj) = fdisj ).
fof(tsy_c_fequal_res,axiom,
! [X_a] : ti(fun(X_a,fun(X_a,bool)),fequal(X_a)) = fequal(X_a) ).
fof(tsy_c_fimplies_res,axiom,
ti(fun(bool,fun(bool,bool)),fimplies) = fimplies ).
fof(tsy_c_hAPP_arg1,axiom,
! [X_a,X_c,B_1_1,B_2_1] : hAPP(X_a,X_c,ti(fun(X_a,X_c),B_1_1),B_2_1) = hAPP(X_a,X_c,B_1_1,B_2_1) ).
fof(tsy_c_hAPP_arg2,axiom,
! [X_a,X_c,B_1_1,B_2_1] : hAPP(X_a,X_c,B_1_1,ti(X_a,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1) ).
fof(tsy_c_hAPP_res,axiom,
! [X_c,X_a,B_1_1,B_2_1] : ti(X_c,hAPP(X_a,X_c,B_1_1,B_2_1)) = hAPP(X_a,X_c,B_1_1,B_2_1) ).
fof(tsy_c_hBOOL_arg1,axiom,
! [B_1_1] :
( hBOOL(ti(bool,B_1_1))
<=> hBOOL(B_1_1) ) ).
fof(tsy_c_member_res,axiom,
! [X_a] : ti(fun(X_a,fun(fun(X_a,bool),bool)),member(X_a)) = member(X_a) ).
fof(tsy_v_a_res,hypothesis,
ti(int,a) = a ).
fof(tsy_v_b_res,hypothesis,
ti(int,b) = b ).
fof(tsy_v_p_res,hypothesis,
ti(int,p) = p ).
fof(tsy_v_q_res,hypothesis,
ti(int,q) = q ).
%----Relevant facts (4992)
fof(fact_0_xzgcda__linear__aux1,axiom,
! [A_1,R_4,B,M_1,C,D,N_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_4),B))),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),C),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_4),D))),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),N_1))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),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),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D),N_1)))) ).
fof(fact_1_mult__diff__mult,axiom,
! [X_a] :
( ring(X_a)
=> ! [X,Y,A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),A_1)),B)) ) ).
fof(fact_2_mult_Oprod__diff__prod,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X,Y,A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),A_1)),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))) ) ).
fof(fact_3_eq__add__iff2,axiom,
! [X_a] :
( ring(X_a)
=> ! [Aa,E_2,C_1,Ba,D_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),E_2)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),E_2)),D_1)
<=> ti(X_a,C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Ba),Aa)),E_2)),D_1) ) ) ).
fof(fact_4_eq__add__iff1,axiom,
! [X_a] :
( ring(X_a)
=> ! [Aa,E_2,C_1,Ba,D_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),E_2)),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),E_2)),D_1)
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba)),E_2)),C_1) = ti(X_a,D_1) ) ) ).
fof(fact_5_is__sum2sq__def,axiom,
! [X_1] :
( hBOOL(hAPP(int,bool,twoSqu1091635170sum2sq,X_1))
<=> ? [A_3,B_3] : hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_3),B_3)) = X_1 ) ).
fof(fact_6_Int2_Oaux1,axiom,
! [A_1,B,C] :
( hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B) = C
=> A_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),C),B) ) ).
fof(fact_7_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_8_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_9_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_10_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_11_diff__add__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),B) = ti(X_a,A_1) ) ).
fof(fact_12_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [X_a] :
( ab_semigroup_mult(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)) ) ).
fof(fact_13_add__right__imp__eq,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [B,A_1,C] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_1)
=> ti(X_a,B) = ti(X_a,C) ) ) ).
fof(fact_14_add__imp__eq,axiom,
! [X_a] :
( cancel146912293up_add(X_a)
=> ! [A_1,B,C] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)
=> ti(X_a,B) = ti(X_a,C) ) ) ).
fof(fact_15_add__left__imp__eq,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [A_1,B,C] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)
=> ti(X_a,B) = ti(X_a,C) ) ) ).
fof(fact_16_add__right__cancel,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [Ba,Aa,C_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ba),Aa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),Aa)
<=> ti(X_a,Ba) = ti(X_a,C_1) ) ) ).
fof(fact_17_add__left__cancel,axiom,
! [X_a] :
( cancel_semigroup_add(X_a)
=> ! [Aa,Ba,C_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Ba) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),C_1)
<=> ti(X_a,Ba) = ti(X_a,C_1) ) ) ).
fof(fact_18_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [X_a] :
( ab_semigroup_add(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C)) ) ).
fof(fact_19_diff__eq__diff__eq,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C_1),D_1)
=> ( ti(X_a,Aa) = ti(X_a,Ba)
<=> ti(X_a,C_1) = ti(X_a,D_1) ) ) ) ).
fof(fact_20_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_21_zmult__commute,axiom,
! [Z,W] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z),W) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z) ).
fof(fact_22_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_23_zadd__left__commute,axiom,
! [X,Y,Z] : 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)) = 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)) ).
fof(fact_24_zadd__commute,axiom,
! [Z,W] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),W) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),Z) ).
fof(fact_25_combine__common__factor,axiom,
! [X_a] :
( semiring(X_a)
=> ! [A_1,E_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),E_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),E_1)),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),E_1)),C) ) ).
fof(fact_26_mult__left_Oadd,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X,Y,Ya] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),Ya) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Ya)) ) ).
fof(fact_27_mult_Oadd__left,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [A_1,A_5,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),A_5)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_5),B)) ) ).
fof(fact_28_comm__semiring__class_Odistrib,axiom,
! [X_a] :
( comm_semiring(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)) ) ).
fof(fact_29_mult__right_Oadd,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Xa_1,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_1),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_1),Y)) ) ).
fof(fact_30_mult_Oadd__right,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [A_1,B,B_4] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),B_4)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B_4)) ) ).
fof(fact_31_mult__left_Odiff,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X,Y,Ya] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),Ya) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Ya)) ) ).
fof(fact_32_mult_Odiff__left,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [A_1,A_5,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),A_5)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_5),B)) ) ).
fof(fact_33_mult__right_Odiff,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Xa_1,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_1),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_1),Y)) ) ).
fof(fact_34_mult_Odiff__right,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [A_1,B,B_4] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),B_4)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B_4)) ) ).
fof(fact_35_add__diff__add,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,C,B,D] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C),D)) ) ).
fof(fact_36_add__diff__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),B) = ti(X_a,A_1) ) ).
fof(fact_37_crossproduct__eq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [W_1,Y_1,X_1,Z_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W_1),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Z_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W_1),Z_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Y_1))
<=> ( ti(X_a,W_1) = ti(X_a,X_1)
| ti(X_a,Y_1) = ti(X_a,Z_1) ) ) ) ).
fof(fact_38_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),M_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),M_1) ) ).
fof(fact_39_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)) ) ).
fof(fact_40_crossproduct__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [C_1,D_1,Aa,Ba] :
( ( ti(X_a,Aa) != ti(X_a,Ba)
& ti(X_a,C_1) != ti(X_a,D_1) )
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),D_1)) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),D_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),C_1)) ) ) ).
fof(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z)) ) ).
fof(fact_42_minus__apply,axiom,
! [X_b,X_a] :
( minus(X_a)
=> ! [A_2,B_1,X_1] : hAPP(X_b,X_a,hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),minus_minus(fun(X_b,X_a)),A_2),B_1),X_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_b,X_a,A_2,X_1)),hAPP(X_b,X_a,B_1,X_1)) ) ).
fof(fact_43_fun__diff__def,axiom,
! [X_a,X_b] :
( minus(X_b)
=> ! [A_2,B_1,X_2] : hAPP(X_a,X_b,hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),minus_minus(fun(X_a,X_b)),A_2),B_1),X_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,A_2,X_2)),hAPP(X_a,X_b,B_1,X_2)) ) ).
fof(fact_44_Pair__inject,axiom,
! [X_b,X_a,A_1,B,A_5,B_4] :
( hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_1),B) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_5),B_4)
=> ~ ( ti(X_a,A_1) = ti(X_a,A_5)
=> ti(X_b,B) != ti(X_b,B_4) ) ) ).
fof(fact_45_Pair__eq,axiom,
! [X_b,X_a,Aa,Ba,A_7,B_6] :
( hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Aa),Ba) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_7),B_6)
<=> ( ti(X_a,Aa) = ti(X_a,A_7)
& ti(X_b,Ba) = ti(X_b,B_6) ) ) ).
fof(fact_46_split__paired__All,axiom,
! [X_a,X_b,P_1] :
( ! [X1] : hBOOL(hAPP(product_prod(X_a,X_b),bool,P_1,X1))
<=> ! [A_3,B_3] : hBOOL(hAPP(product_prod(X_a,X_b),bool,P_1,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_3))) ) ).
fof(fact_47_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C,D] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D)) ) ).
fof(fact_48_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),B) ) ).
fof(fact_49_times_Oidem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1) = ti(X_a,A_1) ) ).
fof(fact_50_mult__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X) = ti(X_a,X) ) ).
fof(fact_51_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1) ) ).
fof(fact_52_mult__left__idem,axiom,
! [X_a] :
( ab_sem1668676832m_mult(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) ) ).
fof(fact_53_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ry)) ) ).
fof(fact_54_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),Ry) ) ).
fof(fact_55_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Rx) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),Rx)) ) ).
fof(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Rx) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),Ly) ) ).
fof(fact_57_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry))) ) ).
fof(fact_58_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),Ry)) ) ).
fof(fact_59_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Lx,Ly,Rx,Ry] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Ly)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Rx),Ry)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Lx),Rx)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ly),Ry)) ) ).
fof(fact_60_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_1) ) ).
fof(fact_61_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),D)) ) ).
fof(fact_62_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,C,D] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),D) ) ).
fof(fact_63_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C)) ) ).
fof(fact_64_prod_Orecs,axiom,
! [X_b,X_c,X_a,F1,Aa,Ba] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_rec(X_b,X_c,X_a),F1),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),Aa),Ba)) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F1,Aa),Ba) ).
fof(fact_65_split__paired__Ex,axiom,
! [X_a,X_b,P_1] :
( ? [X1] : hBOOL(hAPP(product_prod(X_a,X_b),bool,P_1,X1))
<=> ? [A_3,B_3] : hBOOL(hAPP(product_prod(X_a,X_b),bool,P_1,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_3))) ) ).
fof(fact_66_prod__induct6,axiom,
! [X_a,X_b,X_c,X_d,X_e,X_f,X_1,P_1] :
( ! [A_3,B_3,C_4,D_3,E,F_3] : hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),bool,P_1,hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),A_3),hAPP(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),B_3),hAPP(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),hAPP(X_c,fun(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_Pair(X_c,product_prod(X_d,product_prod(X_e,X_f))),C_4),hAPP(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f)),hAPP(X_d,fun(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f))),product_Pair(X_d,product_prod(X_e,X_f)),D_3),hAPP(X_f,product_prod(X_e,X_f),hAPP(X_e,fun(X_f,product_prod(X_e,X_f)),product_Pair(X_e,X_f),E),F_3)))))))
=> hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),bool,P_1,X_1)) ) ).
fof(fact_67_prod__cases6,axiom,
! [X_a,X_b,X_c,X_d,X_e,X_f,Y] :
~ ! [A_3,B_3,C_4,D_3,E,F_3] : ti(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),Y) != hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),A_3),hAPP(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),product_prod(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),B_3),hAPP(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f))),hAPP(X_c,fun(product_prod(X_d,product_prod(X_e,X_f)),product_prod(X_c,product_prod(X_d,product_prod(X_e,X_f)))),product_Pair(X_c,product_prod(X_d,product_prod(X_e,X_f))),C_4),hAPP(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f)),hAPP(X_d,fun(product_prod(X_e,X_f),product_prod(X_d,product_prod(X_e,X_f))),product_Pair(X_d,product_prod(X_e,X_f)),D_3),hAPP(X_f,product_prod(X_e,X_f),hAPP(X_e,fun(X_f,product_prod(X_e,X_f)),product_Pair(X_e,X_f),E),F_3))))) ).
fof(fact_68_prod__induct5,axiom,
! [X_a,X_b,X_c,X_d,X_e,X_1,P_1] :
( ! [A_3,B_3,C_4,D_3,E] : hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),bool,P_1,hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),A_3),hAPP(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,X_e))),B_3),hAPP(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e)),hAPP(X_c,fun(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e))),product_Pair(X_c,product_prod(X_d,X_e)),C_4),hAPP(X_e,product_prod(X_d,X_e),hAPP(X_d,fun(X_e,product_prod(X_d,X_e)),product_Pair(X_d,X_e),D_3),E))))))
=> hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),bool,P_1,X_1)) ) ).
fof(fact_69_prod__cases5,axiom,
! [X_a,X_b,X_c,X_d,X_e,Y] :
~ ! [A_3,B_3,C_4,D_3,E] : ti(product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),Y) != hAPP(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),product_prod(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),A_3),hAPP(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e))),hAPP(X_b,fun(product_prod(X_c,product_prod(X_d,X_e)),product_prod(X_b,product_prod(X_c,product_prod(X_d,X_e)))),product_Pair(X_b,product_prod(X_c,product_prod(X_d,X_e))),B_3),hAPP(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e)),hAPP(X_c,fun(product_prod(X_d,X_e),product_prod(X_c,product_prod(X_d,X_e))),product_Pair(X_c,product_prod(X_d,X_e)),C_4),hAPP(X_e,product_prod(X_d,X_e),hAPP(X_d,fun(X_e,product_prod(X_d,X_e)),product_Pair(X_d,X_e),D_3),E)))) ).
fof(fact_70_prod__induct4,axiom,
! [X_a,X_b,X_c,X_d,X_1,P_1] :
( ! [A_3,B_3,C_4,D_3] : hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),bool,P_1,hAPP(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d)))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,X_d))),A_3),hAPP(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d)),hAPP(X_b,fun(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d))),product_Pair(X_b,product_prod(X_c,X_d)),B_3),hAPP(X_d,product_prod(X_c,X_d),hAPP(X_c,fun(X_d,product_prod(X_c,X_d)),product_Pair(X_c,X_d),C_4),D_3)))))
=> hBOOL(hAPP(product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),bool,P_1,X_1)) ) ).
fof(fact_71_prod__cases4,axiom,
! [X_a,X_b,X_c,X_d,Y] :
~ ! [A_3,B_3,C_4,D_3] : ti(product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),Y) != hAPP(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d))),hAPP(X_a,fun(product_prod(X_b,product_prod(X_c,X_d)),product_prod(X_a,product_prod(X_b,product_prod(X_c,X_d)))),product_Pair(X_a,product_prod(X_b,product_prod(X_c,X_d))),A_3),hAPP(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d)),hAPP(X_b,fun(product_prod(X_c,X_d),product_prod(X_b,product_prod(X_c,X_d))),product_Pair(X_b,product_prod(X_c,X_d)),B_3),hAPP(X_d,product_prod(X_c,X_d),hAPP(X_c,fun(X_d,product_prod(X_c,X_d)),product_Pair(X_c,X_d),C_4),D_3))) ).
fof(fact_72_prod__induct3,axiom,
! [X_a,X_b,X_c,X_1,P_1] :
( ! [A_3,B_3,C_4] : hBOOL(hAPP(product_prod(X_a,product_prod(X_b,X_c)),bool,P_1,hAPP(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c)),hAPP(X_a,fun(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c))),product_Pair(X_a,product_prod(X_b,X_c)),A_3),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),B_3),C_4))))
=> hBOOL(hAPP(product_prod(X_a,product_prod(X_b,X_c)),bool,P_1,X_1)) ) ).
fof(fact_73_prod__cases3,axiom,
! [X_a,X_b,X_c,Y] :
~ ! [A_3,B_3,C_4] : ti(product_prod(X_a,product_prod(X_b,X_c)),Y) != hAPP(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c)),hAPP(X_a,fun(product_prod(X_b,X_c),product_prod(X_a,product_prod(X_b,X_c))),product_Pair(X_a,product_prod(X_b,X_c)),A_3),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),B_3),C_4)) ).
fof(fact_74_PairE,axiom,
! [X_a,X_b,P_4] :
~ ! [X_2,Y_2] : ti(product_prod(X_a,X_b),P_4) != hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_2),Y_2) ).
fof(fact_75_ext,axiom,
! [X_a,X_b,F,G] :
( ! [X_2] : hAPP(X_a,X_b,F,X_2) = hAPP(X_a,X_b,G,X_2)
=> ti(fun(X_a,X_b),F) = ti(fun(X_a,X_b),G) ) ).
fof(fact_76_mem__def,axiom,
! [X_a,X_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
<=> hBOOL(hAPP(X_a,bool,A_2,X_1)) ) ).
fof(fact_77_Collect__def,axiom,
! [X_a,P_1] : hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1) = ti(fun(X_a,bool),P_1) ).
fof(fact_78_prod_Oexhaust,axiom,
! [X_a,X_b,Y] :
~ ! [A_3,B_3] : ti(product_prod(X_a,X_b),Y) != hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_3) ).
fof(fact_79_internal__split__conv,axiom,
! [X_b,X_c,X_a,C_1,Aa,Ba] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),produc1605651328_split(X_b,X_c,X_a),C_1),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),Aa),Ba)) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),C_1,Aa),Ba) ).
fof(fact_80_curryI,axiom,
! [X_a,X_b,F,Aa,Ba] :
( hBOOL(hAPP(product_prod(X_a,X_b),bool,F,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Aa),Ba)))
=> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),product_curry(X_a,X_b,bool),F),Aa),Ba)) ) ).
fof(fact_81_le__add__iff1,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [Aa,E_2,C_1,Ba,D_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),E_2)),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),E_2)),D_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba)),E_2)),C_1)),D_1)) ) ) ).
fof(fact_82_le__add__iff2,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [Aa,E_2,C_1,Ba,D_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),E_2)),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),E_2)),D_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Ba),Aa)),E_2)),D_1))) ) ) ).
fof(fact_83_real__squared__diff__one__factored,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),one_one(X_a)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),one_one(X_a))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),one_one(X_a))) ) ).
fof(fact_84_less__add__iff1,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [Aa,E_2,C_1,Ba,D_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),E_2)),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),E_2)),D_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba)),E_2)),C_1)),D_1)) ) ) ).
fof(fact_85_less__add__iff2,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [Aa,E_2,C_1,Ba,D_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),E_2)),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),E_2)),D_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Ba),Aa)),E_2)),D_1))) ) ) ).
fof(fact_86_add__number__of__diff1,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),W)),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W))),C) ) ).
fof(fact_87_curry__conv,axiom,
! [X_c,X_a,X_b,F,Aa,Ba] : hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),hAPP(fun(product_prod(X_b,X_c),X_a),fun(X_b,fun(X_c,X_a)),product_curry(X_b,X_c,X_a),F),Aa),Ba) = hAPP(product_prod(X_b,X_c),X_a,F,hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),Aa),Ba)) ).
fof(fact_88_bounded__bilinear_Oprod__diff__prod,axiom,
! [X_a,X_c,X_b] :
( ( real_normed_vector(X_b)
& real_normed_vector(X_c)
& real_normed_vector(X_a) )
=> ! [X_1,Y_1,Aa,Ba,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,X_1),Y_1)),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),plus_plus(X_c),hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),plus_plus(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),Aa)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Y_1),Ba))),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),Aa)),Ba))),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Y_1),Ba))) ) ) ).
fof(fact_89_in__rel__def,axiom,
! [X_b,X_a,R_3,X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),in_rel(X_a,X_b),R_3),X_1),Y_1))
<=> hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1)),R_3)) ) ).
fof(fact_90_curryE,axiom,
! [X_b,X_a,F,Aa,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),product_curry(X_a,X_b,bool),F),Aa),Ba))
=> hBOOL(hAPP(product_prod(X_a,X_b),bool,F,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Aa),Ba))) ) ).
fof(fact_91_one__reorient,axiom,
! [X_a] :
( one(X_a)
=> ! [X_1] :
( one_one(X_a) = ti(X_a,X_1)
<=> ti(X_a,X_1) = one_one(X_a) ) ) ).
fof(fact_92_zle__add1__eq__le,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),one_one(int))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z_1)) ) ).
fof(fact_93_number__of__reorient,axiom,
! [X_a] :
( number(X_a)
=> ! [W_1,X_1] :
( hAPP(int,X_a,number_number_of(X_a),W_1) = ti(X_a,X_1)
<=> ti(X_a,X_1) = hAPP(int,X_a,number_number_of(X_a),W_1) ) ) ).
fof(fact_94_le__number__of__eq__not__less,axiom,
! [X_a] :
( ( number(X_a)
& linorder(X_a) )
=> ! [V_1,W_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),V_1)),hAPP(int,X_a,number_number_of(X_a),W_1)))
<=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),hAPP(int,X_a,number_number_of(X_a),V_1))) ) ) ).
fof(fact_95_eq__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& ring_char_0(X_a) )
=> ! [X_1,Y_1] :
( hAPP(int,X_a,number_number_of(X_a),X_1) = hAPP(int,X_a,number_number_of(X_a),Y_1)
<=> X_1 = Y_1 ) ) ).
fof(fact_96_less__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),X_1)),hAPP(int,X_a,number_number_of(X_a),Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Y_1)) ) ) ).
fof(fact_97_le__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),X_1)),hAPP(int,X_a,number_number_of(X_a),Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),Y_1)) ) ) ).
fof(fact_98_add1__zle__eq,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_1),one_one(int))),Z_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_1)) ) ).
fof(fact_99_linorder__neqE__linordered__idom,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] :
( ti(X_a,X) != ti(X_a,Y)
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).
fof(fact_100_zless__imp__add1__zle,axiom,
! [W,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W),Z))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),one_one(int))),Z)) ) ).
fof(fact_101_order__le__neq__implies__less,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( ti(X_a,X) != ti(X_a,Y)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ) ).
fof(fact_102_zle__diff1__eq,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_1),one_one(int))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_1)) ) ).
fof(fact_103_less__1__mult,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,M_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),M_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),M_1),N_1))) ) ) ) ).
fof(fact_104_less__add__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a)))) ) ).
fof(fact_105_add__le__less__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D))) ) ) ) ).
fof(fact_106_add__less__le__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D))) ) ) ) ).
fof(fact_107_zadd__zless__mono,axiom,
! [Z_4,Z,W_3,W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_3),W))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_4),Z))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_3),Z_4)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),Z))) ) ) ).
fof(fact_108_zless__add1__eq,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_1),one_one(int))))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_1))
| W_1 = Z_1 ) ) ).
fof(fact_109_add__less__cancel__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [Aa,C_1,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ba),C_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba)) ) ) ).
fof(fact_110_add__less__cancel__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C_1,Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),Aa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),Ba)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba)) ) ) ).
fof(fact_111_add__strict__right__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C))) ) ) ).
fof(fact_112_add__strict__left__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B))) ) ) ).
fof(fact_113_add__strict__mono,axiom,
! [X_a] :
( ordere223160158up_add(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D))) ) ) ) ).
fof(fact_114_add__less__imp__less__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_1,C,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_115_add__less__imp__less__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_116_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_117_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_118_mult__1__left,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_119_mult__1,axiom,
! [X_a] :
( comm_monoid_mult(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),one_one(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_120_mult__1__right,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_121_mult_Ocomm__neutral,axiom,
! [X_a] :
( comm_monoid_mult(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_122_diff__eq__diff__less,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C_1),D_1)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_1)) ) ) ) ).
fof(fact_123_add__le__cancel__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [Aa,C_1,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ba),C_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba)) ) ) ).
fof(fact_124_add__le__cancel__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C_1,Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),Aa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C_1),Ba)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba)) ) ) ).
fof(fact_125_add__right__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C))) ) ) ).
fof(fact_126_add__left__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B))) ) ) ).
fof(fact_127_add__mono,axiom,
! [X_a] :
( ordere779506340up_add(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),D))) ) ) ) ).
fof(fact_128_add__le__imp__le__right,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [A_1,C,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).
fof(fact_129_add__le__imp__le__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),B)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).
fof(fact_130_diff__eq__diff__less__eq,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C_1),D_1)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_1)) ) ) ) ).
fof(fact_131_times__numeral__code_I5_J,axiom,
! [V,W] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),V)),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),W)) ).
fof(fact_132_plus__numeral__code_I9_J,axiom,
! [V,W] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),V)),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),W)) ).
fof(fact_133_zadd__strict__right__mono,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_1),K_4)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),J_2),K_4))) ) ).
fof(fact_134_zadd__left__mono,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),I_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),J_2))) ) ).
fof(fact_135_zmult__1,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),one_one(int)),Z) = Z ).
fof(fact_136_zmult__1__right,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z),one_one(int)) = Z ).
fof(fact_137_bounded__bilinear_Oadd__right,axiom,
! [X_a,X_c,X_b] :
( ( real_normed_vector(X_b)
& real_normed_vector(X_c)
& real_normed_vector(X_a) )
=> ! [Aa,Ba,B_6,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),Ba),B_6)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),plus_plus(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),B_6)) ) ) ).
fof(fact_138_bounded__bilinear_Oadd__left,axiom,
! [X_a,X_c,X_b] :
( ( real_normed_vector(X_b)
& real_normed_vector(X_c)
& real_normed_vector(X_a) )
=> ! [Aa,A_7,Ba,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),A_7)),Ba) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),plus_plus(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,A_7),Ba)) ) ) ).
fof(fact_139_bounded__bilinear_Odiff__right,axiom,
! [X_a,X_c,X_b] :
( ( real_normed_vector(X_b)
& real_normed_vector(X_c)
& real_normed_vector(X_a) )
=> ! [Aa,Ba,B_6,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Ba),B_6)) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),B_6)) ) ) ).
fof(fact_140_bounded__bilinear_Odiff__left,axiom,
! [X_a,X_c,X_b] :
( ( real_normed_vector(X_b)
& real_normed_vector(X_c)
& real_normed_vector(X_a) )
=> ! [Aa,A_7,Ba,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),A_7)),Ba) = hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,A_7),Ba)) ) ) ).
fof(fact_141_right__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ! [V,B,C] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),B),C)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),B)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),C)) ) ).
fof(fact_142_left__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& semiring(X_b) )
=> ! [A_1,B,V] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),A_1),B)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_1),hAPP(int,X_b,number_number_of(X_b),V))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B),hAPP(int,X_b,number_number_of(X_b),V))) ) ).
fof(fact_143_right__diff__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ring(X_b) )
=> ! [V,B,C] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),B),C)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),B)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),C)) ) ).
fof(fact_144_left__diff__distrib__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ring(X_b) )
=> ! [A_1,B,V] : hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),A_1),B)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),A_1),hAPP(int,X_b,number_number_of(X_b),V))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),B),hAPP(int,X_b,number_number_of(X_b),V))) ) ).
fof(fact_145_number__of__mult,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) ) ).
fof(fact_146_arith__simps_I32_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W)) ) ).
fof(fact_147_mult__number__of__left,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W)),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),W))),Z) ) ).
fof(fact_148_number__of__add,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) ) ).
fof(fact_149_add__number__of__eq,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W)) ) ).
fof(fact_150_add__number__of__left,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),W)),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),W))),Z) ) ).
fof(fact_151_number__of__diff,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),V),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) ) ).
fof(fact_152_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M_1),M_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),M_1) ) ).
fof(fact_153_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M_1,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),M_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))),M_1) ) ).
fof(fact_154_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),M_1)),M_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),one_one(X_a))),M_1) ) ).
fof(fact_155_curryD,axiom,
! [X_b,X_a,F,Aa,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),product_curry(X_a,X_b,bool),F),Aa),Ba))
=> hBOOL(hAPP(product_prod(X_a,X_b),bool,F,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Aa),Ba))) ) ).
fof(fact_156_order__refl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),X)) ) ).
fof(fact_157_xt1_I8_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X)) ) ) ) ).
fof(fact_158_order__le__less__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z)) ) ) ) ).
fof(fact_159_xt1_I7_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X)) ) ) ) ).
fof(fact_160_order__less__le__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z)) ) ) ) ).
fof(fact_161_xt1_I11_J,axiom,
! [X_a] :
( order(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( ti(X_a,A_1) != ti(X_a,B)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).
fof(fact_162_order__le__neq__trans,axiom,
! [X_a] :
( order(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( ti(X_a,A_1) != ti(X_a,B)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_163_order__le__imp__less__or__eq,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
| ti(X_a,X) = ti(X_a,Y) ) ) ) ).
fof(fact_164_linorder__antisym__conv2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_165_order__less__imp__le,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).
fof(fact_166_leD,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).
fof(fact_167_xt1_I12_J,axiom,
! [X_a] :
( order(X_a)
=> ! [A_1,B] :
( ti(X_a,A_1) != ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).
fof(fact_168_zle__refl,axiom,
! [W] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),W)) ).
fof(fact_169_number__of__is__id,axiom,
! [K_4] : hAPP(int,int,number_number_of(int),K_4) = K_4 ).
fof(fact_170_zle__linear,axiom,
! [Z,W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),W))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ).
fof(fact_171_zless__le,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),W_1))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),W_1))
& Z_1 != W_1 ) ) ).
fof(fact_172_zless__linear,axiom,
! [X,Y] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),Y))
| X = Y
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),X)) ) ).
fof(fact_173_less__number__of__int__code,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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(int,fun(int,bool),ord_less(int),K),L)) ) ).
fof(fact_174_less__eq__number__of__int__code,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).
fof(fact_175_zle__trans,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),J_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_2),K_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_1),K_4)) ) ) ).
fof(fact_176_zle__antisym,axiom,
! [Z,W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),W))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W),Z))
=> Z = W ) ) ).
fof(fact_177_le__fun__def,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
<=> ! [X_2] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,G,X_2))) ) ) ).
fof(fact_178_linorder__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).
fof(fact_179_order__eq__iff,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y_1] :
( ti(X_a,X_1) = ti(X_a,Y_1)
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1)) ) ) ) ).
fof(fact_180_order__eq__refl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( X = Y
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).
fof(fact_181_le__funD,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [X_1,F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) ) ) ).
fof(fact_182_order__antisym__conv,axiom,
! [X_a] :
( order(X_a)
=> ! [Y_1,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_183_ord__eq__le__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C,A_1,B] :
( ti(X_a,A_1) = ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C)) ) ) ) ).
fof(fact_184_xt1_I3_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C,A_1,B] :
( ti(X_a,A_1) = ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_1)) ) ) ) ).
fof(fact_185_ord__le__eq__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( ti(X_a,B) = ti(X_a,C)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C)) ) ) ) ).
fof(fact_186_xt1_I4_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( ti(X_a,B) = ti(X_a,C)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),A_1)) ) ) ) ).
fof(fact_187_order__antisym,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ).
fof(fact_188_order__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z)) ) ) ) ).
fof(fact_189_xt1_I5_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ).
fof(fact_190_xt1_I6_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),X)) ) ) ) ).
fof(fact_191_le__funE,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [X_1,F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,X_b,G,X_1))) ) ) ).
fof(fact_192_linorder__le__cases,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).
fof(fact_193_order__less__irrefl,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),X)) ) ).
fof(fact_194_linorder__neq__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( ti(X_a,X_1) != ti(X_a,Y_1)
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1)) ) ) ) ).
fof(fact_195_not__less__iff__gr__or__eq,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))
| ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_196_linorder__less__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
| ti(X_a,X) = ti(X_a,Y)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_197_linorder__antisym__conv3,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y_1,X_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_198_linorder__neqE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ti(X_a,X) != ti(X_a,Y)
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).
fof(fact_199_less__imp__neq,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ti(X_a,X) != ti(X_a,Y) ) ) ).
fof(fact_200_order__less__not__sym,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_201_order__less__imp__not__less,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_202_order__less__imp__not__eq,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ti(X_a,X) != ti(X_a,Y) ) ) ).
fof(fact_203_order__less__imp__not__eq2,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ti(X_a,Y) != ti(X_a,X) ) ) ).
fof(fact_204_order__less__imp__triv,axiom,
! [X_a] :
( preorder(X_a)
=> ! [P_1,X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1))
=> hBOOL(P_1) ) ) ) ).
fof(fact_205_order__less__asym_H,axiom,
! [X_a] :
( preorder(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ).
fof(fact_206_xt1_I9_J,axiom,
! [X_a] :
( order(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ).
fof(fact_207_ord__eq__less__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C,A_1,B] :
( ti(X_a,A_1) = ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C)) ) ) ) ).
fof(fact_208_xt1_I1_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C,A_1,B] :
( ti(X_a,A_1) = ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),A_1)) ) ) ) ).
fof(fact_209_ord__less__eq__trans,axiom,
! [X_a] :
( ord(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( ti(X_a,B) = ti(X_a,C)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),C)) ) ) ) ).
fof(fact_210_xt1_I2_J,axiom,
! [X_a] :
( order(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ( ti(X_a,B) = ti(X_a,C)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),A_1)) ) ) ) ).
fof(fact_211_order__less__trans,axiom,
! [X_a] :
( preorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Z)) ) ) ) ).
fof(fact_212_xt1_I10_J,axiom,
! [X_a] :
( order(X_a)
=> ! [Z,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),X)) ) ) ) ).
fof(fact_213_order__less__asym,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_214_linorder__cases,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( ti(X_a,X) != ti(X_a,Y)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ) ).
fof(fact_215_linorder__not__less,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1)) ) ) ).
fof(fact_216_linorder__not__le,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1)) ) ) ).
fof(fact_217_linorder__le__less__linear,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),X)) ) ) ).
fof(fact_218_order__less__le,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
& ti(X_a,X_1) != ti(X_a,Y_1) ) ) ) ).
fof(fact_219_less__le__not__le,axiom,
! [X_a] :
( preorder(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
& ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1)) ) ) ) ).
fof(fact_220_order__le__less,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
| ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_221_leI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X)) ) ) ).
fof(fact_222_not__leE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).
fof(fact_223_linorder__antisym__conv1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Y_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ) ).
fof(fact_224_order__neq__le__trans,axiom,
! [X_a] :
( order(X_a)
=> ! [A_1,B] :
( ti(X_a,A_1) != ti(X_a,B)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_225_norR__mem__unique__aux,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),one_one(int))))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B)) ) ).
fof(fact_226_int__le__induct,axiom,
! [P_1,I,K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),K))
=> ( hBOOL(hAPP(int,bool,P_1,K))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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)) ) ) ) ).
fof(fact_227_int__less__induct,axiom,
! [P_1,I,K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),K))
=> ( hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K),one_one(int))))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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)) ) ) ) ).
fof(fact_228_int__ge__induct,axiom,
! [P_1,K,I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I))
=> ( hBOOL(hAPP(int,bool,P_1,K))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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)) ) ) ) ).
fof(fact_229_int__gr__induct,axiom,
! [P_1,K,I] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I))
=> ( hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K),one_one(int))))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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)) ) ) ) ).
fof(fact_230_dense__le,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Z,Y] :
( ! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Z)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z)) ) ) ).
fof(fact_231_dense__le__bounded,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( ! [W_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),W_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),W_2),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W_2),Z)) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),Z)) ) ) ) ).
fof(fact_232_xt5,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [C_1,F,Ba,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),Aa))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),C_1),hAPP(X_a,X_b,F,Ba)))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_2),X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,Y_2)),hAPP(X_a,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C_1),hAPP(X_a,X_b,F,Aa))) ) ) ) ) ).
fof(fact_233_xt7,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [C_1,F,Ba,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),Aa))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C_1),hAPP(X_a,X_b,F,Ba)))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_2),X_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Y_2)),hAPP(X_a,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C_1),hAPP(X_a,X_b,F,Aa))) ) ) ) ) ).
fof(fact_234_order__less__le__subst2,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [F,C_1,Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,Ba)),C_1))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Y_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,F,Y_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,Aa)),C_1)) ) ) ) ) ).
fof(fact_235_order__le__less__subst2,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [F,C_1,Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,Ba)),C_1))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Y_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,F,Y_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),hAPP(X_a,X_b,F,Aa)),C_1)) ) ) ) ) ).
fof(fact_236_less__fun__def,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less(fun(X_a,X_b)),F),G))
<=> ( hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G))
& ~ hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),G),F)) ) ) ) ).
fof(fact_237_le__funI,axiom,
! [X_a,X_b] :
( ord(X_b)
=> ! [F,G] :
( ! [X_2] : hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,G,X_2)))
=> hBOOL(hAPP(fun(X_a,X_b),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,X_b),bool),ord_less_eq(fun(X_a,X_b)),F),G)) ) ) ).
fof(fact_238_xt6,axiom,
! [X_b,X_a] :
( ( order(X_a)
& ord(X_b) )
=> ! [C_1,F,Ba,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Ba)),Aa))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),C_1),Ba))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Y_2),X_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Y_2)),hAPP(X_b,X_a,F,X_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,C_1)),Aa)) ) ) ) ) ).
fof(fact_239_xt4,axiom,
! [X_b,X_a] :
( ( order(X_a)
& order(X_b) )
=> ! [C_1,F,Ba,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,Ba)),Aa))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),C_1),Ba))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Y_2),X_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,Y_2)),hAPP(X_b,X_a,F,X_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,C_1)),Aa)) ) ) ) ) ).
fof(fact_240_order__le__less__subst1,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [C_1,Aa,F,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),hAPP(X_b,X_a,F,Ba)))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),Ba),C_1))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),X_2),Y_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_b,X_a,F,X_2)),hAPP(X_b,X_a,F,Y_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),hAPP(X_b,X_a,F,C_1))) ) ) ) ) ).
fof(fact_241_order__less__le__subst1,axiom,
! [X_a,X_b] :
( ( order(X_b)
& order(X_a) )
=> ! [C_1,Aa,F,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),hAPP(X_b,X_a,F,Ba)))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),Ba),C_1))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),X_2),Y_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_b,X_a,F,X_2)),hAPP(X_b,X_a,F,Y_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),hAPP(X_b,X_a,F,C_1))) ) ) ) ) ).
fof(fact_242_int__induct,axiom,
! [I,P_1,K] :
( hBOOL(hAPP(int,bool,P_1,K))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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)) ) ) ) ).
fof(fact_243_Euler_Oaux2,axiom,
! [B,A_1,C] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),C))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),C))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),B))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),A_1)) ) ) ) ).
fof(fact_244_d22set__induct__old,axiom,
! [X_1,P_1] :
( ! [A_3] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_3))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_3),one_one(int)))) )
=> hBOOL(hAPP(int,bool,P_1,A_3)) )
=> hBOOL(hAPP(int,bool,P_1,X_1)) ) ).
fof(fact_245_convex__bound__lt,axiom,
! [X_a] :
( linord626643107strict(X_a)
=> ! [V,U,Y,X,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V))
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),U),V) = one_one(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),U),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),V),Y))),A_1)) ) ) ) ) ) ) ).
fof(fact_246_number__of__le__ceiling,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [V_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),V_1)),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V_1)),one_one(X_a))),X_1)) ) ) ).
fof(fact_247_ceiling__less__number__of,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1,V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)),hAPP(int,int,number_number_of(int),V_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V_1)),one_one(X_a)))) ) ) ).
fof(fact_248_zero__less__ceiling,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X_1)) ) ) ).
fof(fact_249_ceiling__le__zero,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)),zero_zero(int)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),zero_zero(X_a))) ) ) ).
fof(fact_250_ceiling__zero,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> hAPP(X_a,int,archimedean_ceiling(X_a),zero_zero(X_a)) = zero_zero(int) ) ).
fof(fact_251_zero__reorient,axiom,
! [X_a] :
( zero(X_a)
=> ! [X_1] :
( zero_zero(X_a) = ti(X_a,X_1)
<=> ti(X_a,X_1) = zero_zero(X_a) ) ) ).
fof(fact_252_ceiling__less__one,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)),one_one(int)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),zero_zero(X_a))) ) ) ).
fof(fact_253_one__le__ceiling,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X_1)) ) ) ).
fof(fact_254_divisors__zero,axiom,
! [X_a] :
( no_zero_divisors(X_a)
=> ! [A_1,B] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) = zero_zero(X_a)
=> ( ti(X_a,A_1) = zero_zero(X_a)
| ti(X_a,B) = zero_zero(X_a) ) ) ) ).
fof(fact_255_no__zero__divisors,axiom,
! [X_a] :
( no_zero_divisors(X_a)
=> ! [B,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) != zero_zero(X_a) ) ) ) ).
fof(fact_256_mult__eq__0__iff,axiom,
! [X_a] :
( ring_n68954251visors(X_a)
=> ! [Aa,Ba] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Ba) = zero_zero(X_a)
<=> ( ti(X_a,Aa) = zero_zero(X_a)
| ti(X_a,Ba) = zero_zero(X_a) ) ) ) ).
fof(fact_257_mult__right_Ozero,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_258_mult_Ozero__right,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_259_mult__zero__right,axiom,
! [X_a] :
( mult_zero(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_260_mult__left_Ozero,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),Y) = zero_zero(X_a) ) ).
fof(fact_261_mult_Ozero__left,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),B) = zero_zero(X_a) ) ).
fof(fact_262_mult__zero__left,axiom,
! [X_a] :
( mult_zero(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).
fof(fact_263_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_264_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).
fof(fact_265_double__eq__0__iff,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Aa) = zero_zero(X_a)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_266_add_Ocomm__neutral,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_267_add__0__right,axiom,
! [X_a] :
( monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_268_double__zero__sym,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( zero_zero(X_a) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Aa)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_269_add__0,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_270_add__0__left,axiom,
! [X_a] :
( monoid_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_271_add__0__iff,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [Ba,Aa] :
( ti(X_a,Ba) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Ba),Aa)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_272_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_273_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),A_1) = ti(X_a,A_1) ) ).
fof(fact_274_zero__neq__one,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> zero_zero(X_a) != one_one(X_a) ) ).
fof(fact_275_one__neq__zero,axiom,
! [X_a] :
( zero_neq_one(X_a)
=> one_one(X_a) != zero_zero(X_a) ) ).
fof(fact_276_right__minus__eq,axiom,
! [X_a] :
( group_add(X_a)
=> ! [Aa,Ba] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba) = zero_zero(X_a)
<=> ti(X_a,Aa) = ti(X_a,Ba) ) ) ).
fof(fact_277_eq__iff__diff__eq__0,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [Aa,Ba] :
( ti(X_a,Aa) = ti(X_a,Ba)
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba) = zero_zero(X_a) ) ) ).
fof(fact_278_diff__self,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),A_1) = zero_zero(X_a) ) ).
fof(fact_279_diff__0__right,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_280_int__0__neq__1,axiom,
zero_zero(int) != one_one(int) ).
fof(fact_281_zadd__0__right,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),zero_zero(int)) = Z ).
fof(fact_282_zadd__0,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),zero_zero(int)),Z) = Z ).
fof(fact_283_ceiling__less__cancel,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),hAPP(X_a,int,archimedean_ceiling(X_a),Y)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).
fof(fact_284_ceiling__mono,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),Y)),hAPP(X_a,int,archimedean_ceiling(X_a),X))) ) ) ).
fof(fact_285_ceiling__one,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> hAPP(X_a,int,archimedean_ceiling(X_a),one_one(X_a)) = one_one(int) ) ).
fof(fact_286_bounded__bilinear_Ozero__right,axiom,
! [X_b,X_a,X_c] :
( ( real_normed_vector(X_c)
& real_normed_vector(X_a)
& real_normed_vector(X_b) )
=> ! [Aa,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),zero_zero(X_b)) = zero_zero(X_c) ) ) ).
fof(fact_287_bounded__bilinear_Ozero__left,axiom,
! [X_b,X_a,X_c] :
( ( real_normed_vector(X_c)
& real_normed_vector(X_a)
& real_normed_vector(X_b) )
=> ! [Ba,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,zero_zero(X_a)),Ba) = zero_zero(X_c) ) ) ).
fof(fact_288_ceiling__number__of,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [V] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(int,X_a,number_number_of(X_a),V)) = hAPP(int,int,number_number_of(int),V) ) ).
fof(fact_289_zero__le__square,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1))) ) ).
fof(fact_290_zero__le__mult__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Ba)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Ba)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),zero_zero(X_a))) ) ) ) ) ).
fof(fact_291_mult__le__0__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Ba)),zero_zero(X_a)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),zero_zero(X_a))) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Ba)) ) ) ) ) ).
fof(fact_292_mult__nonneg__nonneg,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).
fof(fact_293_mult__nonneg__nonpos,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_294_mult__nonneg__nonpos2,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_295_mult__nonpos__nonneg,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_296_mult__nonpos__nonpos,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).
fof(fact_297_mult__right__mono,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))) ) ) ) ).
fof(fact_298_mult__left__mono,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_299_comm__mult__left__mono,axiom,
! [X_a] :
( ordere1490568538miring(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_300_mult__right__mono__neg,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))) ) ) ) ).
fof(fact_301_mult__left__mono__neg,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_302_mult__mono_H,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_303_mult__mono,axiom,
! [X_a] :
( ordered_semiring(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_304_split__mult__pos__le,axiom,
! [X_a] :
( ordered_ring(X_a)
=> ! [B,A_1] :
( ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ).
fof(fact_305_split__mult__neg__le,axiom,
! [X_a] :
( ordere453448008miring(X_a)
=> ! [B,A_1] :
( ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B)) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ).
fof(fact_306_not__square__less__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1)),zero_zero(X_a))) ) ).
fof(fact_307_mult__less__cancel__right__disj,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Aa,C_1,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),C_1)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),Aa)) ) ) ) ) ).
fof(fact_308_mult__less__cancel__left__disj,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [C_1,Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Aa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Ba)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),Aa)) ) ) ) ) ).
fof(fact_309_mult__less__cancel__left__pos,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Aa,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Aa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Ba)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba)) ) ) ) ).
fof(fact_310_mult__pos__pos,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).
fof(fact_311_mult__pos__neg,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_312_mult__pos__neg2,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),zero_zero(X_a))) ) ) ) ).
fof(fact_313_zero__less__mult__pos,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) ) ) ) ).
fof(fact_314_zero__less__mult__pos2,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B)) ) ) ) ).
fof(fact_315_mult__less__cancel__left__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Aa,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Aa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Ba)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),Aa)) ) ) ) ).
fof(fact_316_mult__neg__pos,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_317_mult__neg__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ) ) ).
fof(fact_318_mult__strict__right__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))) ) ) ) ).
fof(fact_319_mult__strict__left__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_320_comm__mult__strict__left__mono,axiom,
! [X_a] :
( linord893533164strict(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_321_mult__strict__right__mono__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))) ) ) ) ).
fof(fact_322_mult__strict__left__mono__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))) ) ) ) ).
fof(fact_323_zero__le__double__add__iff__zero__le__single__add,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa)) ) ) ).
fof(fact_324_double__add__le__zero__iff__single__add__le__zero,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Aa)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_325_add__nonneg__nonneg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).
fof(fact_326_add__nonneg__eq__0__iff,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [Y_1,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y_1))
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Y_1) = zero_zero(X_a)
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ) ) ).
fof(fact_327_add__increasing,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C))) ) ) ) ).
fof(fact_328_add__increasing2,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C))) ) ) ) ).
fof(fact_329_add__nonpos__nonpos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_330_zero__less__double__add__iff__zero__less__single__add,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Aa)) ) ) ).
fof(fact_331_even__less__0__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Aa)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_332_double__add__less__zero__iff__single__add__less__zero,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Aa)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_333_add__pos__pos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).
fof(fact_334_pos__add__strict,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [B,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C))) ) ) ) ).
fof(fact_335_add__neg__neg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_336_zero__le__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),one_one(X_a))) ) ).
fof(fact_337_not__one__le__zero,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),zero_zero(X_a))) ) ).
fof(fact_338_zero__less__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),one_one(X_a))) ) ).
fof(fact_339_not__one__less__zero,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),zero_zero(X_a))) ) ).
fof(fact_340_le__iff__diff__le__0,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba)),zero_zero(X_a))) ) ) ).
fof(fact_341_add__scale__eq__noteq,axiom,
! [X_a] :
( semiri456707255roduct(X_a)
=> ! [C,D,A_1,B,R_4] :
( ti(X_a,R_4) != zero_zero(X_a)
=> ( ( ti(X_a,A_1) = ti(X_a,B)
& ti(X_a,C) != ti(X_a,D) )
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),R_4),C)) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),R_4),D)) ) ) ) ).
fof(fact_342_less__iff__diff__less__0,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba)),zero_zero(X_a))) ) ) ).
fof(fact_343_Euler_Oaux1,axiom,
! [A_1,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),A_1))
=> ( X != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),one_one(int))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),one_one(int)))) ) ) ) ).
fof(fact_344_int__0__less__1,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),one_one(int))) ).
fof(fact_345_zmult__zless__mono2,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),I_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),J_2))) ) ) ).
fof(fact_346_odd__nonzero,axiom,
! [Z] : 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)),Z) != zero_zero(int) ).
fof(fact_347_less__bin__lemma,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K),L)),zero_zero(int))) ) ).
fof(fact_348_one__less__ceiling,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),X_1)) ) ) ).
fof(fact_349_ceiling__le__one,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)),one_one(int)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),one_one(X_a))) ) ) ).
fof(fact_350_number__of__less__ceiling,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [V_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),V_1)),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),V_1)),X_1)) ) ) ).
fof(fact_351_ceiling__le__number__of,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1,V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)),hAPP(int,int,number_number_of(int),V_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(int,X_a,number_number_of(X_a),V_1))) ) ) ).
fof(fact_352_ceiling__add__one,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),one_one(X_a))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),one_one(int)) ) ).
fof(fact_353_ceiling__add__number__of,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,V] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),hAPP(int,int,number_number_of(int),V)) ) ).
fof(fact_354_ceiling__diff__one,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),one_one(X_a))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),one_one(int)) ) ).
fof(fact_355_ceiling__diff__number__of,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,V] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),hAPP(int,int,number_number_of(int),V)) ) ).
fof(fact_356_mult__left__le__imp__le,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ) ).
fof(fact_357_mult__right__le__imp__le,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [A_1,C,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ) ).
fof(fact_358_mult__less__imp__less__left,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_359_mult__left__less__imp__less,axiom,
! [X_a] :
( linordered_semiring(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_360_mult__less__imp__less__right,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [A_1,C,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_361_mult__right__less__imp__less,axiom,
! [X_a] :
( linordered_semiring(X_a)
=> ! [A_1,C,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_362_mult__le__less__imp__less,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_363_mult__less__le__imp__less,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),D))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_364_mult__strict__mono_H,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_365_mult__strict__mono,axiom,
! [X_a] :
( linord20386208strict(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),D))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ) ) ).
fof(fact_366_mult__le__cancel__left__neg,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Aa,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Aa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Ba)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),Aa)) ) ) ) ).
fof(fact_367_mult__le__cancel__left__pos,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [Aa,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Aa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Ba)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba)) ) ) ) ).
fof(fact_368_add__nonpos__neg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_369_add__neg__nonpos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),zero_zero(X_a))) ) ) ) ).
fof(fact_370_add__strict__increasing2,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C))) ) ) ) ).
fof(fact_371_add__strict__increasing,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,C,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C))) ) ) ) ).
fof(fact_372_add__nonneg__pos,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).
fof(fact_373_add__pos__nonneg,axiom,
! [X_a] :
( ordere216010020id_add(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))) ) ) ) ).
fof(fact_374_mult__left__le__one__le,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),X)),X)) ) ) ) ) ).
fof(fact_375_mult__right__le__one__le,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),X)) ) ) ) ) ).
fof(fact_376_zero__less__two,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)))) ) ).
fof(fact_377_int__one__le__iff__zero__less,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) ) ).
fof(fact_378_pos__zmult__eq__1__iff,axiom,
! [N,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N) = one_one(int)
<=> ( M = one_one(int)
& N = one_one(int) ) ) ) ).
fof(fact_379_odd__less__0,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_1)),Z_1)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),zero_zero(int))) ) ).
fof(fact_380_convex__bound__le,axiom,
! [X_a] :
( linord1278240602ring_1(X_a)
=> ! [V,U,Y,X,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),U))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),V))
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),U),V) = one_one(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),U),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),V),Y))),A_1)) ) ) ) ) ) ) ).
fof(fact_381_le__imp__0__less,axiom,
! [Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z))) ) ).
fof(fact_382_self__quotient__aux1,axiom,
! [R_4,Q_2,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( A_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),R_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),A_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),Q_2)) ) ) ) ).
fof(fact_383_self__quotient__aux2,axiom,
! [R_4,Q_2,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( A_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),R_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Q_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_2),one_one(int))) ) ) ) ).
fof(fact_384_zdiv__mono2__neg__lemma,axiom,
! [B,Q_2,R_4,B_4,Q_5,R_5] :
( hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_2)),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_4),Q_5)),R_5)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_4),Q_5)),R_5)),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_5))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_4),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_5),Q_2)) ) ) ) ) ) ) ).
fof(fact_385_unique__quotient__lemma__neg,axiom,
! [B,Q_5,R_5,Q_2,R_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),Q_5)),R_5)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_2)),R_4)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_4),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_5))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_2),Q_5)) ) ) ) ) ).
fof(fact_386_zdiv__mono2__lemma,axiom,
! [B,Q_2,R_4,B_4,Q_5,R_5] :
( hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_2)),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_4),Q_5)),R_5)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_4),Q_5)),R_5)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_5),B_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_4),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_2),Q_5)) ) ) ) ) ) ) ).
fof(fact_387_unique__quotient__lemma,axiom,
! [B,Q_5,R_5,Q_2,R_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),Q_5)),R_5)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_2)),R_4)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_5))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_5),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_5),Q_2)) ) ) ) ) ).
fof(fact_388_q__neg__lemma,axiom,
! [B_4,Q_5,R_5] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_4),Q_5)),R_5)),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_5))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Q_5),zero_zero(int))) ) ) ) ).
fof(fact_389_q__pos__lemma,axiom,
! [B_4,Q_5,R_5] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_4),Q_5)),R_5)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_5),B_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q_5)) ) ) ) ).
fof(fact_390_sum__squares__gt__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y_1),Y_1))))
<=> ( ti(X_a,X_1) != zero_zero(X_a)
| ti(X_a,Y_1) != zero_zero(X_a) ) ) ) ).
fof(fact_391_not__sum__squares__lt__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X,Y] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Y))),zero_zero(X_a))) ) ).
fof(fact_392_sum__squares__le__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y_1),Y_1))),zero_zero(X_a)))
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ).
fof(fact_393_sum__squares__eq__zero__iff,axiom,
! [X_a] :
( linord581940658strict(X_a)
=> ! [X_1,Y_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y_1),Y_1)) = zero_zero(X_a)
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ).
fof(fact_394_sum__squares__ge__zero,axiom,
! [X_a] :
( linordered_ring(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Y)))) ) ).
fof(fact_395_xzgcd__linear,axiom,
! [M_1,R_4,S_4,T_6,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N_1))
=> ( hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),xzgcd,M_1),N_1) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),R_4),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),S_4),T_6))
=> 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_4),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_6),N_1)) ) ) ).
fof(fact_396_xzgcda__linear,axiom,
! [M_1,N_1,R_5,S_5,S_4,T_7,T_6,Rn,Sn_1,Tn_1,R_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),R_4))
=> ( hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M_1),N_1),R_5),R_4),S_5),S_4),T_7),T_6) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),Rn),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Sn_1),Tn_1))
=> ( R_5 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_5),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_7),N_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_4),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_6),N_1))
=> Rn = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Sn_1),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Tn_1),N_1)) ) ) ) ) ).
fof(fact_397_Nat__Transfer_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y))) ) ) ).
fof(fact_398_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y))) ) ) ).
fof(fact_399_pos__zmult__pos,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B)) ) ) ).
fof(fact_400_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),one_one(int))) ).
fof(fact_401_decr__mult__lemma,axiom,
! [K,P_1,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> ( ! [X_2] :
( hBOOL(hAPP(int,bool,P_1,X_2))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),D_1))) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ! [X_2] :
( hBOOL(hAPP(int,bool,P_1,X_2))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),D_1)))) ) ) ) ) ).
fof(fact_402_incr__mult__lemma,axiom,
! [K,P_1,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> ( ! [X_2] :
( hBOOL(hAPP(int,bool,P_1,X_2))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),D_1))) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ! [X_2] :
( hBOOL(hAPP(int,bool,P_1,X_2))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),D_1)))) ) ) ) ) ).
fof(fact_403_field__le__mult__one__interval,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [X,Y] :
( ! [Z_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Z_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z_2),X)),Y)) ) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).
fof(fact_404_divmod__int__relI,axiom,
! [A_1,B,Q_2,R_4] :
( A_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),Q_2)),R_4)
=> ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B)) ) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_4))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_4),zero_zero(int))) ) ) )
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),R_4))) ) ) ).
fof(fact_405_xzgcd__def,axiom,
! [M_1,N_1] : hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),xzgcd,M_1),N_1) = hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M_1),N_1),M_1),N_1),one_one(int)),zero_zero(int)),zero_zero(int)),one_one(int)) ).
fof(fact_406_divmod__int__rel__0,axiom,
! [B] :
( B != zero_zero(int)
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,zero_zero(int)),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)))) ) ).
fof(fact_407_self__remainder,axiom,
! [A_1,Q_2,R_4] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),A_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),R_4)))
=> ( A_1 != zero_zero(int)
=> R_4 = zero_zero(int) ) ) ).
fof(fact_408_unique__quotient,axiom,
! [Q_5,R_5,A_1,B,Q_2,R_4] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),R_4)))
=> ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_5),R_5)))
=> ( B != zero_zero(int)
=> Q_2 = Q_5 ) ) ) ).
fof(fact_409_unique__remainder,axiom,
! [Q_5,R_5,A_1,B,Q_2,R_4] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),R_4)))
=> ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_5),R_5)))
=> ( B != zero_zero(int)
=> R_4 = R_5 ) ) ) ).
fof(fact_410_self__quotient,axiom,
! [A_1,Q_2,R_4] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),A_1),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),R_4)))
=> ( A_1 != zero_zero(int)
=> Q_2 = one_one(int) ) ) ).
fof(fact_411_imp__le__cong,axiom,
! [P_2,P_1,X_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hBOOL(P_1) )
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hBOOL(P_2) ) ) ) ).
fof(fact_412_conj__le__cong,axiom,
! [P_2,P_1,X_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(P_1)
<=> hBOOL(P_2) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
& hBOOL(P_1) )
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
& hBOOL(P_2) ) ) ) ).
fof(fact_413_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),zero_zero(int))) ).
fof(fact_414_field__le__epsilon,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [X,Y] :
( ! [E] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),E))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),E))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ).
fof(fact_415_minf_I8_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_6] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Z_2))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),T_6),X_2)) ) ) ).
fof(fact_416_pinf_I8_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_6] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),T_6),X_2)) ) ) ).
fof(fact_417_minf_I6_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_6] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Z_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),T_6)) ) ) ).
fof(fact_418_pinf_I6_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [T_6] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_2))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),T_6)) ) ) ).
fof(fact_419_minusinfinity,axiom,
! [P_1,P1,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> ( ! [X_2,K_1] :
( hBOOL(hAPP(int,bool,P1,X_2))
<=> hBOOL(hAPP(int,bool,P1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_1)))) )
=> ( ? [Z_2] :
! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),Z_2))
=> ( hBOOL(hAPP(int,bool,P_1,X_2))
<=> hBOOL(hAPP(int,bool,P1,X_2)) ) )
=> ( ? [X1] : hBOOL(hAPP(int,bool,P1,X1))
=> ? [X1] : hBOOL(hAPP(int,bool,P_1,X1)) ) ) ) ) ).
fof(fact_420_plusinfinity,axiom,
! [P_1,P_2,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> ( ! [X_2,K_1] :
( hBOOL(hAPP(int,bool,P_2,X_2))
<=> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_1)))) )
=> ( ? [Z_2] :
! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),X_2))
=> ( hBOOL(hAPP(int,bool,P_1,X_2))
<=> hBOOL(hAPP(int,bool,P_2,X_2)) ) )
=> ( ? [X1] : hBOOL(hAPP(int,bool,P_2,X1))
=> ? [X1] : hBOOL(hAPP(int,bool,P_1,X1)) ) ) ) ) ).
fof(fact_421_tsub__def,axiom,
! [Y,X] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),X))
=> hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),X))
=> hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y) = zero_zero(int) ) ) ).
fof(fact_422_zfact_Osimps,axiom,
! [N_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),zero_zero(int)))
=> hAPP(int,int,zfact,N_1) = one_one(int) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),zero_zero(int)))
=> hAPP(int,int,zfact,N_1) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N_1),hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),N_1),one_one(int)))) ) ) ).
fof(fact_423_floor__le__number__of,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1,V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)),hAPP(int,int,number_number_of(int),V_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V_1)),one_one(X_a)))) ) ) ).
fof(fact_424_floor__zero,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> hAPP(X_a,int,archim2085082626_floor(X_a),zero_zero(X_a)) = zero_zero(int) ) ).
fof(fact_425_floor__less__cancel,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(X_a,int,archim2085082626_floor(X_a),Y)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ).
fof(fact_426_floor__mono,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(X_a,int,archim2085082626_floor(X_a),Y))) ) ) ).
fof(fact_427_floor__one,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> hAPP(X_a,int,archim2085082626_floor(X_a),one_one(X_a)) = one_one(int) ) ).
fof(fact_428_floor__number__of,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [V] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(int,X_a,number_number_of(X_a),V)) = hAPP(int,int,number_number_of(int),V) ) ).
fof(fact_429_floor__less__zero,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)),zero_zero(int)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),zero_zero(X_a))) ) ) ).
fof(fact_430_zero__le__floor,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1)) ) ) ).
fof(fact_431_zero__less__floor,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),X_1)) ) ) ).
fof(fact_432_floor__le__zero,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)),zero_zero(int)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),one_one(X_a))) ) ) ).
fof(fact_433_floor__less__one,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)),one_one(int)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),one_one(X_a))) ) ) ).
fof(fact_434_one__le__floor,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),X_1)) ) ) ).
fof(fact_435_floor__less__number__of,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1,V_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)),hAPP(int,int,number_number_of(int),V_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),hAPP(int,X_a,number_number_of(X_a),V_1))) ) ) ).
fof(fact_436_number__of__le__floor,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [V_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),V_1)),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),V_1)),X_1)) ) ) ).
fof(fact_437_floor__add__one,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),one_one(X_a))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),one_one(int)) ) ).
fof(fact_438_floor__add__number__of,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,V] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(int,int,number_number_of(int),V)) ) ).
fof(fact_439_floor__diff__one,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),one_one(X_a))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),one_one(int)) ) ).
fof(fact_440_floor__diff__number__of,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,V] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,number_number_of(X_a),V))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),hAPP(int,int,number_number_of(int),V)) ) ).
fof(fact_441_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y))) ) ) ).
fof(fact_442_tsub__eq,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),X))
=> hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y) ) ).
fof(fact_443_number__of__less__floor,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [V_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),V_1)),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V_1)),one_one(X_a))),X_1)) ) ) ).
fof(fact_444_le__mult__floor,axiom,
! [B,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(real,int,archim2085082626_floor(real),A_1)),hAPP(real,int,archim2085082626_floor(real),B))),hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B)))) ) ) ).
fof(fact_445_posDivAlg__correct,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B))) ) ) ).
fof(fact_446_incr__lemma,axiom,
! [Z,X,D] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z),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))),one_one(int))),D)))) ) ).
fof(fact_447_decr__lemma,axiom,
! [X,Z,D] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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))),one_one(int))),D))),Z)) ) ).
fof(fact_448_Ints__odd__less__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),ring_1_Ints(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),Aa)),Aa)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a))) ) ) ) ).
fof(fact_449_zminus1__lemma,axiom,
! [Aa,Ba,Qa,R_1] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,Aa),Ba),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Qa),R_1)))
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,uminus_uminus(int),Aa)),Ba),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),R_1),zero_zero(int))),hAPP(int,int,uminus_uminus(int),Qa)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,uminus_uminus(int),Qa)),one_one(int)))),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),R_1),zero_zero(int))),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Ba),R_1))))) ) ).
fof(fact_450_d22set__mem,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ba))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),Aa))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),d22set,Aa))) ) ) ).
fof(fact_451_zero__le__ceiling,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X_1)) ) ) ).
fof(fact_452_ceiling__less__zero,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)),zero_zero(int)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(int,X_a,number_number_of(X_a),min))) ) ) ).
fof(fact_453_split__zmod,axiom,
! [P_1,N,K] :
( hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),N),K)))
<=> ( ( K = zero_zero(int)
=> hBOOL(hAPP(int,bool,P_1,N)) )
& ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ! [I_2,J_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),K))
& N = 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(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ! [I_2,J_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),J_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),zero_zero(int)))
& N = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) )
=> hBOOL(hAPP(int,bool,P_1,J_1)) ) ) ) ) ).
fof(fact_454_real__mult__le__cancel__iff1,axiom,
! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X_1),Z_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y_1),Z_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1)) ) ) ).
fof(fact_455_real__mult__le__cancel__iff2,axiom,
! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z_1),X_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z_1),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1)) ) ) ).
fof(fact_456_real__le__eq__diff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X_1),Y_1)),zero_zero(real))) ) ).
fof(fact_457_zmod__zminus2__not__zero,axiom,
! [K_4,L_4] :
( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),hAPP(int,int,uminus_uminus(int),L_4)) != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4) != zero_zero(int) ) ).
fof(fact_458_zmod__zminus1__not__zero,axiom,
! [K_4,L_4] :
( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),K_4)),L_4) != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4) != zero_zero(int) ) ).
fof(fact_459_real__mult__left__cancel,axiom,
! [Aa,Ba,C_1] :
( C_1 != zero_zero(real)
=> ( hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C_1),Aa) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C_1),Ba)
<=> Aa = Ba ) ) ).
fof(fact_460_real__mult__right__cancel,axiom,
! [Aa,Ba,C_1] :
( C_1 != zero_zero(real)
=> ( hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Aa),C_1) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Ba),C_1)
<=> Aa = Ba ) ) ).
fof(fact_461_real__two__squares__add__zero__iff,axiom,
! [X_1,Y_1] :
( hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X_1),X_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y_1),Y_1)) = zero_zero(real)
<=> ( X_1 = zero_zero(real)
& Y_1 = zero_zero(real) ) ) ).
fof(fact_462_real__mult__less__mono2,axiom,
! [X,Y,Z] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z),Y))) ) ) ).
fof(fact_463_real__mult__order,axiom,
! [Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y))) ) ) ).
fof(fact_464_real__mult__less__iff1,axiom,
! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Z_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X_1),Z_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y_1),Z_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1)) ) ) ).
fof(fact_465_real__zero__not__eq__one,axiom,
zero_zero(real) != one_one(real) ).
fof(fact_466_mod__minus__cong,axiom,
! [X_a] :
( ring_div(X_a)
=> ! [A_1,B,A_5] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_5),B)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_5)),B) ) ) ).
fof(fact_467_mod__mod__trivial,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) ) ).
fof(fact_468_zminus__zmod,axiom,
! [X,M_1] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M_1))),M_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),X)),M_1) ).
fof(fact_469_zmod__zminus__zminus,axiom,
! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) ).
fof(fact_470_mod__minus__eq,axiom,
! [X_a] :
( ring_div(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))),B) ) ).
fof(fact_471_zmod__zminus2,axiom,
! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),B)) ).
fof(fact_472_arith__simps_I30_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,uminus_uminus(int),W)) ) ).
fof(fact_473_number__of__minus,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,uminus_uminus(int),W)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),W)) ) ).
fof(fact_474_Int_OMin__def,axiom,
min = hAPP(int,int,uminus_uminus(int),one_one(int)) ).
fof(fact_475_mult__Min,axiom,
! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),min),K_4) = hAPP(int,int,uminus_uminus(int),K_4) ).
fof(fact_476_real__le__refl,axiom,
! [W] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),W)) ).
fof(fact_477_real__le__linear,axiom,
! [Z,W] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z),W))
| hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z)) ) ).
fof(fact_478_real__add__left__mono,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Z),X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Z),Y))) ) ).
fof(fact_479_real__le__trans,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_1),J_2))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),J_2),K_4))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),I_1),K_4)) ) ) ).
fof(fact_480_real__le__antisym,axiom,
! [Z,W] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Z),W))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),W),Z))
=> Z = W ) ) ).
fof(fact_481_abs__ge__minus__self,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),A_1))) ) ).
fof(fact_482_abs__le__iff,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),Aa)),Ba))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),Ba)) ) ) ) ).
fof(fact_483_abs__leI,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),B)) ) ) ) ).
fof(fact_484_abs__le__D2,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B)) ) ) ).
fof(fact_485_compl__eq__compl__iff,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [X_1,Y_1] :
( hAPP(X_a,X_a,uminus_uminus(X_a),X_1) = hAPP(X_a,X_a,uminus_uminus(X_a),Y_1)
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ).
fof(fact_486_uminus__apply,axiom,
! [X_b,X_a] :
( uminus(X_a)
=> ! [A_2,X_1] : hAPP(X_b,X_a,hAPP(fun(X_b,X_a),fun(X_b,X_a),uminus_uminus(fun(X_b,X_a)),A_2),X_1) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_b,X_a,A_2,X_1)) ) ).
fof(fact_487_double__compl,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [X] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = ti(X_a,X) ) ).
fof(fact_488_fun__Compl__def,axiom,
! [X_a,X_b] :
( uminus(X_b)
=> ! [A_2,X_2] : hAPP(X_a,X_b,hAPP(fun(X_a,X_b),fun(X_a,X_b),uminus_uminus(fun(X_a,X_b)),A_2),X_2) = hAPP(X_b,X_b,uminus_uminus(X_b),hAPP(X_a,X_b,A_2,X_2)) ) ).
fof(fact_489_Ints__minus,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),ring_1_Ints(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),ring_1_Ints(X_a))) ) ) ).
fof(fact_490_real__add__mult__distrib,axiom,
! [Z1,Z2,W] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Z1),Z2)),W) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z1),W)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z2),W)) ).
fof(fact_491_real__mult__assoc,axiom,
! [Z1,Z2,Z3] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z1),Z2)),Z3) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z2),Z3)) ).
fof(fact_492_neg__equal__iff__equal,axiom,
! [X_a] :
( group_add(X_a)
=> ! [Aa,Ba] :
( hAPP(X_a,X_a,uminus_uminus(X_a),Aa) = hAPP(X_a,X_a,uminus_uminus(X_a),Ba)
<=> ti(X_a,Aa) = ti(X_a,Ba) ) ) ).
fof(fact_493_minus__equation__iff,axiom,
! [X_a] :
( group_add(X_a)
=> ! [Aa,Ba] :
( hAPP(X_a,X_a,uminus_uminus(X_a),Aa) = ti(X_a,Ba)
<=> hAPP(X_a,X_a,uminus_uminus(X_a),Ba) = ti(X_a,Aa) ) ) ).
fof(fact_494_equation__minus__iff,axiom,
! [X_a] :
( group_add(X_a)
=> ! [Aa,Ba] :
( ti(X_a,Aa) = hAPP(X_a,X_a,uminus_uminus(X_a),Ba)
<=> ti(X_a,Ba) = hAPP(X_a,X_a,uminus_uminus(X_a),Aa) ) ) ).
fof(fact_495_real__mult__commute,axiom,
! [Z,W] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Z),W) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),W),Z) ).
fof(fact_496_abs__minus__cancel,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = hAPP(X_a,X_a,abs_abs(X_a),A_1) ) ).
fof(fact_497_abs__idempotent,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)) = hAPP(X_a,X_a,abs_abs(X_a),A_1) ) ).
fof(fact_498_zminus__zminus,axiom,
! [Z] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,uminus_uminus(int),Z)) = Z ).
fof(fact_499_minus__minus,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = ti(X_a,A_1) ) ).
fof(fact_500_real__mult__1,axiom,
! [Z] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),one_one(real)),Z) = Z ).
fof(fact_501_abs__less__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),Aa)),Ba))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),Ba)) ) ) ) ).
fof(fact_502_less__eq__real__def,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1))
<=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1))
| X_1 = Y_1 ) ) ).
fof(fact_503_real__less__def,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1))
<=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1))
& X_1 != Y_1 ) ) ).
fof(fact_504_abs__minus__le__zero,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1))),zero_zero(X_a))) ) ).
fof(fact_505_abs__of__nonpos,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a)))
=> hAPP(X_a,X_a,abs_abs(X_a),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),A_1) ) ) ).
fof(fact_506_abs__of__neg,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> hAPP(X_a,X_a,abs_abs(X_a),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),A_1) ) ) ).
fof(fact_507_abs__if,axiom,
! [X_a] :
( abs_if(X_a)
=> ! [A_1] :
( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> hAPP(X_a,X_a,abs_abs(X_a),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),A_1) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> hAPP(X_a,X_a,abs_abs(X_a),A_1) = ti(X_a,A_1) ) ) ) ).
fof(fact_508_abs__minus__one,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> hAPP(X_a,X_a,abs_abs(X_a),hAPP(int,X_a,number_number_of(X_a),min)) = one_one(X_a) ) ).
fof(fact_509_minus__number__of__mult,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),W))),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,uminus_uminus(int),W))),Z) ) ).
fof(fact_510_mult__minus1,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),min)),Z) = hAPP(X_a,X_a,uminus_uminus(X_a),Z) ) ).
fof(fact_511_mult__minus1__right,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),hAPP(int,X_a,number_number_of(X_a),min)) = hAPP(X_a,X_a,uminus_uminus(X_a),Z) ) ).
fof(fact_512_number__of__Min,axiom,
! [X_a] :
( number_ring(X_a)
=> hAPP(int,X_a,number_number_of(X_a),min) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ).
fof(fact_513_arith__simps_I31_J,axiom,
! [X_a] :
( number_ring(X_a)
=> hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),min) ) ).
fof(fact_514_zabs__def,axiom,
! [I_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),zero_zero(int)))
=> hAPP(int,int,abs_abs(int),I_1) = hAPP(int,int,uminus_uminus(int),I_1) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),zero_zero(int)))
=> hAPP(int,int,abs_abs(int),I_1) = I_1 ) ) ).
fof(fact_515_zmod__minus1__right,axiom,
! [A_1] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,number_number_of(int),min)) = zero_zero(int) ).
fof(fact_516_zmod__zminus1__eq__if,axiom,
! [A_1,B] :
( ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),B) = zero_zero(int) )
& ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,uminus_uminus(int),A_1)),B) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) ) ) ).
fof(fact_517_zmod__zminus2__eq__if,axiom,
! [A_1,B] :
( ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = zero_zero(int) )
& ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B) ) ) ).
fof(fact_518_abs__eq__1__iff,axiom,
! [Z_1] :
( hAPP(int,int,abs_abs(int),Z_1) = one_one(int)
<=> ( Z_1 = one_one(int)
| Z_1 = hAPP(int,int,number_number_of(int),min) ) ) ).
fof(fact_519_mod__0,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).
fof(fact_520_mod__by__0,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),zero_zero(X_a)) = ti(X_a,A_1) ) ).
fof(fact_521_mod__self,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),A_1) = zero_zero(X_a) ) ).
fof(fact_522_abs__eq__0,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [Aa] :
( hAPP(X_a,X_a,abs_abs(X_a),Aa) = zero_zero(X_a)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_523_abs__zero,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> hAPP(X_a,X_a,abs_abs(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_524_abs__le__D1,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ).
fof(fact_525_abs__ge__self,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),hAPP(X_a,X_a,abs_abs(X_a),A_1))) ) ).
fof(fact_526_mod__mult__cong,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [B,B_4,A_1,C,A_5] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_5),C)
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_4),C)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_5),B_4)),C) ) ) ) ).
fof(fact_527_zmod__simps_I4_J,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,C,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C)),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C) ) ).
fof(fact_528_mod__mult__mult2,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,C,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),C) ) ).
fof(fact_529_mod__mult__mult1,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [C,A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)) ) ).
fof(fact_530_mod__mult__eq,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C))),C) ) ).
fof(fact_531_mod__mult__left__eq,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C)),B)),C) ) ).
fof(fact_532_mod__mult__right__eq,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C))),C) ) ).
fof(fact_533_mod__add__cong,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [B,B_4,A_1,C,A_5] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_5),C)
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_4),C)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_5),B_4)),C) ) ) ) ).
fof(fact_534_zmod__simps_I1_J,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,C,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C)),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) ) ).
fof(fact_535_zmod__simps_I2_J,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C))),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) ) ).
fof(fact_536_mod__add__eq,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C))),C) ) ).
fof(fact_537_mod__add__left__eq,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C)),B)),C) ) ).
fof(fact_538_mod__add__right__eq,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C))),C) ) ).
fof(fact_539_mod__add__self1,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [B,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) ) ).
fof(fact_540_mod__add__self2,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) ) ).
fof(fact_541_abs__mult__self,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),A_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1) ) ).
fof(fact_542_abs__mult,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ).
fof(fact_543_abs__add__abs,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ).
fof(fact_544_mod__diff__cong,axiom,
! [X_a] :
( ring_div(X_a)
=> ! [B,B_4,A_1,C,A_5] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_5),C)
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B_4),C)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_5),B_4)),C) ) ) ) ).
fof(fact_545_mod__diff__eq,axiom,
! [X_a] :
( ring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C))),C) ) ).
fof(fact_546_mod__diff__left__eq,axiom,
! [X_a] :
( ring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C)),B)),C) ) ).
fof(fact_547_mod__diff__right__eq,axiom,
! [X_a] :
( ring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),C))),C) ) ).
fof(fact_548_abs__one,axiom,
! [X_a] :
( linordered_idom(X_a)
=> hAPP(X_a,X_a,abs_abs(X_a),one_one(X_a)) = one_one(X_a) ) ).
fof(fact_549_abs__minus__commute,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)) ) ).
fof(fact_550_floor__minus,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(int,int,uminus_uminus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)) ) ).
fof(fact_551_ceiling__minus,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(int,int,uminus_uminus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)) ) ).
fof(fact_552_ceiling__def,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] : hAPP(X_a,int,archimedean_ceiling(X_a),X) = hAPP(int,int,uminus_uminus(int),hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X))) ) ).
fof(fact_553_neg__equal__zero,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( hAPP(X_a,X_a,uminus_uminus(X_a),Aa) = ti(X_a,Aa)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_554_neg__equal__0__iff__equal,axiom,
! [X_a] :
( group_add(X_a)
=> ! [Aa] :
( hAPP(X_a,X_a,uminus_uminus(X_a),Aa) = zero_zero(X_a)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_555_equal__neg__zero,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( ti(X_a,Aa) = hAPP(X_a,X_a,uminus_uminus(X_a),Aa)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_556_neg__0__equal__iff__equal,axiom,
! [X_a] :
( group_add(X_a)
=> ! [Aa] :
( zero_zero(X_a) = hAPP(X_a,X_a,uminus_uminus(X_a),Aa)
<=> zero_zero(X_a) = ti(X_a,Aa) ) ) ).
fof(fact_557_minus__zero,axiom,
! [X_a] :
( group_add(X_a)
=> hAPP(X_a,X_a,uminus_uminus(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_558_compl__le__swap2,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y)) ) ) ).
fof(fact_559_compl__le__swap1,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),hAPP(X_a,X_a,uminus_uminus(X_a),X)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y))) ) ) ).
fof(fact_560_compl__mono,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),hAPP(X_a,X_a,uminus_uminus(X_a),X))) ) ) ).
fof(fact_561_compl__le__compl__iff,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X_1)),hAPP(X_a,X_a,uminus_uminus(X_a),Y_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),X_1)) ) ) ).
fof(fact_562_le__imp__neg__le,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B)),hAPP(X_a,X_a,uminus_uminus(X_a),A_1))) ) ) ).
fof(fact_563_neg__le__iff__le,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Ba,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Ba)),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba)) ) ) ).
fof(fact_564_minus__le__iff,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),Ba))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Ba)),Aa)) ) ) ).
fof(fact_565_le__minus__iff,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),hAPP(X_a,X_a,uminus_uminus(X_a),Ba)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),hAPP(X_a,X_a,uminus_uminus(X_a),Aa))) ) ) ).
fof(fact_566_compl__less__swap2,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y)),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y)) ) ) ).
fof(fact_567_compl__less__swap1,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),hAPP(X_a,X_a,uminus_uminus(X_a),X)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y))) ) ) ).
fof(fact_568_compl__less__compl__iff,axiom,
! [X_a] :
( boolean_algebra(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X_1)),hAPP(X_a,X_a,uminus_uminus(X_a),Y_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),X_1)) ) ) ).
fof(fact_569_neg__less__iff__less,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Ba,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Ba)),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba)) ) ) ).
fof(fact_570_minus__less__iff,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),Ba))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Ba)),Aa)) ) ) ).
fof(fact_571_less__minus__iff,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),hAPP(X_a,X_a,uminus_uminus(X_a),Ba)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),hAPP(X_a,X_a,uminus_uminus(X_a),Aa))) ) ) ).
fof(fact_572_square__eq__iff,axiom,
! [X_a] :
( idom(X_a)
=> ! [Aa,Ba] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Aa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),Ba)
<=> ( ti(X_a,Aa) = ti(X_a,Ba)
| ti(X_a,Aa) = hAPP(X_a,X_a,uminus_uminus(X_a),Ba) ) ) ) ).
fof(fact_573_minus__mult__minus,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) ) ).
fof(fact_574_minus__mult__commute,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_575_mult__left_Ominus,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) ) ).
fof(fact_576_mult_Ominus__left,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) ) ).
fof(fact_577_mult__right_Ominus,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Xa_1,X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_1),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_1),X)) ) ).
fof(fact_578_mult_Ominus__right,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) ) ).
fof(fact_579_minus__mult__left,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) ) ).
fof(fact_580_minus__mult__right,axiom,
! [X_a] :
( ring(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_581_minus__add__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)) = ti(X_a,B) ) ).
fof(fact_582_add__minus__cancel,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B)) = ti(X_a,B) ) ).
fof(fact_583_minus__add,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),B)),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) ) ).
fof(fact_584_minus__add__distrib,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_585_zmod__self,axiom,
! [A_1] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),A_1) = zero_zero(int) ).
fof(fact_586_zmod__zero,axiom,
! [B] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),zero_zero(int)),B) = zero_zero(int) ).
fof(fact_587_minus__diff__minus,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) ) ).
fof(fact_588_minus__diff__eq,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1) ) ).
fof(fact_589_zmod__simps_I3_J,axiom,
! [A_1,B,C] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C))),C) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C) ).
fof(fact_590_zmod__zmult1__eq,axiom,
! [A_1,B,C] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C))),C) ).
fof(fact_591_zminus__0,axiom,
hAPP(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).
fof(fact_592_zdiff__zmod__left,axiom,
! [X,M_1,Y] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M_1)),Y)),M_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y)),M_1) ).
fof(fact_593_zdiff__zmod__right,axiom,
! [X,Y,M_1] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Y),M_1))),M_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y)),M_1) ).
fof(fact_594_zmult__zminus,axiom,
! [Z,W] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,uminus_uminus(int),Z)),W) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z),W)) ).
fof(fact_595_rel__simps_I7_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),min)) ).
fof(fact_596_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_597_rel__simps_I24_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),min)) ).
fof(fact_598_zminus__zadd__distrib,axiom,
! [Z,W] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),W)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,uminus_uminus(int),Z)),hAPP(int,int,uminus_uminus(int),W)) ).
fof(fact_599_bounded__bilinear_Ominus__left,axiom,
! [X_a,X_c,X_b] :
( ( real_normed_vector(X_b)
& real_normed_vector(X_c)
& real_normed_vector(X_a) )
=> ! [Aa,Ba,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),Ba) = hAPP(X_c,X_c,uminus_uminus(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)) ) ) ).
fof(fact_600_bounded__bilinear_Ominus__right,axiom,
! [X_a,X_c,X_b] :
( ( real_normed_vector(X_b)
& real_normed_vector(X_c)
& real_normed_vector(X_a) )
=> ! [Aa,Ba,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),hAPP(X_b,X_b,uminus_uminus(X_b),Ba)) = hAPP(X_c,X_c,uminus_uminus(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)) ) ) ).
fof(fact_601_abs__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X] :
( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),X)),zero_zero(X_a)))
=> hAPP(X_a,X_a,abs_abs(X_a),hAPP(int,X_a,number_number_of(X_a),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,number_number_of(X_a),X)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),X)),zero_zero(X_a)))
=> hAPP(X_a,X_a,abs_abs(X_a),hAPP(int,X_a,number_number_of(X_a),X)) = hAPP(int,X_a,number_number_of(X_a),X) ) ) ) ).
fof(fact_602_Ints__0,axiom,
! [X_a] :
( ring_1(X_a)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),zero_zero(X_a)),ring_1_Ints(X_a))) ) ).
fof(fact_603_Ints__mult,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [Ba,Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),ring_1_Ints(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ba),ring_1_Ints(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Ba)),ring_1_Ints(X_a))) ) ) ) ).
fof(fact_604_Ints__add,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [Ba,Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),ring_1_Ints(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ba),ring_1_Ints(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Ba)),ring_1_Ints(X_a))) ) ) ) ).
fof(fact_605_Ints__number__of,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),ring_1_Ints(X_a))) ) ).
fof(fact_606_Ints__1,axiom,
! [X_a] :
( ring_1(X_a)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),one_one(X_a)),ring_1_Ints(X_a))) ) ).
fof(fact_607_Ints__diff,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [Ba,Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),ring_1_Ints(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ba),ring_1_Ints(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba)),ring_1_Ints(X_a))) ) ) ) ).
fof(fact_608_mod__mult__self1__is__0,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [B,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),B) = zero_zero(X_a) ) ).
fof(fact_609_mod__mult__self2__is__0,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),B) = zero_zero(X_a) ) ).
fof(fact_610_abs__of__nonneg,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> hAPP(X_a,X_a,abs_abs(X_a),A_1) = ti(X_a,A_1) ) ) ).
fof(fact_611_abs__le__zero__iff,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),Aa)),zero_zero(X_a)))
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_612_abs__ge__zero,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,abs_abs(X_a),A_1))) ) ).
fof(fact_613_abs__of__pos,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hAPP(X_a,X_a,abs_abs(X_a),A_1) = ti(X_a,A_1) ) ) ).
fof(fact_614_zero__less__abs__iff,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,abs_abs(X_a),Aa)))
<=> ti(X_a,Aa) != zero_zero(X_a) ) ) ).
fof(fact_615_abs__not__less__zero,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),zero_zero(X_a))) ) ).
fof(fact_616_mod__by__1,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),one_one(X_a)) = zero_zero(X_a) ) ).
fof(fact_617_abs__mult__less,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [B,D,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),B)),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),D))) ) ) ) ).
fof(fact_618_mod__mult__self1,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,C,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) ) ).
fof(fact_619_mod__mult__self2,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B) ) ).
fof(fact_620_abs__triangle__ineq,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))) ) ).
fof(fact_621_abs__triangle__ineq3,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) ) ).
fof(fact_622_abs__triangle__ineq2,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) ) ).
fof(fact_623_abs__triangle__ineq2__sym,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)))) ) ).
fof(fact_624_abs__diff__less__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,Aa,R_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),Aa))),R_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),R_1)),X_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),R_1))) ) ) ) ).
fof(fact_625_d22set__le__swap,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),Ba))
=> ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),d22set,Aa))) ) ).
fof(fact_626_minus__le__self__iff,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),Aa))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa)) ) ) ).
fof(fact_627_neg__le__0__iff__le,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa)) ) ) ).
fof(fact_628_le__minus__self__iff,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_629_neg__0__le__iff__le,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_630_d22set__le,axiom,
! [Ba,Aa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),d22set,Aa)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),Aa)) ) ).
fof(fact_631_neg__less__nonneg,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),Aa))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Aa)) ) ) ).
fof(fact_632_neg__less__0__iff__less,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Aa)) ) ) ).
fof(fact_633_less__minus__self__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_634_neg__0__less__iff__less,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_635_right__minus,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = zero_zero(X_a) ) ).
fof(fact_636_eq__neg__iff__add__eq__0,axiom,
! [X_a] :
( group_add(X_a)
=> ! [Aa,Ba] :
( ti(X_a,Aa) = hAPP(X_a,X_a,uminus_uminus(X_a),Ba)
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Ba) = zero_zero(X_a) ) ) ).
fof(fact_637_left__minus,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),A_1) = zero_zero(X_a) ) ).
fof(fact_638_ab__left__minus,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),A_1) = zero_zero(X_a) ) ).
fof(fact_639_add__eq__0__iff,axiom,
! [X_a] :
( group_add(X_a)
=> ! [X_1,Y_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Y_1) = zero_zero(X_a)
<=> ti(X_a,Y_1) = hAPP(X_a,X_a,uminus_uminus(X_a),X_1) ) ) ).
fof(fact_640_minus__unique,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) = zero_zero(X_a)
=> hAPP(X_a,X_a,uminus_uminus(X_a),A_1) = ti(X_a,B) ) ) ).
fof(fact_641_diff__0,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),zero_zero(X_a)),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),A_1) ) ).
fof(fact_642_square__eq__1__iff,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [X_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),X_1) = one_one(X_a)
<=> ( ti(X_a,X_1) = one_one(X_a)
| ti(X_a,X_1) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ) ).
fof(fact_643_comm__ring__1__class_Onormalizing__ring__rules_I1_J,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X] : hAPP(X_a,X_a,uminus_uminus(X_a),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) ) ).
fof(fact_644_diff__minus__eq__add,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B) ) ).
fof(fact_645_ab__diff__minus,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_646_Groups_Odiff__def,axiom,
! [X_a] :
( group_add(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_647_comm__ring__1__class_Onormalizing__ring__rules_I2_J,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) ) ).
fof(fact_648_pos__mod__bound,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B)) ) ).
fof(fact_649_neg__mod__bound,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))) ) ).
fof(fact_650_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y))) ) ) ).
fof(fact_651_zmod__le__nonneg__dividend,axiom,
! [K_4,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M_1),K_4)),M_1)) ) ).
fof(fact_652_zmod__eq__0__iff,axiom,
! [M,D_1] :
( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M),D_1) = zero_zero(int)
<=> ? [Q_3] : M = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_1),Q_3) ) ).
fof(fact_653_abs__zmult__eq__1,axiom,
! [M_1,N_1] :
( hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),N_1)) = one_one(int)
=> hAPP(int,int,abs_abs(int),M_1) = one_one(int) ) ).
fof(fact_654_zadd__zminus__inverse2,axiom,
! [Z] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,uminus_uminus(int),Z)),Z) = zero_zero(int) ).
fof(fact_655_bin__less__0__simps_I2_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),zero_zero(int))) ).
fof(fact_656_diff__int__def__symmetric,axiom,
! [Z,W] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),hAPP(int,int,uminus_uminus(int),W)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),W) ).
fof(fact_657_diff__int__def,axiom,
! [Z,W] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),W) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),hAPP(int,int,uminus_uminus(int),W)) ).
fof(fact_658_zmod__minus1,axiom,
! [B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),min)),B) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),one_one(int)) ) ).
fof(fact_659_abs__eq__mult,axiom,
! [X_a] :
( ordered_ring_abs(X_a)
=> ! [B,A_1] :
( ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),zero_zero(X_a))) )
& ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),zero_zero(X_a))) ) )
=> hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ) ).
fof(fact_660_abs__mult__pos,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,abs_abs(X_a),Y)),X) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),X)) ) ) ).
fof(fact_661_abs__diff__triangle__ineq,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B,C,D] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D)))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),C))),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),D))))) ) ).
fof(fact_662_abs__triangle__ineq4,axiom,
! [X_a] :
( ordere142940540dd_abs(X_a)
=> ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)))) ) ).
fof(fact_663_Ints__double__eq__0__iff,axiom,
! [X_a] :
( ring_char_0(X_a)
=> ! [Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),ring_1_Ints(X_a)))
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Aa) = zero_zero(X_a)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ) ).
fof(fact_664_zabs__less__one__iff,axiom,
! [Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,abs_abs(int),Z_1)),one_one(int)))
<=> Z_1 = zero_zero(int) ) ).
fof(fact_665_mod__neg__neg__trivial,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = A_1 ) ) ).
fof(fact_666_neg__mod__conj,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),zero_zero(int)))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))) ) ) ).
fof(fact_667_neg__mod__sign,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),zero_zero(int))) ) ).
fof(fact_668_mod__pos__pos__trivial,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = A_1 ) ) ).
fof(fact_669_pos__mod__conj,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B)) ) ) ).
fof(fact_670_pos__mod__sign,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))) ) ).
fof(fact_671_posDivAlg__0,axiom,
! [B] : hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,zero_zero(int)),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)) ).
fof(fact_672_diff__number__of__eq,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),W))) ) ).
fof(fact_673_pos__zmult__eq__1__iff__lemma,axiom,
! [M_1,N_1] :
( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),N_1) = one_one(int)
=> ( M_1 = one_one(int)
| M_1 = hAPP(int,int,number_number_of(int),min) ) ) ).
fof(fact_674_zmult__eq__1__iff,axiom,
! [M,N] :
( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N) = one_one(int)
<=> ( ( M = one_one(int)
& N = one_one(int) )
| ( M = hAPP(int,int,number_number_of(int),min)
& N = hAPP(int,int,number_number_of(int),min) ) ) ) ).
fof(fact_675_minus__numeral__code_I6_J,axiom,
! [V,W] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,number_number_of(int),V)),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),hAPP(int,int,uminus_uminus(int),W))) ).
fof(fact_676_divmod__int__rel__mod,axiom,
! [A_1,B,Q_2,R_4] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),R_4)))
=> ( B != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = R_4 ) ) ).
fof(fact_677_d22set__g__1,axiom,
! [Ba,Aa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),d22set,Aa)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ba)) ) ).
fof(fact_678_mod__pos__neg__trivial,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),zero_zero(int)))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B) ) ) ).
fof(fact_679_add__number__of__diff2,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V,C,W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),C),hAPP(int,X_a,number_number_of(X_a),W))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),W)))),C) ) ).
fof(fact_680_Ints__odd__nonzero,axiom,
! [X_a] :
( ring_char_0(X_a)
=> ! [Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),ring_1_Ints(X_a)))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),Aa)),Aa) != zero_zero(X_a) ) ) ).
fof(fact_681_divmod__int__rel__mod__eq,axiom,
! [A_8,B_7,Q_4,Y] :
( A_8 = 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_4)),Y)
=> ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_7))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),B_7)) ) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_7))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_7),Y))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int))) ) ) )
=> ( B_7 != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_8),B_7) = Y ) ) ) ).
fof(fact_682_zmult2__lemma__aux2,axiom,
! [Q_2,B,R_4,C] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_4),zero_zero(int)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_2),C))),R_4)),zero_zero(int))) ) ) ) ).
fof(fact_683_zmult2__lemma__aux1,axiom,
! [Q_2,B,R_4,C] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),R_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_4),zero_zero(int)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_2),C))),R_4))) ) ) ) ).
fof(fact_684_zmult2__lemma__aux4,axiom,
! [Q_2,B,R_4,C] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_2),C))),R_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C))) ) ) ) ).
fof(fact_685_zmult2__lemma__aux3,axiom,
! [Q_2,B,R_4,C] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_4),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_2),C))),R_4))) ) ) ) ).
fof(fact_686_zmod__eq__0D,axiom,
! [M_6,D_4] :
( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M_6),D_4) = zero_zero(int)
=> ? [Q_3] : M_6 = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_4),Q_3) ) ).
fof(fact_687_pdivmod__posDivAlg,axiom,
! [K_4,L_4] :
( ( L_4 = zero_zero(int)
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),hAPP(int,int,abs_abs(int),K_4)) )
& ( L_4 != zero_zero(int)
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,abs_abs(int),L_4)) ) ) ).
fof(fact_688_z3mod__def,axiom,
! [K_4,L_4] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4))
=> hAPP(int,int,hAPP(int,fun(int,int),z3mod,K_4),L_4) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4))
=> hAPP(int,int,hAPP(int,fun(int,int),z3mod,K_4),L_4) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),hAPP(int,int,uminus_uminus(int),L_4)) ) ) ).
fof(fact_689_xzgcda_Osimps,axiom,
! [M_1,N_1,R_5,S_5,S_4,T_7,T_6,R_4] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_4),zero_zero(int)))
=> hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M_1),N_1),R_5),R_4),S_5),S_4),T_7),T_6) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),R_5),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),S_5),T_7)) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_4),zero_zero(int)))
=> hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M_1),N_1),R_5),R_4),S_5),S_4),T_7),T_6) = hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M_1),N_1),R_4),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),R_5),R_4)),S_4),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),S_5),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_5),R_4)),S_4))),T_6),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),T_7),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_5),R_4)),T_6))) ) ) ).
fof(fact_690_nat0__intermed__int__val,axiom,
! [K,F,N] :
( ! [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),N))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,N)))
=> ? [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),N))
& hAPP(nat,int,F,I_2) = K ) ) ) ) ).
fof(fact_691_int__val__lemma,axiom,
! [K,F,N] :
( ! [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),N))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,zero_zero(nat))),K))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,N)))
=> ? [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),N))
& hAPP(nat,int,F,I_2) = K ) ) ) ) ).
fof(fact_692_nat__intermed__int__val,axiom,
! [K,F,N,M] :
( ! [I_2] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),I_2))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),N)) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,F,M)),K))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),hAPP(nat,int,F,N)))
=> ? [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),I_2))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),N))
& hAPP(nat,int,F,I_2) = K ) ) ) ) ) ).
fof(fact_693_negDivAlg__minus1,axiom,
! [B] : hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),min)),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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),one_one(int))) ).
fof(fact_694_zmult2__lemma,axiom,
! [C,A_1,B,Q_2,R_4] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),R_4)))
=> ( B != zero_zero(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C))
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Q_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),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Q_2),C))),R_4)))) ) ) ) ).
fof(fact_695_not__real__square__gt__zero,axiom,
! [X_1] :
( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X_1),X_1)))
<=> X_1 = zero_zero(real) ) ).
fof(fact_696_mod__le__divisor,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)),N_1)) ) ).
fof(fact_697_mod__eq__0__iff,axiom,
! [M,D_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),D_1) = zero_zero(nat)
<=> ? [Q_3] : M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),D_1),Q_3) ) ).
fof(fact_698_mult__eq__if,axiom,
! [N_1,M_1] :
( ( M_1 = zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = zero_zero(nat) )
& ( M_1 != zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),one_one(nat))),N_1)) ) ) ).
fof(fact_699_mod__if,axiom,
! [M_1,N_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1) = M_1 )
& ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),N_1) ) ) ).
fof(fact_700_mod__geq,axiom,
! [M_1,N_1] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),N_1) ) ).
fof(fact_701_mod__less__divisor,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)),N_1)) ) ).
fof(fact_702_mod__less,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1) = M_1 ) ).
fof(fact_703_mod__lemma,axiom,
! [Q_2,R_4,B,C] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_4),B))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Q_2),C))),R_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C))) ) ) ).
fof(fact_704_split__mod,axiom,
! [P_1,N,K] :
( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),N),K)))
<=> ( ( K = zero_zero(nat)
=> hBOOL(hAPP(nat,bool,P_1,N)) )
& ( K != zero_zero(nat)
=> ! [I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),K))
=> ( N = 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_705_real__abs__def,axiom,
! [R_4] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_4),zero_zero(real)))
=> hAPP(real,real,abs_abs(real),R_4) = hAPP(real,real,uminus_uminus(real),R_4) )
& ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_4),zero_zero(real)))
=> hAPP(real,real,abs_abs(real),R_4) = R_4 ) ) ).
fof(fact_706_abs__real__def,axiom,
! [A_1] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),zero_zero(real)))
=> hAPP(real,real,abs_abs(real),A_1) = hAPP(real,real,uminus_uminus(real),A_1) )
& ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),zero_zero(real)))
=> hAPP(real,real,abs_abs(real),A_1) = A_1 ) ) ).
fof(fact_707_mod__mult__self3,axiom,
! [K_4,N_1,M_1] : hAPP(nat,nat,hAPP(nat,fun(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_4),N_1)),M_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1) ).
fof(fact_708_le__mod__geq,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),N_1) ) ).
fof(fact_709_mod__less__eq__dividend,axiom,
! [M_1,N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)),M_1)) ).
fof(fact_710_abs__sum__triangle__ineq,axiom,
! [X,Y,L_4,M_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,uminus_uminus(real),L_4)),hAPP(real,real,uminus_uminus(real),M_1))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,uminus_uminus(real),L_4)))),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Y),hAPP(real,real,uminus_uminus(real),M_1)))))) ).
fof(fact_711_abs__le__interval__iff,axiom,
! [X_1,R_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X_1)),R_1))
<=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),R_1)),X_1))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),R_1)) ) ) ).
fof(fact_712_termination__basic__simps_I3_J,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_713_termination__basic__simps_I4_J,axiom,
! [Y,X,Z] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Z))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_714_real__diff__def,axiom,
! [R_4,S_4] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),R_4),S_4) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_4),hAPP(real,real,uminus_uminus(real),S_4)) ).
fof(fact_715_minus__real__def,axiom,
! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,uminus_uminus(real),Y)) ).
fof(fact_716_abs__minus__add__cancel,axiom,
! [X,Y] : hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,uminus_uminus(real),Y))) = hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Y),hAPP(real,real,uminus_uminus(real),X))) ).
fof(fact_717_abs__add__one__not__less__self,axiom,
! [X] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,abs_abs(real),X)),one_one(real))),X)) ).
fof(fact_718_termination__basic__simps_I5_J,axiom,
! [X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),Y)) ) ).
fof(fact_719_termination__basic__simps_I2_J,axiom,
! [Y,X,Z] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Z))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_720_termination__basic__simps_I1_J,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y),Z))) ) ).
fof(fact_721_zdiv__zero,axiom,
! [B] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),zero_zero(int)),B) = zero_zero(int) ).
fof(fact_722_abs__add__one__gt__zero,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(real,real,abs_abs(real),X)))) ).
fof(fact_723_real__0__less__add__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X_1),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),X_1)),Y_1)) ) ).
fof(fact_724_real__add__less__0__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X_1),Y_1)),zero_zero(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),hAPP(real,real,uminus_uminus(real),X_1))) ) ).
fof(fact_725_real__add__eq__0__iff,axiom,
! [X_1,Y_1] :
( hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X_1),Y_1) = zero_zero(real)
<=> Y_1 = hAPP(real,real,uminus_uminus(real),X_1) ) ).
fof(fact_726_real__add__minus__iff,axiom,
! [X_1,Aa] :
( hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X_1),hAPP(real,real,uminus_uminus(real),Aa)) = zero_zero(real)
<=> X_1 = Aa ) ).
fof(fact_727_zdiv__zminus2,axiom,
! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),B) ).
fof(fact_728_zdiv__zminus__zminus,axiom,
! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) ).
fof(fact_729_div__by__0,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_730_div__0,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).
fof(fact_731_div__by__1,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_732_div__neg__pos__less0,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),zero_zero(int))) ) ) ).
fof(fact_733_neg__imp__zdiv__neg__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ba),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Aa),Ba)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Aa)) ) ) ).
fof(fact_734_pos__imp__zdiv__neg__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Aa),Ba)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),zero_zero(int))) ) ) ).
fof(fact_735_Divides_Otransfer__nat__int__function__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y))) ) ) ).
fof(fact_736_zdiv__self,axiom,
! [A_1] :
( A_1 != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),A_1) = one_one(int) ) ).
fof(fact_737_zmod__zdiv__trivial,axiom,
! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),B) = zero_zero(int) ).
fof(fact_738_DIVISION__BY__ZERO,axiom,
! [A_1] :
( hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),zero_zero(int)) = zero_zero(int)
& hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),zero_zero(int)) = A_1 ) ).
fof(fact_739_real__add__le__0__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X_1),Y_1)),zero_zero(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),hAPP(real,real,uminus_uminus(real),X_1))) ) ).
fof(fact_740_real__0__le__add__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X_1),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),X_1)),Y_1)) ) ).
fof(fact_741_real__minus__mult__self__le,axiom,
! [U,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),U),U))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X))) ).
fof(fact_742_zdiv__zadd1__eq,axiom,
! [A_1,B,C] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),C) = 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_1),C)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),C))),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,hAPP(int,fun(int,int),div_mod(int),A_1),C)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C))),C)) ).
fof(fact_743_pdivmod__def,axiom,
! [K_4,L_4] : hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,abs_abs(int),L_4))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,abs_abs(int),L_4))) ).
fof(fact_744_div__mult__mult1__if,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] :
( ( ti(X_a,C) = zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B)) = zero_zero(X_a) )
& ( ti(X_a,C) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B) ) ) ) ).
fof(fact_745_div__mult__self2__is__id,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),B) = ti(X_a,A_1) ) ) ).
fof(fact_746_div__mult__self1__is__id,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1)),B) = ti(X_a,A_1) ) ) ).
fof(fact_747_div__mult__mult2,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] :
( ti(X_a,C) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B) ) ) ).
fof(fact_748_div__mult__mult1,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] :
( ti(X_a,C) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B) ) ) ).
fof(fact_749_div__self,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),A_1) = one_one(X_a) ) ) ).
fof(fact_750_mod__div__trivial,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),B) = zero_zero(X_a) ) ).
fof(fact_751_negDivAlg__div__mod,axiom,
! [L_4,K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_4),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),L_4))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),L_4)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4)) ) ) ).
fof(fact_752_zdiv__eq__0__iff,axiom,
! [I,K] :
( hAPP(int,int,hAPP(int,fun(int,int),div_div(int),I),K) = zero_zero(int)
<=> ( K = zero_zero(int)
| ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),K)) )
| ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int)))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),I)) ) ) ) ).
fof(fact_753_pos__imp__zdiv__nonneg__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Aa),Ba)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Aa)) ) ) ).
fof(fact_754_pos__imp__zdiv__pos__iff,axiom,
! [I,K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),I),K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),I)) ) ) ).
fof(fact_755_nonneg1__imp__zdiv__pos__iff,axiom,
! [Ba,Aa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Aa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Aa),Ba)))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),Aa))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba)) ) ) ) ).
fof(fact_756_zdiv__mono2,axiom,
! [B,B_4,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_4),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B_4))) ) ) ) ).
fof(fact_757_div__nonneg__neg__le0,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),zero_zero(int))) ) ) ).
fof(fact_758_div__pos__pos__trivial,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = zero_zero(int) ) ) ).
fof(fact_759_neg__imp__zdiv__nonneg__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ba),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Aa),Ba)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Aa),zero_zero(int))) ) ) ).
fof(fact_760_div__nonpos__pos__le0,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),zero_zero(int))) ) ) ).
fof(fact_761_zdiv__mono2__neg,axiom,
! [B,B_4,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_4),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B_4)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))) ) ) ) ).
fof(fact_762_div__neg__neg__trivial,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = zero_zero(int) ) ) ).
fof(fact_763_zdiv__mono1,axiom,
! [B,A_1,A_5] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),A_5))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_5),B))) ) ) ).
fof(fact_764_zdiv__mono1__neg,axiom,
! [B,A_1,A_5] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),A_5))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_5),B)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))) ) ) ).
fof(fact_765_int__div__less__self,axiom,
! [K_4,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),K_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),K_4)),X)) ) ) ).
fof(fact_766_zdiv__zmult2__eq,axiom,
! [A_1,B,C] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),C) ) ).
fof(fact_767_div__prop1,axiom,
! [X,Y,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Y),Z)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Z)),Y)) ) ) ).
fof(fact_768_zmod__zdiv__equality,axiom,
! [A_1,B] : A_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) ).
fof(fact_769_zdiv__zmult1__eq,axiom,
! [A_1,B,C] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),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),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),C))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),C))),C)) ).
fof(fact_770_zdiv__zmod__equality,axiom,
! [B,A_1,K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))),K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),K_4) ).
fof(fact_771_zdiv__zmod__equality2,axiom,
! [A_1,B,K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B))),K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),K_4) ).
fof(fact_772_zmod__zdiv__equality_H,axiom,
! [M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M_1),N_1) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),M_1),N_1)),N_1)) ).
fof(fact_773_zmult__div__cancel,axiom,
! [N_1,M_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N_1),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),M_1),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M_1),N_1)) ).
fof(fact_774_zdiv__minus1__right,axiom,
! [A_1] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,number_number_of(int),min)) = hAPP(int,int,uminus_uminus(int),A_1) ).
fof(fact_775_div__mult__self2,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,C,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)) ) ) ).
fof(fact_776_div__mult__self1,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,C,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B))),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)) ) ) ).
fof(fact_777_divmod__int__rel__div,axiom,
! [A_1,B,Q_2,R_4] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),R_4)))
=> ( B != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = Q_2 ) ) ).
fof(fact_778_div__add__self2,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),one_one(X_a)) ) ) ).
fof(fact_779_div__add__self1,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),A_1)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),one_one(X_a)) ) ) ).
fof(fact_780_semiring__div__class_Omod__div__equality_H,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)) = ti(X_a,A_1) ) ).
fof(fact_781_mod__div__equality2,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [B,A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)) = ti(X_a,A_1) ) ).
fof(fact_782_mod__div__equality,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)) = ti(X_a,A_1) ) ).
fof(fact_783_div__mod__equality2,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [B,A_1,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C) ) ).
fof(fact_784_div__mod__equality,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B))),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),C) ) ).
fof(fact_785_zdiv__leq__prop,axiom,
! [X,Y] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_786_div__eq__minus1,axiom,
! [B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),min)),B) = hAPP(int,int,number_number_of(int),min) ) ).
fof(fact_787_zcong__zmod__aux,axiom,
! [A_1,B,M_1] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),M_1)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),M_1)))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),M_1)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),B),M_1))) ).
fof(fact_788_xzgcda__linear__aux2,axiom,
! [R_4,S_4,T_6,R_5,S_5,M_1,T_7,N_1] :
( R_5 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),S_5),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_7),N_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_4),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_6),N_1))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),R_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),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),S_5),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_5),R_4)),S_4))),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),T_7),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),R_5),R_4)),T_6))),N_1)) ) ) ).
fof(fact_789_divmod__int__rel__div__mod,axiom,
! [A_1,B] :
( B != zero_zero(int)
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)))) ) ).
fof(fact_790_negDivAlg__correct,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B))) ) ) ).
fof(fact_791_split__zdiv,axiom,
! [P_1,N,K] :
( hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),K)))
<=> ( ( K = zero_zero(int)
=> hBOOL(hAPP(int,bool,P_1,zero_zero(int))) )
& ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ! [I_2,J_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),K))
& N = 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(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ! [I_2,J_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),J_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),zero_zero(int)))
& N = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) )
=> hBOOL(hAPP(int,bool,P_1,I_2)) ) ) ) ) ).
fof(fact_792_divmod__int__rel__div__eq,axiom,
! [A_8,B_7,Y,R_6] :
( A_8 = 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_6)
=> ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_7))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R_6))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),R_6),B_7)) ) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_7))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_7),R_6))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),R_6),zero_zero(int))) ) ) )
=> ( B_7 != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_8),B_7) = Y ) ) ) ).
fof(fact_793_div__prop2,axiom,
! [X,Y,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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)),Z)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Z)),Y)) ) ) ).
fof(fact_794_zmod__zmult2__eq,axiom,
! [A_1,B,C] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),C))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)),C))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)) ) ).
fof(fact_795_zdiv__zminus2__eq__if,axiom,
! [A_1,B] :
( B != zero_zero(int)
=> ( ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)) )
& ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,uminus_uminus(int),B)) = 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_1),B))),one_one(int)) ) ) ) ).
fof(fact_796_zdiv__zminus1__eq__if,axiom,
! [A_1,B] :
( B != zero_zero(int)
=> ( ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),B) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B)) )
& ( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,uminus_uminus(int),A_1)),B) = 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_1),B))),one_one(int)) ) ) ) ).
fof(fact_797_zadd1__lemma,axiom,
! [B,Bq,Br,A_1,C,Aq,Ar] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),C),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Aq),Ar)))
=> ( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,B),C),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Bq),Br)))
=> ( C != zero_zero(int)
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),C),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Ar),Br)),C)))) ) ) ) ).
fof(fact_798_posDivAlg__div__mod,axiom,
! [L_4,K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,K_4),L_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),L_4)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),K_4),L_4)) ) ) ).
fof(fact_799_div__pos__neg__trivial,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),zero_zero(int)))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(int,int,number_number_of(int),min) ) ) ).
fof(fact_800_split__pos__lemma,axiom,
! [P_1,N,K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),K)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),N),K)))
<=> ! [I_2,J_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),J_1),K))
& N = 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(int,fun(int,bool),P_1,I_2),J_1)) ) ) ) ).
fof(fact_801_split__neg__lemma,axiom,
! [P_1,N,K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N),K)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),N),K)))
<=> ! [I_2,J_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),J_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),J_1),zero_zero(int)))
& N = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),I_2)),J_1) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,I_2),J_1)) ) ) ) ).
fof(fact_802_zmult1__lemma,axiom,
! [A_1,B,C,Q_2,R_4] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,B),C),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),R_4)))
=> ( C != zero_zero(int)
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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_1),Q_2)),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),R_4)),C))),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),R_4)),C)))) ) ) ).
fof(fact_803_z3div__def,axiom,
! [K_4,L_4] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4))
=> hAPP(int,int,hAPP(int,fun(int,int),z3div,K_4),L_4) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),L_4) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L_4))
=> hAPP(int,int,hAPP(int,fun(int,int),z3div,K_4),L_4) = hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),K_4),hAPP(int,int,uminus_uminus(int),L_4))) ) ) ).
fof(fact_804_le0,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1)) ).
fof(fact_805_less__zeroE,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat))) ).
fof(fact_806_mod__div__decomp,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] : ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),A_1),B)),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)) ) ).
fof(fact_807_sin__bound__lemma,axiom,
! [U,V,X,Y] :
( X = Y
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),U)),V))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),U)),Y))),V)) ) ) ).
fof(fact_808_nat__less__add__iff2,axiom,
! [U_1,M,N,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),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),U_1)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(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),I)),U_1)),N))) ) ) ).
fof(fact_809_nat__less__add__iff1,axiom,
! [U_1,M,N,J,I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),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),U_1)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),J)),U_1)),M)),N)) ) ) ).
fof(fact_810_rabs__ratiotest__lemma,axiom,
! [X,Y,C] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),zero_zero(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(real,real,abs_abs(real),Y))))
=> X = zero_zero(real) ) ) ).
fof(fact_811_mult__le__cancel1,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
fof(fact_812_nat__mult__div__cancel__disj,axiom,
! [M_1,N_1,K_4] :
( ( K_4 = 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_4),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N_1)) = zero_zero(nat) )
& ( K_4 != 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_4),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1) ) ) ).
fof(fact_813_div__mult2__eq,axiom,
! [A_1,B,C] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),B)),C) ).
fof(fact_814_mod__mult__distrib2,axiom,
! [K_4,M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N_1)) ).
fof(fact_815_mod__mult__distrib,axiom,
! [M_1,N_1,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),K_4)) ).
fof(fact_816_div__mod__equality_H,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)) ).
fof(fact_817_mult__div__cancel,axiom,
! [N_1,M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)) ).
fof(fact_818_Divides_Omod__div__equality_H,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1)),N_1)) ).
fof(fact_819_diff__mult__distrib2,axiom,
! [K_4,M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N_1)) ).
fof(fact_820_diff__mult__distrib,axiom,
! [M_1,N_1,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),K_4)) ).
fof(fact_821_diff__commute,axiom,
! [I_1,J_2,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J_2)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),K_4)),J_2) ).
fof(fact_822_nat__mult__commute,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),M_1) ).
fof(fact_823_nat__mult__assoc,axiom,
! [M_1,N_1,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),K_4)) ).
fof(fact_824_div__add1__eq,axiom,
! [A_1,B,C] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_1),B)),C) = 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_1),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),C))),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,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),C))),C)) ).
fof(fact_825_div__mult1__eq,axiom,
! [A_1,B,C] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),C))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),B),C))),C)) ).
fof(fact_826_mod__mult2__eq,axiom,
! [A_1,B,C] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),B)),C))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),B)) ).
fof(fact_827_div__le__mono,axiom,
! [K_4,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N_1),K_4))) ) ).
fof(fact_828_div__le__dividend,axiom,
! [M_1,N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1)),M_1)) ).
fof(fact_829_nat__mult__div__cancel1,axiom,
! [M_1,N_1,K_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1) ) ).
fof(fact_830_div__mult__self__is__m,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)),N_1) = M_1 ) ).
fof(fact_831_div__mult__self1__is__m,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),M_1)),N_1) = M_1 ) ).
fof(fact_832_div__less,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1) = zero_zero(nat) ) ).
fof(fact_833_split__div,axiom,
! [P_1,N,K] :
( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N),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(nat,fun(nat,bool),ord_less(nat),J_1),K))
=> ( N = 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_834_div__le__mono2,axiom,
! [K_4,N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),K_4),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),K_4),M_1))) ) ) ).
fof(fact_835_div__less__dividend,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1)),M_1)) ) ) ).
fof(fact_836_mult__0,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N_1) = zero_zero(nat) ).
fof(fact_837_mult__0__right,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),zero_zero(nat)) = zero_zero(nat) ).
fof(fact_838_mult__is__0,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat)
<=> ( M = zero_zero(nat)
| N = zero_zero(nat) ) ) ).
fof(fact_839_nat__mult__eq__cancel__disj,axiom,
! [K,M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)
<=> ( K = zero_zero(nat)
| M = N ) ) ).
fof(fact_840_mult__cancel1,axiom,
! [K,M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)
<=> ( M = N
| K = zero_zero(nat) ) ) ).
fof(fact_841_mult__cancel2,axiom,
! [M,K,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K)
<=> ( M = N
| K = zero_zero(nat) ) ) ).
fof(fact_842_diffs0__imp__equal,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) = zero_zero(nat)
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),M_1) = zero_zero(nat)
=> M_1 = N_1 ) ) ).
fof(fact_843_diff__self__eq__0,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),M_1) = zero_zero(nat) ).
fof(fact_844_minus__nat_Odiff__0,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),zero_zero(nat)) = M_1 ).
fof(fact_845_diff__0__eq__0,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N_1) = zero_zero(nat) ).
fof(fact_846_nat__less__cases,axiom,
! [P_1,M,N] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,N),M)) )
=> ( ( M = N
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,N),M)) )
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,N),M)) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,N),M)) ) ) ) ).
fof(fact_847_less__not__refl3,axiom,
! [S_4,T_6] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S_4),T_6))
=> S_4 != T_6 ) ).
fof(fact_848_less__not__refl2,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),M_1))
=> M_1 != N_1 ) ).
fof(fact_849_less__irrefl__nat,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1)) ).
fof(fact_850_linorder__neqE__nat,axiom,
! [X,Y] :
( X != Y
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).
fof(fact_851_nat__neq__iff,axiom,
! [M,N] :
( M != N
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ).
fof(fact_852_less__not__refl,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_1)) ).
fof(fact_853_diff__less__mono2,axiom,
! [L_4,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),L_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),M_1))) ) ) ).
fof(fact_854_less__imp__diff__less,axiom,
! [N_1,J_2,K_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_2),K_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),N_1)),K_4)) ) ).
fof(fact_855_diff__add__inverse2,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)),N_1) = M_1 ).
fof(fact_856_diff__add__inverse,axiom,
! [N_1,M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M_1)),N_1) = M_1 ).
fof(fact_857_diff__diff__left,axiom,
! [I_1,J_2,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J_2)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),K_4)) ).
fof(fact_858_Nat_Odiff__cancel,axiom,
! [K_4,M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) ).
fof(fact_859_diff__cancel2,axiom,
! [M_1,K_4,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) ).
fof(fact_860_nat__add__right__cancel,axiom,
! [M,K,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K)
<=> M = N ) ).
fof(fact_861_nat__add__left__cancel,axiom,
! [K,M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N)
<=> M = N ) ).
fof(fact_862_nat__add__assoc,axiom,
! [M_1,N_1,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),K_4)) ).
fof(fact_863_nat__add__left__commute,axiom,
! [X,Y,Z] : 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)) = 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)) ).
fof(fact_864_nat__add__commute,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M_1) ).
fof(fact_865_add__mult__distrib2,axiom,
! [K_4,M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N_1)) ).
fof(fact_866_add__mult__distrib,axiom,
! [M_1,N_1,K_4] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),K_4)) ).
fof(fact_867_left__add__mult__distrib,axiom,
! [I_1,U,J_2,K_4] : 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)),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)),K_4)) = 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_1),J_2)),U)),K_4) ).
fof(fact_868_Nat_Odiff__le__self,axiom,
! [M_1,N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),M_1)) ).
fof(fact_869_diff__le__mono2,axiom,
! [L_4,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),L_4),M_1))) ) ).
fof(fact_870_diff__le__mono,axiom,
! [L_4,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),L_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),L_4))) ) ).
fof(fact_871_diff__diff__cancel,axiom,
! [I_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),I_1)) = I_1 ) ).
fof(fact_872_eq__diff__iff,axiom,
! [N,K,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),K) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),K)
<=> M = N ) ) ) ).
fof(fact_873_Nat_Odiff__diff__eq,axiom,
! [N_1,K_4,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) ) ) ).
fof(fact_874_le__diff__iff,axiom,
! [N,K,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).
fof(fact_875_le__refl,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_1)) ).
fof(fact_876_nat__le__linear,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1)) ) ).
fof(fact_877_eq__imp__le,axiom,
! [M_1,N_1] :
( M_1 = N_1
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_878_le__trans,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),K_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),K_4)) ) ) ).
fof(fact_879_le__antisym,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> M_1 = N_1 ) ) ).
fof(fact_880_mult__le__mono,axiom,
! [K_4,L_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),L_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),L_4))) ) ) ).
fof(fact_881_mult__le__mono2,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),J_2))) ) ).
fof(fact_882_mult__le__mono1,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),K_4))) ) ).
fof(fact_883_le__cube,axiom,
! [M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),M_1)))) ).
fof(fact_884_le__square,axiom,
! [M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),M_1))) ).
fof(fact_885_nat__mult__1,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),one_one(nat)),N_1) = N_1 ).
fof(fact_886_nat__1__eq__mult__iff,axiom,
! [M,N] :
( one_one(nat) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)
<=> ( M = one_one(nat)
& N = one_one(nat) ) ) ).
fof(fact_887_nat__mult__1__right,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),one_one(nat)) = N_1 ).
fof(fact_888_nat__mult__eq__1__iff,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat)
<=> ( M = one_one(nat)
& N = one_one(nat) ) ) ).
fof(fact_889_nat__0__less__mult__iff,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).
fof(fact_890_mult__less__cancel1,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_891_mult__less__cancel2,axiom,
! [M,K,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_892_nat__mult__eq__cancel1,axiom,
! [M,N,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)
<=> M = N ) ) ).
fof(fact_893_nat__mult__less__cancel1,axiom,
! [M,N,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_894_mult__less__mono1,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),J_2),K_4))) ) ) ).
fof(fact_895_mult__less__mono2,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),J_2))) ) ) ).
fof(fact_896_gr0I,axiom,
! [N_1] :
( N_1 != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) ) ).
fof(fact_897_gr__implies__not0,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> N_1 != zero_zero(nat) ) ).
fof(fact_898_less__nat__zero__code,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat))) ).
fof(fact_899_neq0__conv,axiom,
! [N] :
( N != zero_zero(nat)
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
fof(fact_900_not__less0,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),zero_zero(nat))) ).
fof(fact_901_diff__less,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),M_1)) ) ) ).
fof(fact_902_zero__less__diff,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),M)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_903_plus__nat_Oadd__0,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N_1) = N_1 ).
fof(fact_904_Nat_Oadd__0__right,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),zero_zero(nat)) = M_1 ).
fof(fact_905_add__is__0,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat)
<=> ( M = zero_zero(nat)
& N = zero_zero(nat) ) ) ).
fof(fact_906_add__eq__self__zero,axiom,
! [M_1,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = M_1
=> N_1 = zero_zero(nat) ) ).
fof(fact_907_diff__add__0,axiom,
! [N_1,M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M_1)) = zero_zero(nat) ).
fof(fact_908_diff__is__0__eq,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat)
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_909_diff__is__0__eq_H,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) = zero_zero(nat) ) ).
fof(fact_910_less__eq__nat_Osimps_I1_J,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N_1)) ).
fof(fact_911_le__0__eq,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat)))
<=> N = zero_zero(nat) ) ).
fof(fact_912_not__add__less1,axiom,
! [I_1,J_2] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J_2)),I_1)) ).
fof(fact_913_not__add__less2,axiom,
! [J_2,I_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),I_1)),I_1)) ).
fof(fact_914_nat__add__left__cancel__less,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_915_trans__less__add1,axiom,
! [M_1,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),M_1))) ) ).
fof(fact_916_trans__less__add2,axiom,
! [M_1,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),J_2))) ) ).
fof(fact_917_add__less__mono1,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),K_4))) ) ).
fof(fact_918_add__less__mono,axiom,
! [K_4,L_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_4),L_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),L_4))) ) ) ).
fof(fact_919_less__add__eq__less,axiom,
! [M_1,N_1,K_4,L_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_4),L_4))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),L_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),N_1)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ) ).
fof(fact_920_add__lessD1,axiom,
! [I_1,J_2,K_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J_2)),K_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_4)) ) ).
fof(fact_921_add__diff__inverse,axiom,
! [M_1,N_1] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)) = M_1 ) ).
fof(fact_922_less__diff__conv,axiom,
! [I,J,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) ) ).
fof(fact_923_mult__eq__self__implies__10,axiom,
! [M_1,N_1] :
( M_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)
=> ( N_1 = one_one(nat)
| M_1 = zero_zero(nat) ) ) ).
fof(fact_924_less__or__eq__imp__le,axiom,
! [M_1,N_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
| M_1 = N_1 )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_925_le__neq__implies__less,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> ( M_1 != N_1
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ) ).
fof(fact_926_less__imp__le__nat,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_927_le__eq__less__or__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
| M = N ) ) ).
fof(fact_928_nat__less__le,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
& M != N ) ) ).
fof(fact_929_less__diff__iff,axiom,
! [N,K,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).
fof(fact_930_diff__less__mono,axiom,
! [C,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),A_1),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),C),A_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),A_1),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),B),C))) ) ) ).
fof(fact_931_diff__diff__right,axiom,
! [I_1,K_4,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),J_2) ) ).
fof(fact_932_le__diff__conv,axiom,
! [J,K,I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)),I))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ).
fof(fact_933_le__add__diff,axiom,
! [M_1,K_4,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M_1)),K_4))) ) ).
fof(fact_934_le__add__diff__inverse,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)) = M_1 ) ).
fof(fact_935_add__diff__assoc,axiom,
! [I_1,K_4,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),J_2)),K_4) ) ).
fof(fact_936_le__diff__conv2,axiom,
! [I,K,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) ) ) ).
fof(fact_937_le__add__diff__inverse2,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),N_1) = M_1 ) ).
fof(fact_938_le__imp__diff__is__add,axiom,
! [K,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J),I) = K
<=> J = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ).
fof(fact_939_diff__add__assoc,axiom,
! [I_1,K_4,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),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_1),J_2)),K_4) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4)) ) ).
fof(fact_940_add__diff__assoc2,axiom,
! [I_1,K_4,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),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_4)),I_1) = 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_1)),K_4) ) ).
fof(fact_941_diff__add__assoc2,axiom,
! [I_1,K_4,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),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_1)),K_4) = 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_4)),I_1) ) ).
fof(fact_942_nat__eq__add__iff2,axiom,
! [U_1,M,N,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( 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),U_1)),N)
<=> 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),I)),U_1)),N) ) ) ).
fof(fact_943_nat__diff__add__eq2,axiom,
! [U,M_1,N_1,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),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_1),U)),M_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)),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),I_1)),U)),N_1)) ) ).
fof(fact_944_nat__le__add__iff2,axiom,
! [U_1,M,N,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),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),U_1)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(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),I)),U_1)),N))) ) ) ).
fof(fact_945_nat__eq__add__iff1,axiom,
! [U_1,M,N,J,I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( 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),U_1)),N)
<=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I),J)),U_1)),M) = N ) ) ).
fof(fact_946_nat__diff__add__eq1,axiom,
! [U,M_1,N_1,J_2,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J_2),I_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),I_1),U)),M_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)),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),I_1),J_2)),U)),M_1)),N_1) ) ).
fof(fact_947_nat__le__add__iff1,axiom,
! [U_1,M,N,J,I] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),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),U_1)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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),J)),U_1)),M)),N)) ) ) ).
fof(fact_948_add__leE,axiom,
! [M_1,K_4,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_4)),N_1))
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N_1)) ) ) ).
fof(fact_949_add__leD1,axiom,
! [M_1,K_4,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_4)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_950_add__leD2,axiom,
! [M_1,K_4,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_4)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N_1)) ) ).
fof(fact_951_add__le__mono,axiom,
! [K_4,L_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),L_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),L_4))) ) ) ).
fof(fact_952_add__le__mono1,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_4)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),K_4))) ) ).
fof(fact_953_trans__le__add2,axiom,
! [M_1,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),J_2))) ) ).
fof(fact_954_trans__le__add1,axiom,
! [M_1,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),J_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J_2),M_1))) ) ).
fof(fact_955_nat__add__left__cancel__le,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_956_le__iff__add,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
<=> ? [K_1] : N = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K_1) ) ).
fof(fact_957_le__Suc__ex__iff,axiom,
! [K,L] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K),L))
<=> ? [N_2] : L = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),N_2) ) ).
fof(fact_958_le__add1,axiom,
! [N_1,M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),M_1))) ).
fof(fact_959_le__add2,axiom,
! [N_1,M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1))) ).
fof(fact_960_eq__diff__eq_H,axiom,
! [X_1,Y_1,Z_1] :
( X_1 = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Y_1),Z_1)
<=> Y_1 = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X_1),Z_1) ) ).
fof(fact_961_add__gr__0,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).
fof(fact_962_nat__diff__split,axiom,
! [P_1,Aa,Ba] :
( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Aa),Ba)))
<=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Aa),Ba))
=> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) )
& ! [D_3] :
( Aa = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ba),D_3)
=> hBOOL(hAPP(nat,bool,P_1,D_3)) ) ) ) ).
fof(fact_963_nat__diff__split__asm,axiom,
! [P_1,Aa,Ba] :
( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),Aa),Ba)))
<=> ~ ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Aa),Ba))
& ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) )
| ? [D_3] :
( Aa = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ba),D_3)
& ~ hBOOL(hAPP(nat,bool,P_1,D_3)) ) ) ) ).
fof(fact_964_nat__mult__le__cancel1,axiom,
! [M,N,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
fof(fact_965_mult__le__cancel2,axiom,
! [M,K,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
fof(fact_966_ex__least__nat__less,axiom,
! [N,P_1] :
( ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
=> ( hBOOL(hAPP(nat,bool,P_1,N))
=> ? [K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),N))
& ! [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_967_nat__mod__eq__lemma,axiom,
! [X,N_1,Y] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Y),N_1)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
=> ? [Q_3] : 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_1),Q_3)) ) ) ).
fof(fact_968_mod__eqD,axiom,
! [M_1,D,R_4] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),D) = R_4
=> ? [Q_3] : M_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),R_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Q_3),D)) ) ).
fof(fact_969_nat__mod__eq__iff,axiom,
! [X_1,N,Y_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X_1),N) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Y_1),N)
<=> ? [Q1,Q2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),Q1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Y_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),Q2)) ) ).
fof(fact_970_mod__eq__0D,axiom,
! [M_6,D_4] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_6),D_4) = zero_zero(nat)
=> ? [Q_3] : M_6 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),D_4),Q_3) ) ).
fof(fact_971_mono__nat__linear__lb,axiom,
! [M,K,F] :
( ! [M_3,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F,M_3)),hAPP(nat,nat,F,N_2))) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,F,M)),K)),hAPP(nat,nat,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K)))) ) ).
fof(fact_972_ex__least__nat__le,axiom,
! [N,P_1] :
( ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
=> ( hBOOL(hAPP(nat,bool,P_1,N))
=> ? [K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N))
& ! [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_973_less__imp__add__positive,axiom,
! [I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
=> ? [K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_1))
& hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),K_1) = J_2 ) ) ).
fof(fact_974_mult__left__cancel,axiom,
! [N_1,M_1,K_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M_1)
=> N_1 = M_1 ) ) ).
fof(fact_975_divmod__nat__rel__mult1__eq,axiom,
! [A_1,B,C,Q_2,R_4] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,B),C),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_2),R_4)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C))
=> hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)),C),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,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_1),Q_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),R_4)),C))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),R_4)),C)))) ) ) ).
fof(fact_976_divmod__nat__rel__mult2__eq,axiom,
! [C,A_1,B,Q_2,R_4] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,A_1),B),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_2),R_4)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C))
=> hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),C)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),Q_2),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),Q_2),C))),R_4)))) ) ) ) ).
fof(fact_977_lemma__interval,axiom,
! [B,A_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),B))
=> ? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [Y_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y_2))),D_3))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_1),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_2),B)) ) ) ) ) ) ).
fof(fact_978_le__Suc__ex,axiom,
! [K_4,L_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),L_4))
=> ? [N_2] : L_4 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K_4),N_2) ) ).
fof(fact_979_less__mono__imp__le__mono,axiom,
! [I,J,F] :
( ! [I_2,J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),J_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,F,I_2)),hAPP(nat,nat,F,J_1))) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,F,I)),hAPP(nat,nat,F,J))) ) ) ).
fof(fact_980_real__lbound__gt__zero,axiom,
! [D2,D1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
=> ? [E] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),E),D1))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),E),D2)) ) ) ) ).
fof(fact_981_natceiling__add__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),one_one(real))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natceiling,X)),one_one(nat)) ) ).
fof(fact_982_divmod__nat__rel__unique,axiom,
! [Qr_2,M_1,N_1,Qr_1] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M_1),N_1),Qr_1))
=> ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M_1),N_1),Qr_2))
=> Qr_1 = Qr_2 ) ) ).
fof(fact_983_natceiling__number__of__eq,axiom,
! [N_1] : hAPP(real,nat,natceiling,hAPP(int,real,number_number_of(real),N_1)) = hAPP(int,nat,number_number_of(nat),N_1) ).
fof(fact_984_div__eq,axiom,
! [M_1,N_1,Q_2,R_4] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M_1),N_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_2),R_4)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1) = Q_2 ) ).
fof(fact_985_mod__eq,axiom,
! [M_1,N_1,Q_2,R_4] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M_1),N_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_2),R_4)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1) = R_4 ) ).
fof(fact_986_natceiling__zero,axiom,
hAPP(real,nat,natceiling,zero_zero(real)) = zero_zero(nat) ).
fof(fact_987_zero__le__natceiling,axiom,
! [X] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(real,nat,natceiling,X))) ).
fof(fact_988_natceiling__mono,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,X)),hAPP(real,nat,natceiling,Y))) ) ).
fof(fact_989_natceiling__one,axiom,
hAPP(real,nat,natceiling,one_one(real)) = one_one(nat) ).
fof(fact_990_divmod__nat__rel,axiom,
! [M_1,N_1] : hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M_1),N_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)))) ).
fof(fact_991_natceiling__neg,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
=> hAPP(real,nat,natceiling,X) = zero_zero(nat) ) ).
fof(fact_992_natceiling__le__eq__one,axiom,
! [X_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,X_1)),one_one(nat)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),one_one(real))) ) ).
fof(fact_993_divmod__nat__rel__add1__eq,axiom,
! [B,Bq,Br,A_1,C,Aq,Ar] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,A_1),C),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Aq),Ar)))
=> ( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,B),C),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Bq),Br)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C))
=> hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_1),B)),C),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,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))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Ar),Br)),C)))) ) ) ) ).
fof(fact_994_lemma__interval__lt,axiom,
! [B,A_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),B))
=> ? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [Y_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y_2))),D_3))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_2),B)) ) ) ) ) ) ).
fof(fact_995_divmod__nat__rel__ex,axiom,
! [M_1,N_1] :
~ ! [Q_3,R_2] : ~ hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M_1),N_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Q_3),R_2))) ).
fof(fact_996_natfloor__add__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),one_one(real))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,X)),one_one(nat)) ) ).
fof(fact_997_nat__mult__eq__one,axiom,
! [N,M] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),M) = one_one(nat)
<=> ( N = one_one(nat)
& M = one_one(nat) ) ) ).
fof(fact_998_natceiling__eq,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N_1)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),N_1)),one_one(real))))
=> hAPP(real,nat,natceiling,X) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),one_one(nat)) ) ) ).
fof(fact_999_natceiling__add__number__of,axiom,
! [X,N_1] :
( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),N_1)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,number_number_of(real),N_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natceiling,X)),hAPP(int,nat,number_number_of(nat),N_1)) ) ) ).
fof(fact_1000_divmod__int__def,axiom,
! [B,A_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B))
=> ( ( A_1 = zero_zero(int)
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)) )
& ( A_1 != zero_zero(int)
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B))) ) ) ) ) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B))) ) ) ) ) ).
fof(fact_1001_divmod__int__mod__div,axiom,
! [P_4,Q_2] : hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,P_4),Q_2) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),P_4),Q_2)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),P_4),Q_2)) ).
fof(fact_1002_real__of__nat__ge__zero,axiom,
! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,real_1(nat),N_1))) ).
fof(fact_1003_natfloor__real__of__nat,axiom,
! [N_1] : hAPP(real,nat,natfloor,hAPP(nat,real,real_1(nat),N_1)) = N_1 ).
fof(fact_1004_Ints__real__of__nat,axiom,
! [N] : hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(nat,real,real_1(nat),N)),ring_1_Ints(real))) ).
fof(fact_1005_natfloor__number__of__eq,axiom,
! [N_1] : hAPP(real,nat,natfloor,hAPP(int,real,number_number_of(real),N_1)) = hAPP(int,nat,number_number_of(nat),N_1) ).
fof(fact_1006_real__of__nat__inject,axiom,
! [N,M] :
( hAPP(nat,real,real_1(nat),N) = hAPP(nat,real,real_1(nat),M)
<=> N = M ) ).
fof(fact_1007_real__of__nat__number__of,axiom,
! [V] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,real,real_1(nat),hAPP(int,nat,number_number_of(nat),V)) = zero_zero(real) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,real,real_1(nat),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,real,number_number_of(real),V) ) ) ).
fof(fact_1008_real__natfloor__le,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X))),X)) ) ).
fof(fact_1009_le__natfloor,axiom,
! [X,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),X)),A_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X),hAPP(real,nat,natfloor,A_1))) ) ).
fof(fact_1010_abs__real__of__nat__cancel,axiom,
! [X] : hAPP(real,real,abs_abs(real),hAPP(nat,real,real_1(nat),X)) = hAPP(nat,real,real_1(nat),X) ).
fof(fact_1011_not__neg__0,axiom,
~ hBOOL(hAPP(int,bool,nat_neg,zero_zero(int))) ).
fof(fact_1012_not__neg__1,axiom,
~ hBOOL(hAPP(int,bool,nat_neg,one_one(int))) ).
fof(fact_1013_natceiling__real__of__nat,axiom,
! [N_1] : hAPP(real,nat,natceiling,hAPP(nat,real,real_1(nat),N_1)) = N_1 ).
fof(fact_1014_le__natfloor__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Aa),hAPP(real,nat,natfloor,X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),Aa)),X_1)) ) ) ).
fof(fact_1015_real__natfloor__add__one__gt,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X))),one_one(real)))) ).
fof(fact_1016_natfloor__subtract,axiom,
! [A_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),A_1)),X))
=> hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(real,nat,natfloor,X)),A_1) ) ).
fof(fact_1017_real__natfloor__gt__diff__one,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),one_one(real))),hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X)))) ).
fof(fact_1018_real__of__nat__zero,axiom,
hAPP(nat,real,real_1(nat),zero_zero(nat)) = zero_zero(real) ).
fof(fact_1019_real__of__nat__zero__iff,axiom,
! [N] :
( hAPP(nat,real,real_1(nat),N) = zero_zero(real)
<=> N = zero_zero(nat) ) ).
fof(fact_1020_not__real__of__nat__less__zero,axiom,
! [N_1] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N_1)),zero_zero(real))) ).
fof(fact_1021_real__of__nat__less__iff,axiom,
! [N,M] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,real_1(nat),M)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M)) ) ).
fof(fact_1022_real__of__nat__le__iff,axiom,
! [N,M] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,real_1(nat),M)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).
fof(fact_1023_less__natfloor,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(real,nat,natfloor,X)),N_1)) ) ) ).
fof(fact_1024_real__of__nat__mult,axiom,
! [M_1,N_1] : hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),M_1)),hAPP(nat,real,real_1(nat),N_1)) ).
fof(fact_1025_natfloor__add,axiom,
! [A_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,X)),A_1) ) ).
fof(fact_1026_real__of__nat__add,axiom,
! [M_1,N_1] : hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),M_1)),hAPP(nat,real,real_1(nat),N_1)) ).
fof(fact_1027_ge__natfloor__plus__one__imp__gt,axiom,
! [Z,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,Z)),one_one(nat))),N_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Z),hAPP(nat,real,real_1(nat),N_1))) ) ).
fof(fact_1028_natfloor__eq,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N_1)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),N_1)),one_one(real))))
=> hAPP(real,nat,natfloor,X) = N_1 ) ) ).
fof(fact_1029_real__of__nat__1,axiom,
hAPP(nat,real,real_1(nat),one_one(nat)) = one_one(real) ).
fof(fact_1030_le__natfloor__eq__number__of,axiom,
! [X_1,N] :
( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),N)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),N)),hAPP(real,nat,natfloor,X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),N)),X_1)) ) ) ) ).
fof(fact_1031_real__natceiling__ge,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(nat,real,real_1(nat),hAPP(real,nat,natceiling,X)))) ).
fof(fact_1032_neg__def,axiom,
! [Z_5] :
( hBOOL(hAPP(int,bool,nat_neg,Z_5))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_5),zero_zero(int))) ) ).
fof(fact_1033_not__neg__eq__ge__0,axiom,
! [X_1] :
( ~ hBOOL(hAPP(int,bool,nat_neg,X_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) ) ).
fof(fact_1034_neg__imp__number__of__eq__0,axiom,
! [V] :
( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(int,nat,number_number_of(nat),V) = zero_zero(nat) ) ).
fof(fact_1035_neg__number__of__Min,axiom,
hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),min))) ).
fof(fact_1036_natfloor__zero,axiom,
hAPP(real,nat,natfloor,zero_zero(real)) = zero_zero(nat) ).
fof(fact_1037_zero__le__natfloor,axiom,
! [X] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(real,nat,natfloor,X))) ).
fof(fact_1038_natfloor__mono,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natfloor,X)),hAPP(real,nat,natfloor,Y))) ) ).
fof(fact_1039_natfloor__one,axiom,
hAPP(real,nat,natfloor,one_one(real)) = one_one(nat) ).
fof(fact_1040_natfloor__add__number__of,axiom,
! [X,N_1] :
( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),N_1)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,number_number_of(real),N_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natfloor,X)),hAPP(int,nat,number_number_of(nat),N_1)) ) ) ).
fof(fact_1041_real__of__nat__le__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N)),zero_zero(real)))
<=> N = zero_zero(nat) ) ).
fof(fact_1042_real__of__nat__diff,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,real_1(nat),M_1)),hAPP(nat,real,real_1(nat),N_1)) ) ).
fof(fact_1043_natceiling__le,axiom,
! [X,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(nat,real,real_1(nat),A_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,X)),A_1)) ) ).
fof(fact_1044_eq__nat__number__of,axiom,
! [V_1,V_2] :
( hAPP(int,nat,number_number_of(nat),V_1) = hAPP(int,nat,number_number_of(nat),V_2)
<=> ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_1)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),V_2)),zero_zero(int))) )
& ( ~ 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(int,int,number_number_of(int),V_1) = zero_zero(int) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_2)))
=> V_1 = V_2 ) ) ) ) ) ).
fof(fact_1045_nat__number__of__add__left,axiom,
! [V_3,K_4,V] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_3)),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_3)),K_4) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_3)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_3)),K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),K_4) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_3)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_3)),K_4)) = 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),V_3))),K_4) ) ) ) ) ).
fof(fact_1046_natfloor__neg,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
=> hAPP(real,nat,natfloor,X) = zero_zero(nat) ) ).
fof(fact_1047_real__of__nat__gt__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,real_1(nat),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
fof(fact_1048_nat__less__real__le,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),N)),one_one(real))),hAPP(nat,real,real_1(nat),M))) ) ).
fof(fact_1049_negateSnd__eq,axiom,
! [Q_2,R_4] : hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),R_4)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q_2),hAPP(int,int,uminus_uminus(int),R_4)) ).
fof(fact_1050_nat__le__real__less,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),M)),one_one(real)))) ) ).
fof(fact_1051_natceiling__le__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,X_1)),Aa))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(nat,real,real_1(nat),Aa))) ) ) ).
fof(fact_1052_divmod__int__rel__neg,axiom,
! [A_1,B,Qr_1] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)),Qr_1))
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,Qr_1))) ) ).
fof(fact_1053_natceiling__subtract,axiom,
! [A_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),A_1)),X))
=> hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(real,nat,natceiling,X)),A_1) ) ).
fof(fact_1054_divmod__int__correct,axiom,
! [A_1,B] :
( B != zero_zero(int)
=> hBOOL(hAPP(product_prod(int,int),bool,hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,A_1),B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B))) ) ).
fof(fact_1055_le__natfloor__eq__one,axiom,
! [X_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),hAPP(real,nat,natfloor,X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X_1)) ) ).
fof(fact_1056_natceiling__add,axiom,
! [A_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(real,nat,natceiling,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(nat,real,real_1(nat),A_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(real,nat,natceiling,X)),A_1) ) ).
fof(fact_1057_natceiling__le__eq__number__of,axiom,
! [X_1,N] :
( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),N)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(real,nat,natceiling,X_1)),hAPP(int,nat,number_number_of(nat),N)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(int,real,number_number_of(real),N))) ) ) ) ).
fof(fact_1058_le__mult__natfloor,axiom,
! [B,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(real,nat,natfloor,A_1)),hAPP(real,nat,natfloor,B))),hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B)))) ) ) ).
fof(fact_1059_reals__Archimedean6,axiom,
! [R_4] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R_4))
=> ? [N_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_2),one_one(nat)))),R_4))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_4),hAPP(nat,real,real_1(nat),N_2))) ) ) ).
fof(fact_1060_reals__Archimedean3,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ! [Y_2] :
? [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_2),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_2)),X))) ) ).
fof(fact_1061_reals__Archimedean2,axiom,
! [X] :
? [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),N_2))) ).
fof(fact_1062_div__pos__neg__1__number__of,axiom,
! [W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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(product_prod(int,int),int,product_fst(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),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_1063_mod__pos__neg__1__number__of,axiom,
! [W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),zero_zero(int)))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),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_1064_natfloor__div__nat,axiom,
! [Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
=> hAPP(real,nat,natfloor,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(nat,real,real_1(nat),Y))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(real,nat,natfloor,X)),Y) ) ) ).
fof(fact_1065_div__neg__neg,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int)))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) ) ) ).
fof(fact_1066_mod__neg__neg,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int)))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) ) ) ).
fof(fact_1067_prod__eqI,axiom,
! [X_a,X_b,P_4,Q_2] :
( hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),P_4) = hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Q_2)
=> ( hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),P_4) = hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Q_2)
=> ti(product_prod(X_a,X_b),P_4) = ti(product_prod(X_a,X_b),Q_2) ) ) ).
fof(fact_1068_prod__eq__iff,axiom,
! [X_a,X_b,S_3,T_3] :
( ti(product_prod(X_a,X_b),S_3) = ti(product_prod(X_a,X_b),T_3)
<=> ( hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),S_3) = hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),T_3)
& hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),S_3) = hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),T_3) ) ) ).
fof(fact_1069_surjective__pairing,axiom,
! [X_a,X_b,T_6] : ti(product_prod(X_a,X_b),T_6) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),T_6)),hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),T_6)) ).
fof(fact_1070_pair__collapse,axiom,
! [X_b,X_a,P_4] : hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),P_4)),hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),P_4)) = ti(product_prod(X_a,X_b),P_4) ).
fof(fact_1071_divide__zero__left,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),zero_zero(X_a)),A_1) = zero_zero(X_a) ) ).
fof(fact_1072_divide__zero,axiom,
! [X_a] :
( divisi14063676e_zero(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_1073_divide_Ozero,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),zero_zero(X_a)),Y) = zero_zero(X_a) ) ).
fof(fact_1074_times__divide__times__eq,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [X,Y,Z,W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Z),W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),W)) ) ).
fof(fact_1075_times__divide__eq__right,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C) ) ).
fof(fact_1076_add__divide__distrib,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C)) ) ).
fof(fact_1077_divide_Oadd,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [X,Y,Ya] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),Ya) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Ya)) ) ).
fof(fact_1078_divide__1,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),one_one(X_a)) = ti(X_a,A_1) ) ).
fof(fact_1079_diff__divide__distrib,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C)) ) ).
fof(fact_1080_divide_Odiff,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [X,Y,Ya] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),Ya) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Ya)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Ya)) ) ).
fof(fact_1081_divide_Ominus,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)) ) ).
fof(fact_1082_minus__divide__divide,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) ) ).
fof(fact_1083_minus__divide__left,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),B) ) ).
fof(fact_1084_minus__divide__right,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ).
fof(fact_1085_abs__divide,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ).
fof(fact_1086_real__divide__square__eq,axiom,
! [R_4,A_1] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),A_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),R_4)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A_1),R_4) ).
fof(fact_1087_fst__eqD,axiom,
! [X_a,X_b,X,Y,A_1] :
( hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X),Y)) = ti(X_a,A_1)
=> ti(X_a,X) = ti(X_a,A_1) ) ).
fof(fact_1088_fst__conv,axiom,
! [X_a,X_b,A_1,B] : hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_1),B)) = ti(X_a,A_1) ).
fof(fact_1089_snd__eqD,axiom,
! [X_b,X_a,X,Y,A_1] :
( hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),X),Y)) = ti(X_a,A_1)
=> ti(X_a,Y) = ti(X_a,A_1) ) ).
fof(fact_1090_snd__conv,axiom,
! [X_b,X_a,A_1,B] : hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),A_1),B)) = ti(X_a,B) ).
fof(fact_1091_divide__right__mono__neg,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C))) ) ) ) ).
fof(fact_1092_divide__right__mono,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C))) ) ) ) ).
fof(fact_1093_divide__le__0__iff,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Ba)),zero_zero(X_a)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),zero_zero(X_a))) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Ba)) ) ) ) ) ).
fof(fact_1094_zero__le__divide__iff,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Ba)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Ba)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),zero_zero(X_a))) ) ) ) ) ).
fof(fact_1095_zero__less__divide__iff,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Ba)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Ba)) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),zero_zero(X_a))) ) ) ) ) ).
fof(fact_1096_divide__less__0__iff,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Ba)),zero_zero(X_a)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),zero_zero(X_a))) )
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Ba)) ) ) ) ) ).
fof(fact_1097_divide__less__cancel,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa,C_1,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba)) )
& ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),Aa)) )
& ti(X_a,C_1) != zero_zero(X_a) ) ) ) ).
fof(fact_1098_divide__pos__pos,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).
fof(fact_1099_divide__pos__neg,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a))) ) ) ) ).
fof(fact_1100_divide__neg__pos,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a))) ) ) ) ).
fof(fact_1101_divide__neg__neg,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).
fof(fact_1102_divide__strict__right__mono,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C))) ) ) ) ).
fof(fact_1103_divide__strict__right__mono__neg,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C))) ) ) ) ).
fof(fact_1104_eq__divide__imp,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1,B,C] :
( ti(X_a,C) != zero_zero(X_a)
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C) = ti(X_a,B)
=> ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C) ) ) ) ).
fof(fact_1105_divide__eq__imp,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [B,A_1,C] :
( ti(X_a,C) != zero_zero(X_a)
=> ( ti(X_a,B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),C) = ti(X_a,A_1) ) ) ) ).
fof(fact_1106_frac__eq__eq,axiom,
! [X_a] :
( field(X_a)
=> ! [X_1,W_1,Z_1,Y_1] :
( ti(X_a,Y_1) != zero_zero(X_a)
=> ( ti(X_a,Z_1) != zero_zero(X_a)
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X_1),Y_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),W_1),Z_1)
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Z_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W_1),Y_1) ) ) ) ) ).
fof(fact_1107_mult__divide__mult__cancel__left,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [A_1,B,C] :
( ti(X_a,C) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) ) ) ).
fof(fact_1108_mult__divide__mult__cancel__right,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [A_1,B,C] :
( ti(X_a,C) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) ) ) ).
fof(fact_1109_nonzero__divide__eq__eq,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [Ba,Aa,C_1] :
( ti(X_a,C_1) != zero_zero(X_a)
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1) = ti(X_a,Aa)
<=> ti(X_a,Ba) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1) ) ) ) ).
fof(fact_1110_nonzero__eq__divide__eq,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [Aa,Ba,C_1] :
( ti(X_a,C_1) != zero_zero(X_a)
=> ( ti(X_a,Aa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1) = ti(X_a,Ba) ) ) ) ).
fof(fact_1111_divide__eq__eq,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [Ba,C_1,Aa] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1) = ti(X_a,Aa)
<=> ( ( ti(X_a,C_1) != zero_zero(X_a)
=> ti(X_a,Ba) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1) )
& ( ti(X_a,C_1) = zero_zero(X_a)
=> ti(X_a,Aa) = zero_zero(X_a) ) ) ) ) ).
fof(fact_1112_eq__divide__eq,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [Aa,Ba,C_1] :
( ti(X_a,Aa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)
<=> ( ( ti(X_a,C_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1) = ti(X_a,Ba) )
& ( ti(X_a,C_1) = zero_zero(X_a)
=> ti(X_a,Aa) = zero_zero(X_a) ) ) ) ) ).
fof(fact_1113_divide__self__if,axiom,
! [X_a] :
( divisi14063676e_zero(X_a)
=> ! [A_1] :
( ( ti(X_a,A_1) = zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),A_1) = zero_zero(X_a) )
& ( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),A_1) = one_one(X_a) ) ) ) ).
fof(fact_1114_divide__self,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),A_1) = one_one(X_a) ) ) ).
fof(fact_1115_right__inverse__eq,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [Aa,Ba] :
( ti(X_a,Ba) != zero_zero(X_a)
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Ba) = one_one(X_a)
<=> ti(X_a,Aa) = ti(X_a,Ba) ) ) ) ).
fof(fact_1116_nonzero__minus__divide__divide,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) ) ) ).
fof(fact_1117_nonzero__minus__divide__right,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) ) ) ).
fof(fact_1118_nonzero__abs__divide,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(X_a,X_a,abs_abs(X_a),B)) ) ) ).
fof(fact_1119_real__0__le__divide__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_1),Y_1)))
<=> ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),zero_zero(real)))
| hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_1)) )
& ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
| hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),zero_zero(real))) ) ) ) ).
fof(fact_1120_lemma__MVT,axiom,
! [F,Aa,Ba] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,Aa)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,Ba)),hAPP(real,real,F,Aa))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Ba),Aa))),Aa)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,Ba)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,Ba)),hAPP(real,real,F,Aa))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Ba),Aa))),Ba)) ).
fof(fact_1121_mod__int__def,axiom,
! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B)) ).
fof(fact_1122_div__int__def,axiom,
! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,A_1),B)) ).
fof(fact_1123_divide__le__cancel,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa,C_1,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba)) )
& ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),Aa)) ) ) ) ) ).
fof(fact_1124_divide__nonneg__pos,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).
fof(fact_1125_divide__nonneg__neg,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a))) ) ) ) ).
fof(fact_1126_frac__less2,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Z,W,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),W))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),W),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),W))) ) ) ) ) ) ).
fof(fact_1127_frac__less,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Z,W,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),W))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),W))) ) ) ) ) ) ).
fof(fact_1128_frac__le,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Z,W,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),W))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),W),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),W))) ) ) ) ) ) ).
fof(fact_1129_divide__nonpos__pos,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),zero_zero(X_a))) ) ) ) ).
fof(fact_1130_divide__nonpos__neg,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).
fof(fact_1131_divide__strict__left__mono__neg,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C),B))) ) ) ) ) ).
fof(fact_1132_divide__strict__left__mono,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C),B))) ) ) ) ) ).
fof(fact_1133_neg__divide__less__eq,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Ba,Aa,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)),Aa))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),Ba)) ) ) ) ).
fof(fact_1134_neg__less__divide__eq,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Aa,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1))) ) ) ) ).
fof(fact_1135_mult__imp__less__div__pos,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),Y)),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).
fof(fact_1136_mult__imp__div__pos__less,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [X,Z,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),Y)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),Z)) ) ) ) ).
fof(fact_1137_pos__divide__less__eq,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Ba,Aa,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)),Aa))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1))) ) ) ) ).
fof(fact_1138_pos__less__divide__eq,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Aa,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),Ba)) ) ) ) ).
fof(fact_1139_divide__less__eq,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Ba,C_1,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)),Aa))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Aa)) ) ) ) ) ) ) ).
fof(fact_1140_less__divide__eq,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a))) ) ) ) ) ) ) ).
fof(fact_1141_add__frac__eq,axiom,
! [X_a] :
( field(X_a)
=> ! [X,W,Z,Y] :
( ti(X_a,Y) != zero_zero(X_a)
=> ( ti(X_a,Z) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),W),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W),Y))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Z)) ) ) ) ).
fof(fact_1142_divide__add__eq__iff,axiom,
! [X_a] :
( field(X_a)
=> ! [X,Y,Z] :
( ti(X_a,Z) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z)),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),Y))),Z) ) ) ).
fof(fact_1143_add__frac__num,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [X,Z,Y] :
( ti(X_a,Y) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),Y))),Y) ) ) ).
fof(fact_1144_add__divide__eq__iff,axiom,
! [X_a] :
( field(X_a)
=> ! [X,Y,Z] :
( ti(X_a,Z) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),X)),Y)),Z) ) ) ).
fof(fact_1145_add__num__frac,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [Z,X,Y] :
( ti(X_a,Y) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),Y))),Y) ) ) ).
fof(fact_1146_divide__eq__eq__number__of1,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& number(X_a) )
=> ! [Ba,W_1,Aa] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),hAPP(int,X_a,number_number_of(X_a),W_1)) = ti(X_a,Aa)
<=> ( ( hAPP(int,X_a,number_number_of(X_a),W_1) != zero_zero(X_a)
=> ti(X_a,Ba) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),hAPP(int,X_a,number_number_of(X_a),W_1)) )
& ( hAPP(int,X_a,number_number_of(X_a),W_1) = zero_zero(X_a)
=> ti(X_a,Aa) = zero_zero(X_a) ) ) ) ) ).
fof(fact_1147_divide__eq__eq__number__of,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& number(X_a) )
=> ! [Ba,C_1,W_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1) = hAPP(int,X_a,number_number_of(X_a),W_1)
<=> ( ( ti(X_a,C_1) != zero_zero(X_a)
=> ti(X_a,Ba) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C_1) )
& ( ti(X_a,C_1) = zero_zero(X_a)
=> hAPP(int,X_a,number_number_of(X_a),W_1) = zero_zero(X_a) ) ) ) ) ).
fof(fact_1148_eq__divide__eq__number__of,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& number(X_a) )
=> ! [W_1,Ba,C_1] :
( hAPP(int,X_a,number_number_of(X_a),W_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)
<=> ( ( ti(X_a,C_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C_1) = ti(X_a,Ba) )
& ( ti(X_a,C_1) = zero_zero(X_a)
=> hAPP(int,X_a,number_number_of(X_a),W_1) = zero_zero(X_a) ) ) ) ) ).
fof(fact_1149_eq__divide__eq__number__of1,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& number(X_a) )
=> ! [Aa,Ba,W_1] :
( ti(X_a,Aa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),hAPP(int,X_a,number_number_of(X_a),W_1))
<=> ( ( hAPP(int,X_a,number_number_of(X_a),W_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),hAPP(int,X_a,number_number_of(X_a),W_1)) = ti(X_a,Ba) )
& ( hAPP(int,X_a,number_number_of(X_a),W_1) = zero_zero(X_a)
=> ti(X_a,Aa) = zero_zero(X_a) ) ) ) ) ).
fof(fact_1150_diff__frac__eq,axiom,
! [X_a] :
( field(X_a)
=> ! [X,W,Z,Y] :
( ti(X_a,Y) != zero_zero(X_a)
=> ( ti(X_a,Z) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),W),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),W),Y))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Z)) ) ) ) ).
fof(fact_1151_divide__diff__eq__iff,axiom,
! [X_a] :
( field(X_a)
=> ! [X,Y,Z] :
( ti(X_a,Z) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Z)),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),Y))),Z) ) ) ).
fof(fact_1152_diff__divide__eq__iff,axiom,
! [X_a] :
( field(X_a)
=> ! [X,Y,Z] :
( ti(X_a,Z) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),X)),Y)),Z) ) ) ).
fof(fact_1153_gt__half__sum,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)))),B)) ) ) ).
fof(fact_1154_less__half__sum,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))))) ) ) ).
fof(fact_1155_DERIV__mult__lemma,axiom,
! [X_a] :
( real_field(X_a)
=> ! [A_1,B,C,D,H_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C),D))),H_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),D)),H_2))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),C)),H_2)),D)) ) ).
fof(fact_1156_abs__div__pos,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),Y) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)) ) ) ).
fof(fact_1157_divide__minus1,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& number_ring(X_a) )
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),hAPP(int,X_a,number_number_of(X_a),min)) = hAPP(X_a,X_a,uminus_uminus(X_a),X) ) ).
fof(fact_1158_real__of__nat__div4,axiom,
! [N_1,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N_1),X))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(nat,real,real_1(nat),X)))) ).
fof(fact_1159_divide__left__mono__neg,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C),B))) ) ) ) ) ).
fof(fact_1160_divide__left__mono,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [C,B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),C),B))) ) ) ) ) ).
fof(fact_1161_neg__divide__le__eq,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Ba,Aa,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)),Aa))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),Ba)) ) ) ) ).
fof(fact_1162_neg__le__divide__eq,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Aa,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1))) ) ) ) ).
fof(fact_1163_mult__imp__le__div__pos,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),Y)),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y))) ) ) ) ).
fof(fact_1164_mult__imp__div__pos__le,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [X,Z,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),Y)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)),Z)) ) ) ) ).
fof(fact_1165_pos__divide__le__eq,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Ba,Aa,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)),Aa))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1))) ) ) ) ).
fof(fact_1166_pos__le__divide__eq,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [Aa,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),Ba)) ) ) ) ).
fof(fact_1167_divide__le__eq,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Ba,C_1,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)),Aa))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa)) ) ) ) ) ) ) ).
fof(fact_1168_le__divide__eq,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),zero_zero(X_a))) ) ) ) ) ) ) ).
fof(fact_1169_divide__less__eq__number__of1,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number(X_a) )
=> ! [Ba,W_1,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),hAPP(int,X_a,number_number_of(X_a),W_1))),Aa))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W_1)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),hAPP(int,X_a,number_number_of(X_a),W_1)))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W_1)))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),hAPP(int,X_a,number_number_of(X_a),W_1))),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Aa)) ) ) ) ) ) ) ).
fof(fact_1170_divide__less__eq__number__of,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number(X_a) )
=> ! [Ba,C_1,W_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)),hAPP(int,X_a,number_number_of(X_a),W_1)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C_1))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C_1)),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W_1))) ) ) ) ) ) ) ).
fof(fact_1171_less__divide__eq__number__of,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number(X_a) )
=> ! [W_1,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C_1)),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C_1))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),zero_zero(X_a))) ) ) ) ) ) ) ).
fof(fact_1172_less__divide__eq__number__of1,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number(X_a) )
=> ! [Aa,Ba,W_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),hAPP(int,X_a,number_number_of(X_a),W_1))))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W_1)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),hAPP(int,X_a,number_number_of(X_a),W_1))),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W_1)))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),hAPP(int,X_a,number_number_of(X_a),W_1)))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a))) ) ) ) ) ) ) ).
fof(fact_1173_minus1__divide,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& number_ring(X_a) )
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),X)) ) ).
fof(fact_1174_mod__neg__pos,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B)) ) ) ).
fof(fact_1175_div__neg__pos,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B)) ) ) ).
fof(fact_1176_divide__le__eq__number__of1,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number(X_a) )
=> ! [Ba,W_1,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),hAPP(int,X_a,number_number_of(X_a),W_1))),Aa))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W_1)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),hAPP(int,X_a,number_number_of(X_a),W_1)))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W_1)))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),hAPP(int,X_a,number_number_of(X_a),W_1))),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa)) ) ) ) ) ) ) ).
fof(fact_1177_divide__le__eq__number__of,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number(X_a) )
=> ! [Ba,C_1,W_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)),hAPP(int,X_a,number_number_of(X_a),W_1)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C_1))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C_1)),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W_1))) ) ) ) ) ) ) ).
fof(fact_1178_le__divide__eq__number__of,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number(X_a) )
=> ! [W_1,Ba,C_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),C_1)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C_1)),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),C_1))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),C_1))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),zero_zero(X_a))) ) ) ) ) ) ) ).
fof(fact_1179_le__divide__eq__number__of1,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number(X_a) )
=> ! [Aa,Ba,W_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Ba),hAPP(int,X_a,number_number_of(X_a),W_1))))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W_1)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),hAPP(int,X_a,number_number_of(X_a),W_1))),Ba)) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W_1)))
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),hAPP(int,X_a,number_number_of(X_a),W_1)))) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),zero_zero(X_a))) ) ) ) ) ) ) ).
fof(fact_1180_mod__pos__pos,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B)) ) ) ).
fof(fact_1181_div__pos__pos,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B)) ) ) ).
fof(fact_1182_real__of__nat__div2,axiom,
! [N_1,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(nat,real,real_1(nat),X))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N_1),X))))) ).
fof(fact_1183_real__of__nat__div3,axiom,
! [N_1,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(nat,real,real_1(nat),X))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N_1),X)))),one_one(real))) ).
fof(fact_1184_mod__pos__pos__1__number__of,axiom,
! [W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_mod(int),one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W))) ) ).
fof(fact_1185_div__pos__pos__1__number__of,axiom,
! [W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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(product_prod(int,int),int,product_fst(int,int),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W))) ) ).
fof(fact_1186_mod__pos__neg,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B) = hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) ) ) ).
fof(fact_1187_real__of__nat__div__aux,axiom,
! [X,D] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),D))
=> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),X)),hAPP(nat,real,real_1(nat),D)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),D))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),D))),hAPP(nat,real,real_1(nat),D))) ) ).
fof(fact_1188_div__pos__neg,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) = hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(product_prod(int,int),product_prod(int,int),negateSnd,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)))) ) ) ).
fof(fact_1189_conjI__realizer,axiom,
! [X_b,X_a,Q_1,Qa,P_1,Pa] :
( hBOOL(hAPP(X_a,bool,P_1,Pa))
=> ( hBOOL(hAPP(X_b,bool,Q_1,Qa))
=> ( hBOOL(hAPP(X_a,bool,P_1,hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Pa),Qa))))
& hBOOL(hAPP(X_b,bool,Q_1,hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Pa),Qa)))) ) ) ) ).
fof(fact_1190_exI__realizer,axiom,
! [X_b,X_a,P_1,Y_1,X_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Y_1),X_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),X_1),Y_1))),hAPP(product_prod(X_b,X_a),X_b,product_fst(X_b,X_a),hAPP(X_a,product_prod(X_b,X_a),hAPP(X_b,fun(X_a,product_prod(X_b,X_a)),product_Pair(X_b,X_a),X_1),Y_1)))) ) ).
fof(fact_1191_arctan__add,axiom,
! [Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),one_one(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),Y)),one_one(real)))
=> hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,arctan,X)),hAPP(real,real,arctan,Y)) = hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)))) ) ) ).
fof(fact_1192_pair__lessI2,axiom,
! [S_3,T_3,Aa,Ba] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Aa),Ba))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),S_3),T_3))
=> hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Aa),S_3)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Ba),T_3))),pair_less)) ) ) ).
fof(fact_1193_mod__nat__number__of,axiom,
! [V_3,V] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_3)) = zero_zero(nat) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> ( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_3)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_3)) = hAPP(int,nat,number_number_of(nat),V) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_3)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_3)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),V_3))) ) ) ) ) ).
fof(fact_1194_div__nat__number__of,axiom,
! [V_3,V] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_3)) = zero_zero(nat) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_3)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),V_3))) ) ) ).
fof(fact_1195_divmod__nat__div__mod,axiom,
! [M_1,N_1] : hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M_1),N_1) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)) ).
fof(fact_1196_mod__nat__def,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1) = hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M_1),N_1)) ).
fof(fact_1197_div__nat__def,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1) = hAPP(product_prod(nat,nat),nat,product_fst(nat,nat),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M_1),N_1)) ).
fof(fact_1198_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_1199_nat__if__cong,axiom,
! [X_1,Y_1,P_1] :
( ( hBOOL(P_1)
=> hAPP(int,nat,nat_1,X_1) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),P_1),X_1),Y_1)) )
& ( ~ hBOOL(P_1)
=> hAPP(int,nat,nat_1,Y_1) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),P_1),X_1),Y_1)) ) ) ).
fof(fact_1200_arctan__monotone,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,X)),hAPP(real,real,arctan,Y))) ) ).
fof(fact_1201_arctan__zero__zero,axiom,
hAPP(real,real,arctan,zero_zero(real)) = zero_zero(real) ).
fof(fact_1202_arctan__monotone_H,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arctan,X)),hAPP(real,real,arctan,Y))) ) ).
fof(fact_1203_natceiling__def,axiom,
! [X] : hAPP(real,nat,natceiling,X) = hAPP(int,nat,nat_1,hAPP(real,int,archimedean_ceiling(real),X)) ).
fof(fact_1204_nat__0,axiom,
hAPP(int,nat,nat_1,zero_zero(int)) = zero_zero(nat) ).
fof(fact_1205_transfer__nat__int__numerals_I1_J,axiom,
zero_zero(nat) = hAPP(int,nat,nat_1,zero_zero(int)) ).
fof(fact_1206_eq__nat__nat__iff,axiom,
! [Z_3,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_3))
=> ( hAPP(int,nat,nat_1,Z_1) = hAPP(int,nat,nat_1,Z_3)
<=> Z_1 = Z_3 ) ) ) ).
fof(fact_1207_ex__nat,axiom,
! [P_1] :
( ? [X1] : hBOOL(hAPP(nat,bool,P_1,X1))
<=> ? [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
& hBOOL(hAPP(nat,bool,P_1,hAPP(int,nat,nat_1,X_2))) ) ) ).
fof(fact_1208_all__nat,axiom,
! [P_1] :
( ! [X1] : hBOOL(hAPP(nat,bool,P_1,X1))
<=> ! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
=> hBOOL(hAPP(nat,bool,P_1,hAPP(int,nat,nat_1,X_2))) ) ) ).
fof(fact_1209_transfer__nat__int__relations_I1_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
=> ( hAPP(int,nat,nat_1,X_1) = hAPP(int,nat,nat_1,Y_1)
<=> X_1 = Y_1 ) ) ) ).
fof(fact_1210_nat__number__of__def,axiom,
! [V] : hAPP(int,nat,number_number_of(nat),V) = hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),V)) ).
fof(fact_1211_nat__number__of,axiom,
! [W] : hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),W)) = hAPP(int,nat,number_number_of(nat),W) ).
fof(fact_1212_transfer__nat__int__numerals_I2_J,axiom,
one_one(nat) = hAPP(int,nat,nat_1,one_one(int)) ).
fof(fact_1213_neg__nat,axiom,
! [Z] :
( hBOOL(hAPP(int,bool,nat_neg,Z))
=> hAPP(int,nat,nat_1,Z) = zero_zero(nat) ) ).
fof(fact_1214_natfloor__def,axiom,
! [X] : hAPP(real,nat,natfloor,X) = hAPP(int,nat,nat_1,hAPP(real,int,archim2085082626_floor(real),X)) ).
fof(fact_1215_divmod__nat__rel__divmod__nat,axiom,
! [M_1,N_1] : hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M_1),N_1),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M_1),N_1))) ).
fof(fact_1216_divmod__nat__eq,axiom,
! [M_1,N_1,Qr_1] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M_1),N_1),Qr_1))
=> hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M_1),N_1) = Qr_1 ) ).
fof(fact_1217_nat__le__0,axiom,
! [Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int)))
=> hAPP(int,nat,nat_1,Z) = zero_zero(nat) ) ).
fof(fact_1218_nat__0__iff,axiom,
! [I] :
( hAPP(int,nat,nat_1,I) = zero_zero(nat)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int))) ) ).
fof(fact_1219_nat__mono__iff,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,W_1)),hAPP(int,nat,nat_1,Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_1)) ) ) ).
fof(fact_1220_zless__nat__conj,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,W_1)),hAPP(int,nat,nat_1,Z_1)))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_1)) ) ) ).
fof(fact_1221_transfer__nat__int__relations_I3_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,nat_1,X_1)),hAPP(int,nat,nat_1,Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),Y_1)) ) ) ) ).
fof(fact_1222_nat__abs__mult__distrib,axiom,
! [W,Z] : hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),W))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Z))) ).
fof(fact_1223_divmod__nat__rel__def,axiom,
! [M,N,Qr] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N),Qr))
<=> ( M = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(product_prod(nat,nat),nat,product_fst(nat,nat),Qr)),N)),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr))
& ( N = zero_zero(nat)
=> hAPP(product_prod(nat,nat),nat,product_fst(nat,nat),Qr) = zero_zero(nat) )
& ( N != zero_zero(nat)
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)),N)) ) )
& ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(product_prod(nat,nat),nat,product_snd(nat,nat),Qr)),zero_zero(nat))) ) ) ) ) ) ) ).
fof(fact_1224_zero__less__nat__eq,axiom,
! [Z_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,nat_1,Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) ) ).
fof(fact_1225_nat__less__eq__zless,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,W_1)),hAPP(int,nat,nat_1,Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_1)) ) ) ).
fof(fact_1226_transfer__nat__int__relations_I2_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,X_1)),hAPP(int,nat,nat_1,Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Y_1)) ) ) ) ).
fof(fact_1227_nat__le__eq__zle,axiom,
! [Z_1,W_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),W_1))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,nat_1,W_1)),hAPP(int,nat,nat_1,Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z_1)) ) ) ).
fof(fact_1228_nat__mult__distrib,axiom,
! [Z_4,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z),Z_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,Z)),hAPP(int,nat,nat_1,Z_4)) ) ).
fof(fact_1229_Nat__Transfer_Otransfer__nat__int__functions_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)) ) ) ).
fof(fact_1230_nat__add__distrib,axiom,
! [Z_4,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_4))
=> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z),Z_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,Z)),hAPP(int,nat,nat_1,Z_4)) ) ) ).
fof(fact_1231_Nat__Transfer_Otransfer__nat__int__functions_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)) ) ) ).
fof(fact_1232_divmod__nat__zero,axiom,
! [M_1] : hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M_1),zero_zero(nat)) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M_1) ).
fof(fact_1233_nat__diff__distrib,axiom,
! [Z,Z_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_4),Z))
=> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z),Z_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,Z)),hAPP(int,nat,nat_1,Z_4)) ) ) ).
fof(fact_1234_Divides_Otransfer__nat__int__functions_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),Y)) ) ) ).
fof(fact_1235_nat__div__distrib,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hAPP(int,nat,nat_1,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_1,X)),hAPP(int,nat,nat_1,Y)) ) ).
fof(fact_1236_Divides_Otransfer__nat__int__functions_I2_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y)) ) ) ).
fof(fact_1237_nat__mod__distrib,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),Y)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) ) ) ).
fof(fact_1238_pair__lessI1,axiom,
! [S_3,T_3,Aa,Ba] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Aa),Ba))
=> hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Aa),S_3)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Ba),T_3))),pair_less)) ) ).
fof(fact_1239_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,X),Y)) ) ) ).
fof(fact_1240_nat__mult__distrib__neg,axiom,
! [Z_4,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int)))
=> hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Z),Z_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),Z))),hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),Z_4))) ) ).
fof(fact_1241_divmod__nat__base,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M_1),N_1) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M_1) ) ).
fof(fact_1242_nat__aux__def,axiom,
! [I_1,N_1] : hAPP(nat,nat,hAPP(int,fun(nat,nat),nat_aux,I_1),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,I_1)),N_1) ).
fof(fact_1243_transfer__morphism__nat__int,axiom,
hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,nat),fun(fun(int,bool),bool),nat_tr1645093318rphism(int,nat),nat_1),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))) ).
fof(fact_1244_exE__realizer_H,axiom,
! [X_b,X_a,P_1,Pa] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),Pa)),hAPP(product_prod(X_b,X_a),X_b,product_fst(X_b,X_a),Pa)))
=> ~ ! [X_2,Y_2] : ~ hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Y_2),X_2)) ) ).
fof(fact_1245_divmod__nat__step,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M_1),N_1) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,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_1),N_1)),N_1))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),N_1)) ) ) ).
fof(fact_1246_one__mod__nat__number__of,axiom,
! [V_3] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_3)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,number_number_of(nat),V_3)) = hAPP(nat,nat,suc,zero_zero(nat)) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V_3)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,number_number_of(nat),V_3)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),one_one(int)),hAPP(int,int,number_number_of(int),V_3))) ) ) ).
fof(fact_1247_one__div__nat__number__of,axiom,
! [V_3] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,number_number_of(nat),V_3)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),one_one(int)),hAPP(int,int,number_number_of(int),V_3))) ).
fof(fact_1248_arccos__lbound,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,arccos,Y))) ) ) ).
fof(fact_1249_transfer__morphismI,axiom,
! [X_a,X_b,F,A_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),bool),nat_tr1645093318rphism(X_a,X_b),F),A_2)) ).
fof(fact_1250_Suc__mono,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M_1)),hAPP(nat,nat,suc,N_1))) ) ).
fof(fact_1251_lessI,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(nat,nat,suc,N_1))) ).
fof(fact_1252_zero__less__Suc,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,suc,N_1))) ).
fof(fact_1253_Suc__inject,axiom,
! [X,Y] :
( hAPP(nat,nat,suc,X) = hAPP(nat,nat,suc,Y)
=> X = Y ) ).
fof(fact_1254_nat_Oinject,axiom,
! [Nat,Nat_5] :
( hAPP(nat,nat,suc,Nat) = hAPP(nat,nat,suc,Nat_5)
<=> Nat = Nat_5 ) ).
fof(fact_1255_Suc__n__not__n,axiom,
! [N_1] : hAPP(nat,nat,suc,N_1) != N_1 ).
fof(fact_1256_n__not__Suc__n,axiom,
! [N_1] : N_1 != hAPP(nat,nat,suc,N_1) ).
fof(fact_1257_Zero__not__Suc,axiom,
! [M_1] : zero_zero(nat) != hAPP(nat,nat,suc,M_1) ).
fof(fact_1258_nat_Osimps_I2_J,axiom,
! [Nat_4] : zero_zero(nat) != hAPP(nat,nat,suc,Nat_4) ).
fof(fact_1259_Suc__not__Zero,axiom,
! [M_1] : hAPP(nat,nat,suc,M_1) != zero_zero(nat) ).
fof(fact_1260_nat_Osimps_I3_J,axiom,
! [Nat_3] : hAPP(nat,nat,suc,Nat_3) != zero_zero(nat) ).
fof(fact_1261_Zero__neq__Suc,axiom,
! [M_1] : zero_zero(nat) != hAPP(nat,nat,suc,M_1) ).
fof(fact_1262_Suc__neq__Zero,axiom,
! [M_1] : hAPP(nat,nat,suc,M_1) != zero_zero(nat) ).
fof(fact_1263_not__less__eq,axiom,
! [M,N] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,M))) ) ).
fof(fact_1264_less__Suc__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
| M = N ) ) ).
fof(fact_1265_Suc__less__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_1266_not__less__less__Suc__eq,axiom,
! [N,M] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,M)))
<=> N = M ) ) ).
fof(fact_1267_less__antisym,axiom,
! [N_1,M_1] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(nat,nat,suc,M_1)))
=> M_1 = N_1 ) ) ).
fof(fact_1268_less__SucI,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),hAPP(nat,nat,suc,N_1))) ) ).
fof(fact_1269_Suc__lessI,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> ( hAPP(nat,nat,suc,M_1) != N_1
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M_1)),N_1)) ) ) ).
fof(fact_1270_less__trans__Suc,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_2),K_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,I_1)),K_4)) ) ) ).
fof(fact_1271_less__SucE,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),hAPP(nat,nat,suc,N_1)))
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> M_1 = N_1 ) ) ).
fof(fact_1272_Suc__lessD,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M_1)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ).
fof(fact_1273_Suc__less__SucD,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,M_1)),hAPP(nat,nat,suc,N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ).
fof(fact_1274_transfer__morphism__def,axiom,
! [X_a,X_b,F,A_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),bool),nat_tr1645093318rphism(X_a,X_b),F),A_2)) ).
fof(fact_1275_add__Suc__right,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) ).
fof(fact_1276_add__Suc,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) ).
fof(fact_1277_add__Suc__shift,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),hAPP(nat,nat,suc,N_1)) ).
fof(fact_1278_Suc__leD,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M_1)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ).
fof(fact_1279_le__SucE,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,suc,N_1)))
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> M_1 = hAPP(nat,nat,suc,N_1) ) ) ).
fof(fact_1280_le__SucI,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),hAPP(nat,nat,suc,N_1))) ) ).
fof(fact_1281_Suc__le__mono,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),hAPP(nat,nat,suc,M)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).
fof(fact_1282_le__Suc__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,suc,N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
| M = hAPP(nat,nat,suc,N) ) ) ).
fof(fact_1283_not__less__eq__eq,axiom,
! [M,N] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),M)) ) ).
fof(fact_1284_Suc__n__not__le__n,axiom,
! [N_1] : ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N_1)),N_1)) ).
fof(fact_1285_Suc__mult__cancel1,axiom,
! [K,M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),M) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),N)
<=> M = N ) ).
fof(fact_1286_Suc__diff__diff,axiom,
! [M_1,N_1,K_4] : 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_1)),N_1)),hAPP(nat,nat,suc,K_4)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),K_4) ).
fof(fact_1287_diff__Suc__Suc,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M_1)),hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1) ).
fof(fact_1288_mod__Suc__eq__Suc__mod,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1))),N_1) ).
fof(fact_1289_less__Suc__eq__0__disj,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N)))
<=> ( M = zero_zero(nat)
| ? [J_1] :
( M = hAPP(nat,nat,suc,J_1)
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J_1),N)) ) ) ) ).
fof(fact_1290_less__Suc0,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,zero_zero(nat))))
<=> N = zero_zero(nat) ) ).
fof(fact_1291_gr0__conv__Suc,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
<=> ? [M_3] : N = hAPP(nat,nat,suc,M_3) ) ).
fof(fact_1292_one__is__add,axiom,
! [M,N] :
( hAPP(nat,nat,suc,zero_zero(nat)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)
<=> ( ( M = hAPP(nat,nat,suc,zero_zero(nat))
& N = zero_zero(nat) )
| ( M = zero_zero(nat)
& N = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ) ).
fof(fact_1293_add__is__1,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N) = hAPP(nat,nat,suc,zero_zero(nat))
<=> ( ( M = hAPP(nat,nat,suc,zero_zero(nat))
& N = zero_zero(nat) )
| ( M = zero_zero(nat)
& N = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ) ).
fof(fact_1294_mult__eq__1__iff,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N) = hAPP(nat,nat,suc,zero_zero(nat))
<=> ( M = hAPP(nat,nat,suc,zero_zero(nat))
& N = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).
fof(fact_1295_less__add__Suc1,axiom,
! [I_1,M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I_1),M_1)))) ).
fof(fact_1296_less__add__Suc2,axiom,
! [I_1,M_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),I_1)))) ).
fof(fact_1297_less__iff__Suc__add,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_1298_One__nat__def,axiom,
one_one(nat) = hAPP(nat,nat,suc,zero_zero(nat)) ).
fof(fact_1299_Suc__le__lessD,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M_1)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ).
fof(fact_1300_le__less__Suc__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(nat,nat,suc,M)))
<=> N = M ) ) ).
fof(fact_1301_Suc__leI,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M_1)),N_1)) ) ).
fof(fact_1302_le__imp__less__Suc,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),hAPP(nat,nat,suc,N_1))) ) ).
fof(fact_1303_Suc__le__eq,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_1304_less__Suc__eq__le,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,suc,N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_1305_less__eq__Suc__le,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),M)) ) ).
fof(fact_1306_Suc__mult__less__cancel1,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_1307_diff__less__Suc,axiom,
! [M_1,N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),hAPP(nat,nat,suc,M_1))) ).
fof(fact_1308_mult__Suc,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)) ).
fof(fact_1309_mult__Suc__right,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)) ).
fof(fact_1310_Suc__mult__le__cancel1,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,K)),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_1311_Suc__eq__plus1__left,axiom,
! [N_1] : hAPP(nat,nat,suc,N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N_1) ).
fof(fact_1312_Suc__eq__plus1,axiom,
! [N_1] : hAPP(nat,nat,suc,N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),one_one(nat)) ).
fof(fact_1313_Suc__diff__le,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)) ) ).
fof(fact_1314_diff__Suc__1,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,N_1)),one_one(nat)) = N_1 ).
fof(fact_1315_diff__Suc__eq__diff__pred,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),one_one(nat))),N_1) ).
fof(fact_1316_div__1,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),hAPP(nat,nat,suc,zero_zero(nat))) = M_1 ).
fof(fact_1317_mod__Suc,axiom,
! [M_1,N_1] :
( ( hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)) = N_1
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M_1)),N_1) = zero_zero(nat) )
& ( hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)) != N_1
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),N_1)) ) ) ).
fof(fact_1318_mod__1,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),hAPP(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).
fof(fact_1319_one__less__m,axiom,
! [M_1,K_4] :
( M_1 != hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),K_4)
=> ( M_1 != hAPP(nat,nat,suc,zero_zero(nat))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_1)) ) ) ).
fof(fact_1320_one__less__k,axiom,
! [M_1,K_4] :
( M_1 != hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),K_4)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),K_4)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),K_4)) ) ) ).
fof(fact_1321_prod__mn__less__k,axiom,
! [M_1,K_4,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_1))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = K_4
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),K_4)) ) ) ) ) ).
fof(fact_1322_mn__eq__m__one,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1) = M_1
=> N_1 = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).
fof(fact_1323_n__less__m__mult__n,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1))) ) ) ).
fof(fact_1324_n__less__n__mult__m,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),M_1))) ) ) ).
fof(fact_1325_one__less__mult,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1))) ) ) ).
fof(fact_1326_diff__Suc__less,axiom,
! [I_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(nat,nat,suc,I_1))),N_1)) ) ).
fof(fact_1327_Suc__pred,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(nat,nat,suc,zero_zero(nat)))) = N_1 ) ).
fof(fact_1328_one__le__mult__iff,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),M))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),N)) ) ) ).
fof(fact_1329_real__of__nat__Suc__gt__zero,axiom,
! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1)))) ).
fof(fact_1330_nat__1,axiom,
hAPP(int,nat,nat_1,one_one(int)) = hAPP(nat,nat,suc,zero_zero(nat)) ).
fof(fact_1331_real__of__nat__one,axiom,
hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,zero_zero(nat))) = one_one(real) ).
fof(fact_1332_diff__Suc__diff__eq1,axiom,
! [M_1,K_4,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),J_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),J_2),K_4))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_4)),hAPP(nat,nat,suc,J_2)) ) ).
fof(fact_1333_diff__Suc__diff__eq2,axiom,
! [M_1,K_4,J_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),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_4))),M_1) = 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_4),M_1)) ) ).
fof(fact_1334_real__of__nat__Suc,axiom,
! [N_1] : hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,real_1(nat),N_1)),one_one(real)) ).
fof(fact_1335_mod__mult__self4,axiom,
! [K_4,N_1,M_1] : hAPP(nat,nat,hAPP(nat,fun(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_4),N_1)),M_1))),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,M_1)),N_1) ).
fof(fact_1336_Suc__diff__1,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat))) = N_1 ) ).
fof(fact_1337_Suc__pred_H,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> N_1 = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat))) ) ).
fof(fact_1338_add__eq__if,axiom,
! [N_1,M_1] :
( ( M_1 = zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = N_1 )
& ( M_1 != zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),one_one(nat))),N_1)) ) ) ).
fof(fact_1339_div__geq,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),N_1)) ) ) ).
fof(fact_1340_div__if,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1) = zero_zero(nat) )
& ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),N_1)) ) ) ) ).
fof(fact_1341_Suc__times__mod__eq,axiom,
! [M_1,K_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),K_4))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M_1))),K_4) = one_one(nat) ) ).
fof(fact_1342_floor__eq3,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N_1)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1))))
=> hAPP(int,nat,nat_1,hAPP(real,int,archim2085082626_floor(real),X)) = N_1 ) ) ).
fof(fact_1343_one__less__nat__eq,axiom,
! [Z_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(int,nat,nat_1,Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Z_1)) ) ).
fof(fact_1344_expand__Suc,axiom,
! [V] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),V)))
=> hAPP(int,nat,number_number_of(nat),V) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat))) ) ).
fof(fact_1345_Suc__nat__eq__nat__zadd1,axiom,
! [Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hAPP(nat,nat,suc,hAPP(int,nat,nat_1,Z)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ).
fof(fact_1346_split__div__lemma,axiom,
! [Qa,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),Qa)),M))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),hAPP(nat,nat,suc,Qa)))) )
<=> Qa = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N) ) ) ).
fof(fact_1347_split__div_H,axiom,
! [P_1,M,N] :
( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M),N)))
<=> ( ( N = zero_zero(nat)
& hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) )
| ? [Q_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),Q_3)),M))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),hAPP(nat,nat,suc,Q_3))))
& hBOOL(hAPP(nat,bool,P_1,Q_3)) ) ) ) ).
fof(fact_1348_le__div__geq,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),N_1)) ) ) ).
fof(fact_1349_floor__eq4,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N_1)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1))))
=> hAPP(int,nat,nat_1,hAPP(real,int,archim2085082626_floor(real),X)) = N_1 ) ) ).
fof(fact_1350_prod__size__simp,axiom,
! [X_a,X_b,F,G,Pa] : hAPP(product_prod(X_a,X_b),nat,hAPP(fun(X_b,nat),fun(product_prod(X_a,X_b),nat),hAPP(fun(X_a,nat),fun(fun(X_b,nat),fun(product_prod(X_a,X_b),nat)),product_prod_size(X_a,X_b),F),G),Pa) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(X_a,nat,F,hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Pa))),hAPP(X_b,nat,G,hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Pa)))),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_1351_reals__Archimedean4,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ? [N_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_2)),Y)),X))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2))),Y))) ) ) ) ).
fof(fact_1352_reals__Archimedean6a,axiom,
! [R_4] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R_4))
=> ? [N_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,real_1(nat),N_2)),R_4))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_4),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2)))) ) ) ).
fof(fact_1353_div__Suc,axiom,
! [A_1,C] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,A_1)),C) = 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_1),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,zero_zero(nat))),C))),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,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),C)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,zero_zero(nat))),C))),C)) ).
fof(fact_1354_mod__induct__0,axiom,
! [I,P_1,Pa] :
( ! [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),Pa))
=> ( hBOOL(hAPP(nat,bool,P_1,I_2))
=> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,I_2)),Pa))) ) )
=> ( hBOOL(hAPP(nat,bool,P_1,I))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),Pa))
=> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) ) ) ) ).
fof(fact_1355_mod__induct,axiom,
! [J,I,P_1,Pa] :
( ! [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_2),Pa))
=> ( hBOOL(hAPP(nat,bool,P_1,I_2))
=> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,I_2)),Pa))) ) )
=> ( hBOOL(hAPP(nat,bool,P_1,I))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I),Pa))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),J),Pa))
=> hBOOL(hAPP(nat,bool,P_1,J)) ) ) ) ) ).
fof(fact_1356_inc__induct,axiom,
! [P_1,I,J] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( hBOOL(hAPP(nat,bool,P_1,J))
=> ( ! [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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)) ) ) ) ).
fof(fact_1357_nat__lt__two__imp__zero__or__one,axiom,
! [X] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_1358_less__imp__Suc__add,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> ? [K_1] : N_1 = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),K_1)) ) ).
fof(fact_1359_lemma__f__mono__add,axiom,
! [M,No_1,F] :
( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_2)),hAPP(nat,real,F,hAPP(nat,nat,suc,N_2))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,M)),hAPP(nat,real,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),No_1)))) ) ).
fof(fact_1360_gr0__implies__Suc,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ? [M_3] : N_1 = hAPP(nat,nat,suc,M_3) ) ).
fof(fact_1361_lift__Suc__mono__le,axiom,
! [X_a] :
( order(X_a)
=> ! [N,N_5,F] :
( ! [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,N_2)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_2))))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_5))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,N)),hAPP(nat,X_a,F,N_5))) ) ) ) ).
fof(fact_1362_lift__Suc__mono__less,axiom,
! [X_a] :
( order(X_a)
=> ! [N,N_5,F] :
( ! [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N_2)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_2))))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_5))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N)),hAPP(nat,X_a,F,N_5))) ) ) ) ).
fof(fact_1363_lift__Suc__mono__less__iff,axiom,
! [X_a] :
( order(X_a)
=> ! [N,M,F] :
( ! [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N_2)),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_2))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,N)),hAPP(nat,X_a,F,M)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).
fof(fact_1364_Suc__nat__number__of__add,axiom,
! [N_1,V] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N_1) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V))),N_1) ) ) ).
fof(fact_1365_zero__induct__lemma,axiom,
! [I,P_1,K] :
( hBOOL(hAPP(nat,bool,P_1,K))
=> ( ! [N_2] :
( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_2)))
=> hBOOL(hAPP(nat,bool,P_1,N_2)) )
=> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),K),I))) ) ) ).
fof(fact_1366_Suc__le__D,axiom,
! [N_1,M_5] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N_1)),M_5))
=> ? [M_3] : M_5 = hAPP(nat,nat,suc,M_3) ) ).
fof(fact_1367_lessE,axiom,
! [I_1,K_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),K_4))
=> ( K_4 != hAPP(nat,nat,suc,I_1)
=> ~ ! [J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1))
=> K_4 != hAPP(nat,nat,suc,J_1) ) ) ) ).
fof(fact_1368_Suc__lessE,axiom,
! [I_1,K_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,I_1)),K_4))
=> ~ ! [J_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),I_1),J_1))
=> K_4 != hAPP(nat,nat,suc,J_1) ) ) ).
fof(fact_1369_not0__implies__Suc,axiom,
! [N_1] :
( N_1 != zero_zero(nat)
=> ? [M_3] : N_1 = hAPP(nat,nat,suc,M_3) ) ).
fof(fact_1370_succ__def,axiom,
! [K_4] : hAPP(int,int,succ,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),one_one(int)) ).
fof(fact_1371_diff__bin__simps_I2_J,axiom,
! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),min) = hAPP(int,int,succ,K_4) ).
fof(fact_1372_number__of__succ,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [K_4] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,succ,K_4)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),K_4)) ) ).
fof(fact_1373_Suc__nat__number__of,axiom,
! [V] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,suc,hAPP(int,nat,number_number_of(nat),V)) = one_one(nat) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,suc,hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V)) ) ) ).
fof(fact_1374_exists__least__lemma,axiom,
! [P_1] :
( ~ hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
=> ( ? [X1] : hBOOL(hAPP(nat,bool,P_1,X1))
=> ? [N_2] :
( ~ hBOOL(hAPP(nat,bool,P_1,N_2))
& hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_2))) ) ) ) ).
fof(fact_1375_nat_Oexhaust,axiom,
! [Y] :
( Y != zero_zero(nat)
=> ~ ! [Nat_2] : Y != hAPP(nat,nat,suc,Nat_2) ) ).
fof(fact_1376_zero__induct,axiom,
! [P_1,K] :
( hBOOL(hAPP(nat,bool,P_1,K))
=> ( ! [N_2] :
( hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_2)))
=> hBOOL(hAPP(nat,bool,P_1,N_2)) )
=> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) ) ) ).
fof(fact_1377_nat__induct,axiom,
! [N,P_1] :
( hBOOL(hAPP(nat,bool,P_1,zero_zero(nat)))
=> ( ! [N_2] :
( hBOOL(hAPP(nat,bool,P_1,N_2))
=> hBOOL(hAPP(nat,bool,P_1,hAPP(nat,nat,suc,N_2))) )
=> hBOOL(hAPP(nat,bool,P_1,N)) ) ) ).
fof(fact_1378_int__power__div__base,axiom,
! [K_4,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_4))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_4),M_1)),K_4) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(nat,nat,suc,zero_zero(nat)))) ) ) ).
fof(fact_1379_pair__leqI2,axiom,
! [S_3,T_3,Aa,Ba] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Aa),Ba))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),S_3),T_3))
=> hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Aa),S_3)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Ba),T_3))),pair_leq)) ) ) ).
fof(fact_1380_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,Y,Q_2] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)),Q_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_2)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),Q_2)) ) ).
fof(fact_1381_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,P_4,Q_2] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),P_4)),Q_2) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_4),Q_2)) ) ).
fof(fact_1382_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),one_one(nat)) = ti(X_a,X) ) ).
fof(fact_1383_zpower__zpower,axiom,
! [X,Y,Z] : 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) = 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)) ).
fof(fact_1384_zpower__zmod,axiom,
! [X,M_1,Y] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M_1)),Y)),M_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),M_1) ).
fof(fact_1385_Ints__power,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [N,Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),ring_1_Ints(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),N)),ring_1_Ints(X_a))) ) ) ).
fof(fact_1386_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),zero_zero(nat)) = one_one(X_a) ) ).
fof(fact_1387_comm__semiring__1__class_Onormalizing__semiring__rules_I35_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,Q_2] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,Q_2)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_2)) ) ).
fof(fact_1388_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,Q_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_2)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,Q_2)) ) ).
fof(fact_1389_comm__semiring__1__class_Onormalizing__semiring__rules_I28_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,Q_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_2)),X) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,Q_2)) ) ).
fof(fact_1390_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,P_4,Q_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),P_4)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),Q_2)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),P_4),Q_2)) ) ).
fof(fact_1391_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N_1,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N_1))) ) ).
fof(fact_1392_zpower__zadd__distrib,axiom,
! [X,Y,Z] : 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)) = 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)) ).
fof(fact_1393_power__eq__0__iff__number__of,axiom,
! [X_a] :
( ( power(X_a)
& mult_zero(X_a)
& no_zero_divisors(X_a)
& zero_neq_one(X_a) )
=> ! [Aa,W_1] :
( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),hAPP(int,nat,number_number_of(nat),W_1)) = zero_zero(X_a)
<=> ( ti(X_a,Aa) = zero_zero(X_a)
& hAPP(int,nat,number_number_of(nat),W_1) != zero_zero(nat) ) ) ) ).
fof(fact_1394_realpow__two__disj,axiom,
! [X_a] :
( idom(X_a)
=> ! [X_1,Y_1] :
( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))
<=> ( ti(X_a,X_1) = ti(X_a,Y_1)
| ti(X_a,X_1) = hAPP(X_a,X_a,uminus_uminus(X_a),Y_1) ) ) ) ).
fof(fact_1395_zero__le__zpower__abs,axiom,
! [X,N_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_1))) ).
fof(fact_1396_realpow__Suc__le__self,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,R_4] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),R_4))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),R_4),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),R_4),hAPP(nat,nat,suc,N_1))),R_4)) ) ) ) ).
fof(fact_1397_power__0__left__number__of,axiom,
! [X_a] :
( ( power(X_a)
& semiring_0(X_a) )
=> ! [W] :
( ( hAPP(int,nat,number_number_of(nat),W) = zero_zero(nat)
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(int,nat,number_number_of(nat),W)) = one_one(X_a) )
& ( hAPP(int,nat,number_number_of(nat),W) != zero_zero(nat)
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(int,nat,number_number_of(nat),W)) = zero_zero(X_a) ) ) ) ).
fof(fact_1398_lemma__realpow__diff,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [Y,P_4,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),P_4),N_1))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,N_1)),P_4)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),P_4))),Y) ) ) ).
fof(fact_1399_abs__power__minus__one,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [N_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),N_1)) = one_one(X_a) ) ).
fof(fact_1400_zero__less__zpower__abs__iff,axiom,
! [X_1,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,abs_abs(int),X_1)),N)))
<=> ( X_1 != zero_zero(int)
| N = zero_zero(nat) ) ) ).
fof(fact_1401_realpow__two__diff,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) ) ).
fof(fact_1402_realpow__num__eq__if,axiom,
! [X_a] :
( power(X_a)
=> ! [M_1,N_1] :
( ( N_1 = zero_zero(nat)
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),M_1),N_1) = one_one(X_a) )
& ( N_1 != zero_zero(nat)
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),M_1),N_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),M_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat)))) ) ) ) ).
fof(fact_1403_realpow__minus__mult,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat)))),X) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1) ) ) ).
fof(fact_1404_Euler_Oaux__1,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_4))
=> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,P_4)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,P_4)),one_one(nat)))) ) ).
fof(fact_1405_pair__leqI1,axiom,
! [S_3,T_3,Aa,Ba] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),Aa),Ba))
=> hBOOL(hAPP(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool,hAPP(product_prod(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(product_prod(nat,nat),product_prod(nat,nat)),bool),bool),member(product_prod(product_prod(nat,nat),product_prod(nat,nat))),hAPP(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),hAPP(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Aa),S_3)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Ba),T_3))),pair_leq)) ) ).
fof(fact_1406_power__strict__mono,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1))) ) ) ) ) ).
fof(fact_1407_power__diff,axiom,
! [X_a] :
( field(X_a)
=> ! [N_1,M_1,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) ) ) ) ).
fof(fact_1408_one__less__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))) ) ) ) ).
fof(fact_1409_power__increasing__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X_1,Y_1,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ba),X_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ba),Y_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),Y_1)) ) ) ) ).
fof(fact_1410_power__le__imp__le__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1,N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ) ) ).
fof(fact_1411_power__decreasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,N_6] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_6))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_6)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))) ) ) ) ) ).
fof(fact_1412_real__of__nat__power,axiom,
! [M_1,N_1] : hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M_1),N_1)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(nat,real,real_1(nat),M_1)),N_1) ).
fof(fact_1413_power__real__of__nat,axiom,
! [M_1,N_1] : hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(nat,real,real_1(nat),M_1)),N_1) = hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M_1),N_1)) ).
fof(fact_1414_natfloor__power,axiom,
! [N_1,X] :
( X = hAPP(nat,real,real_1(nat),hAPP(real,nat,natfloor,X))
=> hAPP(real,nat,natfloor,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(real,nat,natfloor,X)),N_1) ) ).
fof(fact_1415_nat__power__eq__0__iff,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M),N) = zero_zero(nat)
<=> ( N != zero_zero(nat)
& M = zero_zero(nat) ) ) ).
fof(fact_1416_exp__mono__eq,axiom,
! [X_1,N,Y_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X_1),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y_1),hAPP(nat,nat,suc,N))
<=> X_1 = Y_1 ) ).
fof(fact_1417_nat__power__eq__Suc__0__iff,axiom,
! [X_1,M] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X_1),M) = hAPP(nat,nat,suc,zero_zero(nat))
<=> ( M = zero_zero(nat)
| X_1 = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).
fof(fact_1418_power__Suc__0,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,suc,zero_zero(nat))),N_1) = hAPP(nat,nat,suc,zero_zero(nat)) ).
fof(fact_1419_nat__zero__less__power__iff,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X_1),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X_1))
| N = zero_zero(nat) ) ) ).
fof(fact_1420_nat__power__less__imp__less,axiom,
! [M_1,N_1,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ) ).
fof(fact_1421_nat__one__le__power,axiom,
! [N_1,I_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),I_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),N_1))) ) ).
fof(fact_1422_zero__less__power__nat__eq,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X_1),N)))
<=> ( N = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X_1)) ) ) ).
fof(fact_1423_exp__mono__lt,axiom,
! [X_1,N,Y_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X_1),hAPP(nat,nat,suc,N))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y_1),hAPP(nat,nat,suc,N))))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_1)) ) ).
fof(fact_1424_exp__eq__1,axiom,
! [X_1,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X_1),N) = one_one(nat)
<=> ( X_1 = one_one(nat)
| N = zero_zero(nat) ) ) ).
fof(fact_1425_exp__mono__le,axiom,
! [X_1,N,Y_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X_1),hAPP(nat,nat,suc,N))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y_1),hAPP(nat,nat,suc,N))))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),Y_1)) ) ).
fof(fact_1426_zero__less__power__nat__eq__number__of,axiom,
! [X_1,W_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X_1),hAPP(int,nat,number_number_of(nat),W_1))))
<=> ( hAPP(int,nat,number_number_of(nat),W_1) = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X_1)) ) ) ).
fof(fact_1427_power__eq__if,axiom,
! [P_4,M_1] :
( ( M_1 = zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),M_1) = one_one(nat) )
& ( M_1 != zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),M_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),one_one(nat)))) ) ) ).
fof(fact_1428_Nat__Transfer_Otransfer__nat__int__functions_I4_J,axiom,
! [N_1,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,nat_1,X)),N_1) = hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N_1)) ) ).
fof(fact_1429_nat__power__eq,axiom,
! [N_1,Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,nat_1,Z)),N_1) ) ).
fof(fact_1430_power__nat__number__of__number__of,axiom,
! [W,V] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),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)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),W)) = hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,nat,number_number_of(nat),W))) ) ) ).
fof(fact_1431_power__nat__number__of,axiom,
! [N_1,V] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),zero_zero(nat)),N_1) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(int,nat,number_number_of(nat),V)),N_1) = hAPP(int,nat,nat_1,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),V)),N_1)) ) ) ).
fof(fact_1432_field__power__not__zero,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [N_1,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1) != zero_zero(X_a) ) ) ).
fof(fact_1433_power__commutes,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,N_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) ) ).
fof(fact_1434_power__mult__distrib,axiom,
! [X_a] :
( comm_monoid_mult(X_a)
=> ! [A_1,B,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),N_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1)) ) ).
fof(fact_1435_power__one,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),one_one(X_a)),N_1) = one_one(X_a) ) ).
fof(fact_1436_power__divide,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [A_1,B,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)),N_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1)) ) ).
fof(fact_1437_power__mult,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,M_1,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M_1)),N_1) ) ).
fof(fact_1438_power__abs,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,N_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),N_1) ) ).
fof(fact_1439_power__one__right,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),one_one(nat)) = ti(X_a,A_1) ) ).
fof(fact_1440_zero__le__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))) ) ) ).
fof(fact_1441_power__mono,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1))) ) ) ) ).
fof(fact_1442_zero__less__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))) ) ) ).
fof(fact_1443_one__le__power,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))) ) ) ).
fof(fact_1444_power__inject__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),Aa))
=> ( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),M) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),N)
<=> M = N ) ) ) ).
fof(fact_1445_power__eq__0__iff,axiom,
! [X_a] :
( ( power(X_a)
& mult_zero(X_a)
& no_zero_divisors(X_a)
& zero_neq_one(X_a) )
=> ! [Aa,N] :
( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),N) = zero_zero(X_a)
<=> ( ti(X_a,Aa) = zero_zero(X_a)
& N != zero_zero(nat) ) ) ) ).
fof(fact_1446_power__0__Suc,axiom,
! [X_a] :
( ( power(X_a)
& semiring_0(X_a) )
=> ! [N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(nat,nat,suc,N_1)) = zero_zero(X_a) ) ).
fof(fact_1447_nonzero__power__divide,axiom,
! [X_a] :
( field(X_a)
=> ! [A_1,N_1,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)),N_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1)) ) ) ).
fof(fact_1448_power__0,axiom,
! [X_a] :
( power(X_a)
=> ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),zero_zero(nat)) = one_one(X_a) ) ).
fof(fact_1449_power__Suc,axiom,
! [X_a] :
( power(X_a)
=> ! [A_1,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) ) ).
fof(fact_1450_power__Suc2,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),A_1) ) ).
fof(fact_1451_power__add,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,M_1,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) ) ).
fof(fact_1452_power__one__over,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [A_1,N_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1)),N_1) ) ).
fof(fact_1453_abs__power__minus,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,N_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),N_1)) = hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) ) ).
fof(fact_1454_power__less__imp__less__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B)) ) ) ) ).
fof(fact_1455_power__gt1__lemma,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)))) ) ) ).
fof(fact_1456_power__less__power__Suc,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)))) ) ) ).
fof(fact_1457_power__inject__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,B] :
( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),hAPP(nat,nat,suc,N_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> ti(X_a,A_1) = ti(X_a,B) ) ) ) ) ).
fof(fact_1458_power__le__imp__le__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N_1))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),hAPP(nat,nat,suc,N_1))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B)) ) ) ) ).
fof(fact_1459_power__0__left,axiom,
! [X_a] :
( ( power(X_a)
& semiring_0(X_a) )
=> ! [N_1] :
( ( N_1 = zero_zero(nat)
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),N_1) = one_one(X_a) )
& ( N_1 != zero_zero(nat)
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),N_1) = zero_zero(X_a) ) ) ) ).
fof(fact_1460_zero__le__power__abs,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),N_1))) ) ).
fof(fact_1461_power__gt1,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N_1)))) ) ) ).
fof(fact_1462_power__strict__increasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,N_6] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_6))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_6))) ) ) ) ).
fof(fact_1463_power__less__imp__less__exp,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1,N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ) ) ).
fof(fact_1464_power__strict__increasing__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X_1,Y_1,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ba),X_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Ba),Y_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_1)) ) ) ) ).
fof(fact_1465_power__minus,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [A_1,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),N_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) ) ).
fof(fact_1466_power__increasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,N_6] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),N_6))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_6))) ) ) ) ).
fof(fact_1467_power__Suc__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))) ) ) ) ).
fof(fact_1468_power__Suc__less__one,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,N_1))),one_one(X_a))) ) ) ) ).
fof(fact_1469_power__strict__decreasing,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,N_6] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_6))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_6)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))) ) ) ) ) ).
fof(fact_1470_power__eq__imp__eq__base,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [A_1,N_1,B] :
( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),B),N_1)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ti(X_a,A_1) = ti(X_a,B) ) ) ) ) ) ).
fof(fact_1471_LIMSEQ__inverse__realpow__zero__lemma,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),X)),one_one(real))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),one_one(real))),N_1))) ) ).
fof(fact_1472_realpow__pos__nth,axiom,
! [A_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ? [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
& hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_2),N_1) = A_1 ) ) ) ).
fof(fact_1473_realpow__pos__nth__unique,axiom,
! [A_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ? [X_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_2))
& hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_2),N_1) = A_1
& ! [Y_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_2))
& hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_2),N_1) = A_1 )
=> Y_2 = X_2 ) ) ) ) ).
fof(fact_1474_neg__one__power,axiom,
! [N_1] :
( hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),N_1) = one_one(int)
| hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),N_1) = hAPP(int,int,number_number_of(int),min) ) ).
fof(fact_1475_realpow__pos__nth2,axiom,
! [N_1,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ? [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
& hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_2),hAPP(nat,nat,suc,N_1)) = A_1 ) ) ).
fof(fact_1476_power__power__power,axiom,
! [X_a] :
( power(X_a)
=> power_power(X_a) = hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),one_one(X_a)),times_times(X_a)) ) ).
fof(fact_1477_power_Opower_Opower__0,axiom,
! [X_a,One,Times,Aa] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),One),Times),Aa),zero_zero(nat)) = ti(X_a,One) ).
fof(fact_1478_power_Opower_Opower__Suc,axiom,
! [X_a,One,Times,Aa,N] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),One),Times),Aa),hAPP(nat,nat,suc,N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),Times,Aa),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),hAPP(X_a,fun(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),power_1(X_a),One),Times),Aa),N)) ).
fof(fact_1479_EvenOdd_Oneg__one__even__power,axiom,
! [X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,X_1)) = one_one(int) ) ) ).
fof(fact_1480_EvenOdd_Oneg__one__odd__power,axiom,
! [X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,X_1)) = hAPP(int,int,number_number_of(int),min) ) ) ).
fof(fact_1481_neg__one__power__parity,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
=> ( ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
<=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zEven)) )
=> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,X_1)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(int,nat,nat_1,Y_1)) ) ) ) ).
fof(fact_1482_ceiling__eq2,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N_1)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N_1)),one_one(real))))
=> hAPP(real,int,archimedean_ceiling(real),X) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int)) ) ) ).
fof(fact_1483_neg__odd__power,axiom,
! [Aa,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,uminus_uminus(int),Aa)),hAPP(int,nat,nat_1,X_1)) = hAPP(int,int,uminus_uminus(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Aa),hAPP(int,nat,nat_1,X_1))) ) ) ).
fof(fact_1484_arccos__lt__bounded,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),one_one(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,arccos,Y)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arccos,Y)),pi)) ) ) ) ).
fof(fact_1485_even__odd__conj,axiom,
! [X_1] :
~ ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
& hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven)) ) ).
fof(fact_1486_odd__iff__not__even,axiom,
! [X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
<=> ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven)) ) ).
fof(fact_1487_even__odd__disj,axiom,
! [X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
| hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven)) ) ).
fof(fact_1488_not__odd__impl__even,axiom,
! [X_1] :
( ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven)) ) ).
fof(fact_1489_odd__minus__odd,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zOdd))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),Y_1)),zEven)) ) ) ).
fof(fact_1490_odd__minus__even,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zEven))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),Y_1)),zOdd)) ) ) ).
fof(fact_1491_even__minus__odd,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zOdd))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),Y_1)),zOdd)) ) ) ).
fof(fact_1492_even__plus__odd__prop2,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aa),Ba)),zOdd))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),zEven))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),zOdd)) ) ) ).
fof(fact_1493_even__plus__odd__prop1,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aa),Ba)),zOdd))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),zOdd))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),zEven)) ) ) ).
fof(fact_1494_IntNatAux_Oeven__plus__odd,axiom,
! [Ba,Aa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),zEven))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),zOdd))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aa),Ba)),zOdd)) ) ) ).
fof(fact_1495_IntNatAux_Oodd__plus__odd,axiom,
! [Ba,Aa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),zOdd))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),zOdd))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aa),Ba)),zEven)) ) ) ).
fof(fact_1496_real__of__int__inject,axiom,
! [X_1,Y_1] :
( hAPP(int,real,real_1(int),X_1) = hAPP(int,real,real_1(int),Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_1497_pi__neq__zero,axiom,
pi != zero_zero(real) ).
fof(fact_1498_floor__real__of__int,axiom,
! [N_1] : hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,real_1(int),N_1)) = N_1 ).
fof(fact_1499_real__of__int__floor__cancel,axiom,
! [X_1] :
( hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),X_1)) = X_1
<=> ? [N_2] : X_1 = hAPP(int,real,real_1(int),N_2) ) ).
fof(fact_1500_odd__minus__one__even,axiom,
! [X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),one_one(int))),zEven)) ) ).
fof(fact_1501_ceiling__real__of__int,axiom,
! [N_1] : hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,real_1(int),N_1)) = N_1 ).
fof(fact_1502_real__of__int__ceiling__cancel,axiom,
! [X_1] :
( hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),X_1)) = X_1
<=> ? [N_2] : X_1 = hAPP(int,real,real_1(int),N_2) ) ).
fof(fact_1503_one__not__even,axiom,
~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),one_one(int)),zEven)) ).
fof(fact_1504_EvenOdd_Oeven__product,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X_1),Y_1)),zEven))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
| hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zEven)) ) ) ).
fof(fact_1505_even__times__either,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X_1),Y_1)),zEven)) ) ).
fof(fact_1506_EvenOdd_Oeven__plus__even,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zEven))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Y_1)),zEven)) ) ) ).
fof(fact_1507_even__minus__even,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zEven))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),Y_1)),zEven)) ) ) ).
fof(fact_1508_even__diff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),Y_1)),zEven))
<=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
<=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zEven)) ) ) ).
fof(fact_1509_odd__mult__odd__prop,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X_1),Y_1)),zOdd))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd)) ) ).
fof(fact_1510_EvenOdd_Oodd__times__odd,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zOdd))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X_1),Y_1)),zOdd)) ) ) ).
fof(fact_1511_real__of__int__zero__cancel,axiom,
! [X_1] :
( hAPP(int,real,real_1(int),X_1) = zero_zero(real)
<=> X_1 = zero_zero(int) ) ).
fof(fact_1512_real__of__int__zero,axiom,
hAPP(int,real,real_1(int),zero_zero(int)) = zero_zero(real) ).
fof(fact_1513_real__of__int__less__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),X_1)),hAPP(int,real,real_1(int),Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Y_1)) ) ).
fof(fact_1514_real__of__int__le__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),X_1)),hAPP(int,real,real_1(int),Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),Y_1)) ) ).
fof(fact_1515_real__of__int__mult,axiom,
! [X,Y] : hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),Y)) ).
fof(fact_1516_real__of__one,axiom,
hAPP(int,real,real_1(int),one_one(int)) = one_one(real) ).
fof(fact_1517_real__number__of,axiom,
! [V] : hAPP(int,real,real_1(int),hAPP(int,int,number_number_of(int),V)) = hAPP(int,real,number_number_of(real),V) ).
fof(fact_1518_real__of__int__add,axiom,
! [X,Y] : hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),Y)) ).
fof(fact_1519_real__of__int__power,axiom,
! [X,N_1] : hAPP(int,real,real_1(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N_1)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,real_1(int),X)),N_1) ).
fof(fact_1520_power__real__of__int,axiom,
! [X,N_1] : hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,real_1(int),X)),N_1) = hAPP(int,real,real_1(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),N_1)) ).
fof(fact_1521_real__of__int__diff,axiom,
! [X,Y] : hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),Y)) ).
fof(fact_1522_pi__ge__zero,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),pi)) ).
fof(fact_1523_pi__gt__zero,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),pi)) ).
fof(fact_1524_pi__not__less__zero,axiom,
~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),pi),zero_zero(real))) ).
fof(fact_1525_real__of__int__minus,axiom,
! [X] : hAPP(int,real,real_1(int),hAPP(int,int,uminus_uminus(int),X)) = hAPP(real,real,uminus_uminus(real),hAPP(int,real,real_1(int),X)) ).
fof(fact_1526_Ints__real__of__int,axiom,
! [X_1] : hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(int,real,real_1(int),X_1)),ring_1_Ints(real))) ).
fof(fact_1527_real__of__int__abs,axiom,
! [X] : hAPP(int,real,real_1(int),hAPP(int,int,abs_abs(int),X)) = hAPP(real,real,abs_abs(real),hAPP(int,real,real_1(int),X)) ).
fof(fact_1528_real__of__int__floor__le,axiom,
! [R_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_4))),R_4)) ).
fof(fact_1529_real__of__int__ceiling__ge,axiom,
! [R_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),R_4),hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),R_4)))) ).
fof(fact_1530_floor__ceiling,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [R_4] : hAPP(real,int,archim2085082626_floor(real),hAPP(int,real,real_1(int),hAPP(X_a,int,archimedean_ceiling(X_a),R_4))) = hAPP(X_a,int,archimedean_ceiling(X_a),R_4) ) ).
fof(fact_1531_ceiling__floor,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [R_4] : hAPP(real,int,archimedean_ceiling(real),hAPP(int,real,real_1(int),hAPP(X_a,int,archim2085082626_floor(X_a),R_4))) = hAPP(X_a,int,archim2085082626_floor(X_a),R_4) ) ).
fof(fact_1532_real__of__int__div4,axiom,
! [N_1,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N_1),X))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N_1)),hAPP(int,real,real_1(int),X)))) ).
fof(fact_1533_floor__less__eq,axiom,
! [X_1,Aa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archim2085082626_floor(real),X_1)),Aa))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(int,real,real_1(int),Aa))) ) ).
fof(fact_1534_le__floor,axiom,
! [A_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),A_1)),X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(real,int,archim2085082626_floor(real),X))) ) ).
fof(fact_1535_le__floor__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Aa),hAPP(real,int,archim2085082626_floor(real),X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),Aa)),X_1)) ) ).
fof(fact_1536_real__le__floor,axiom,
! [A_1,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),hAPP(real,int,archim2085082626_floor(real),X)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),A_1)),X)) ) ).
fof(fact_1537_less__ceiling__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),hAPP(real,int,archimedean_ceiling(real),X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),Aa)),X_1)) ) ).
fof(fact_1538_floor__add,axiom,
! [X,A_1] : hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archim2085082626_floor(real),X)),A_1) ).
fof(fact_1539_ceiling__le__eq,axiom,
! [X_1,Aa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),X_1)),Aa))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(int,real,real_1(int),Aa))) ) ).
fof(fact_1540_ceiling__le,axiom,
! [X,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(int,real,real_1(int),A_1)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1)) ) ).
fof(fact_1541_ceiling__le__real,axiom,
! [X,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(int,real,real_1(int),A_1))) ) ).
fof(fact_1542_floor__power,axiom,
! [N_1,X] :
( X = hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),X))
=> hAPP(real,int,archim2085082626_floor(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N_1)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(real,int,archim2085082626_floor(real),X)),N_1) ) ).
fof(fact_1543_floor__subtract,axiom,
! [X,A_1] : hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archim2085082626_floor(real),X)),A_1) ).
fof(fact_1544_ceiling__add,axiom,
! [X,A_1] : hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1) ).
fof(fact_1545_floor__minus__real__of__int,axiom,
! [N_1] : hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,uminus_uminus(real),hAPP(int,real,real_1(int),N_1))) = hAPP(int,int,uminus_uminus(int),N_1) ).
fof(fact_1546_ceiling__subtract,axiom,
! [X,A_1] : hAPP(real,int,archimedean_ceiling(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(int,real,real_1(int),A_1))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(real,int,archimedean_ceiling(real),X)),A_1) ).
fof(fact_1547_power__preserves__even,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X_1),N)),zEven))
<=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven)) ) ) ).
fof(fact_1548_real__of__int__gt__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(int,real,real_1(int),N)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N)) ) ).
fof(fact_1549_real__of__int__lt__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N)),zero_zero(real)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),N),zero_zero(int))) ) ).
fof(fact_1550_real__of__int__le__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N)),zero_zero(real)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),zero_zero(int))) ) ).
fof(fact_1551_real__of__int__ge__zero__cancel__iff,axiom,
! [N] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(int,real,real_1(int),N)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N)) ) ).
fof(fact_1552_number__of__less__real__of__int__iff2,axiom,
! [M,N] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),M)),hAPP(int,real,number_number_of(real),N)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M),hAPP(int,int,number_number_of(int),N))) ) ).
fof(fact_1553_number__of__less__real__of__int__iff,axiom,
! [N,M] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),N)),hAPP(int,real,real_1(int),M)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),N)),M)) ) ).
fof(fact_1554_number__of__le__real__of__int__iff2,axiom,
! [M,N] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),M)),hAPP(int,real,number_number_of(real),N)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M),hAPP(int,int,number_number_of(int),N))) ) ).
fof(fact_1555_number__of__le__real__of__int__iff,axiom,
! [N,M] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),N)),hAPP(int,real,real_1(int),M)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,number_number_of(int),N)),M)) ) ).
fof(fact_1556_int__less__real__le,axiom,
! [N,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),N),M))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N)),one_one(real))),hAPP(int,real,real_1(int),M))) ) ).
fof(fact_1557_int__le__real__less,axiom,
! [N,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N),M))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),M)),one_one(real)))) ) ).
fof(fact_1558_lemma__floor2,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N_1)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),X)),one_one(real))))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),X)) ) ).
fof(fact_1559_real__nat__eq__real,axiom,
! [X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hAPP(nat,real,real_1(nat),hAPP(int,nat,nat_1,X)) = hAPP(int,real,real_1(int),X) ) ).
fof(fact_1560_power__preserves__odd,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X_1),N)),zOdd))
<=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd)) ) ) ).
fof(fact_1561_real__of__int__floor__add__one__ge,axiom,
! [R_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),R_4),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_4))),one_one(real)))) ).
fof(fact_1562_real__of__int__floor__add__one__gt,axiom,
! [R_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_4),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_4))),one_one(real)))) ).
fof(fact_1563_floor__eq,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N_1)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N_1)),one_one(real))))
=> hAPP(real,int,archim2085082626_floor(real),X) = N_1 ) ) ).
fof(fact_1564_real__of__int__floor__ge__diff__one,axiom,
! [R_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),R_4),one_one(real))),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_4)))) ).
fof(fact_1565_real__of__int__floor__gt__diff__one,axiom,
! [R_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),R_4),one_one(real))),hAPP(int,real,real_1(int),hAPP(real,int,archim2085082626_floor(real),R_4)))) ).
fof(fact_1566_real__of__int__ceiling__le__add__one,axiom,
! [R_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),R_4))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_4),one_one(real)))) ).
fof(fact_1567_real__of__int__ceiling__diff__one__le,axiom,
! [R_4] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),hAPP(real,int,archimedean_ceiling(real),R_4))),one_one(real))),R_4)) ).
fof(fact_1568_lemma__floor,axiom,
! [N_1,M_1,R_4] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),M_1)),R_4))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_4),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N_1)),one_one(real))))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M_1),N_1)) ) ) ).
fof(fact_1569_real__of__int__div2,axiom,
! [N_1,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N_1)),hAPP(int,real,real_1(int),X))),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N_1),X))))) ).
fof(fact_1570_real__of__int__div3,axiom,
! [N_1,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N_1)),hAPP(int,real,real_1(int),X))),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N_1),X)))),one_one(real))) ).
fof(fact_1571_real__of__int__div__aux,axiom,
! [X,D] :
( D != zero_zero(int)
=> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),X)),hAPP(int,real,real_1(int),D)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X),D))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),D))),hAPP(int,real,real_1(int),D))) ) ).
fof(fact_1572_floor__eq2,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N_1)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N_1)),one_one(real))))
=> hAPP(real,int,archim2085082626_floor(real),X) = N_1 ) ) ).
fof(fact_1573_less__floor__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),hAPP(real,int,archim2085082626_floor(real),X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),Aa)),one_one(real))),X_1)) ) ).
fof(fact_1574_floor__le__eq,axiom,
! [X_1,Aa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(real,int,archim2085082626_floor(real),X_1)),Aa))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),Aa)),one_one(real)))) ) ).
fof(fact_1575_ceiling__eq3,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),N_1)),one_one(real))),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(int,real,real_1(int),N_1)))
=> hAPP(real,int,archimedean_ceiling(real),X) = N_1 ) ) ).
fof(fact_1576_ceiling__less__eq,axiom,
! [X_1,Aa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(real,int,archimedean_ceiling(real),X_1)),Aa))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),Aa)),one_one(real)))) ) ).
fof(fact_1577_le__ceiling__eq,axiom,
! [Aa,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Aa),hAPP(real,int,archimedean_ceiling(real),X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(int,real,real_1(int),Aa)),one_one(real))),X_1)) ) ).
fof(fact_1578_arccos__ubound,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arccos,Y)),pi)) ) ) ).
fof(fact_1579_ceiling__eq,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,real_1(int),N_1)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(int,real,real_1(int),N_1)),one_one(real))))
=> hAPP(real,int,archimedean_ceiling(real),X) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int)) ) ) ).
fof(fact_1580_neg__even__power,axiom,
! [Aa,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,uminus_uminus(int),Aa)),hAPP(int,nat,nat_1,X_1)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Aa),hAPP(int,nat,nat_1,X_1)) ) ) ).
fof(fact_1581_arccos__bounded,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,arccos,Y)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arccos,Y)),pi)) ) ) ) ).
fof(fact_1582_reals__Archimedean__6c__int,axiom,
! [R_4] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_4),zero_zero(real)))
=> ? [N_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N_2)),R_4))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_4),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_2),one_one(int))))) ) ) ).
fof(fact_1583_reals__Archimedean__6b__int,axiom,
! [R_4] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R_4))
=> ? [N_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N_2)),R_4))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_4),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_2),one_one(int))))) ) ) ).
fof(fact_1584_real__lb__ub__int,axiom,
! [R_4] :
? [N_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,real_1(int),N_2)),R_4))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_4),hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_2),one_one(int))))) ) ).
fof(fact_1585_arccos,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,arccos,Y)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arccos,Y)),pi))
& hAPP(real,real,cos,hAPP(real,real,arccos,Y)) = Y ) ) ) ).
fof(fact_1586_cos__npi2,axiom,
! [N_1] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),pi),hAPP(nat,real,real_1(nat),N_1))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),N_1) ).
fof(fact_1587_cos__npi,axiom,
! [N_1] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),pi)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),N_1) ).
fof(fact_1588_cos__minus,axiom,
! [X] : hAPP(real,real,cos,hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,cos,X) ).
fof(fact_1589_cos__le__one,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,X)),one_one(real))) ).
fof(fact_1590_cos__zero,axiom,
hAPP(real,real,cos,zero_zero(real)) = one_one(real) ).
fof(fact_1591_cos__arctan__not__zero,axiom,
! [X] : hAPP(real,real,cos,hAPP(real,real,arctan,X)) != zero_zero(real) ).
fof(fact_1592_cos__monotone__0__pi_H,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))) ) ) ) ).
fof(fact_1593_cos__ge__minus__one,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),hAPP(real,real,cos,X))) ).
fof(fact_1594_abs__cos__le__one,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,cos,X))),one_one(real))) ).
fof(fact_1595_cos__pi,axiom,
hAPP(real,real,cos,pi) = hAPP(int,real,number_number_of(real),min) ).
fof(fact_1596_cos__periodic__pi,axiom,
! [X] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),pi)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,cos,X)) ).
fof(fact_1597_cos__monotone__0__pi,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))) ) ) ) ).
fof(fact_1598_cos__monotone__minus__pi__0_H,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),pi)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,Y)),hAPP(real,real,cos,X))) ) ) ) ).
fof(fact_1599_arccos__cos,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi))
=> hAPP(real,real,arccos,hAPP(real,real,cos,X)) = X ) ) ).
fof(fact_1600_cos__arccos__abs,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,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_1601_cos__monotone__minus__pi__0,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),pi)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,Y)),hAPP(real,real,cos,X))) ) ) ) ).
fof(fact_1602_cos__arccos,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> hAPP(real,real,cos,hAPP(real,real,arccos,Y)) = Y ) ) ).
fof(fact_1603_arccos__cos2,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,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_1604_cos__total,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> ? [X_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),pi))
& hAPP(real,real,cos,X_2) = Y
& ! [Y_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_2),pi))
& hAPP(real,real,cos,Y_2) = Y )
=> Y_2 = X_2 ) ) ) ) ).
fof(fact_1605_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(real,fun(real,real),plus_plus(real),X),Y)) != zero_zero(real)
=> hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y)))) ) ) ) ).
fof(fact_1606_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(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))) ) ) ).
fof(fact_1607_nat__1__add__number__of,axiom,
! [V] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(int,nat,number_number_of(nat),V)) = one_one(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V)) ) ) ).
fof(fact_1608_nat__number__of__add__1,axiom,
! [V] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat)) = one_one(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,succ,V)) ) ) ).
fof(fact_1609_semiring__number__of__add__1,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,succ,V)) ) ) ).
fof(fact_1610_Pls__def,axiom,
pls = zero_zero(int) ).
fof(fact_1611_tan__zero,axiom,
hAPP(real,real,tan,zero_zero(real)) = zero_zero(real) ).
fof(fact_1612_rel__simps_I2_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),pls)) ).
fof(fact_1613_rel__simps_I19_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),pls)) ).
fof(fact_1614_mult__Pls,axiom,
! [W] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),pls),W) = pls ).
fof(fact_1615_add__Pls,axiom,
! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),K_4) = K_4 ).
fof(fact_1616_add__Pls__right,axiom,
! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),pls) = K_4 ).
fof(fact_1617_diff__bin__simps_I1_J,axiom,
! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),pls) = K_4 ).
fof(fact_1618_rel__simps_I40_J,axiom,
min != pls ).
fof(fact_1619_rel__simps_I37_J,axiom,
pls != min ).
fof(fact_1620_minus__Pls,axiom,
hAPP(int,int,uminus_uminus(int),pls) = pls ).
fof(fact_1621_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_1622_tan__arctan,axiom,
! [Y] : hAPP(real,real,tan,hAPP(real,real,arctan,Y)) = Y ).
fof(fact_1623_semiring__norm_I112_J,axiom,
! [X_a] :
( number_ring(X_a)
=> zero_zero(X_a) = hAPP(int,X_a,number_number_of(X_a),pls) ) ).
fof(fact_1624_number__of__Pls,axiom,
! [X_a] :
( number_ring(X_a)
=> hAPP(int,X_a,number_number_of(X_a),pls) = zero_zero(X_a) ) ).
fof(fact_1625_semiring__numeral__0__eq__0,axiom,
! [X_a] :
( number_semiring(X_a)
=> hAPP(int,X_a,number_number_of(X_a),pls) = zero_zero(X_a) ) ).
fof(fact_1626_add__numeral__0,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),pls)),A_1) = ti(X_a,A_1) ) ).
fof(fact_1627_add__numeral__0__right,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),hAPP(int,X_a,number_number_of(X_a),pls)) = ti(X_a,A_1) ) ).
fof(fact_1628_semiring__norm_I113_J,axiom,
zero_zero(nat) = hAPP(int,nat,number_number_of(nat),pls) ).
fof(fact_1629_nat__number__of__Pls,axiom,
hAPP(int,nat,number_number_of(nat),pls) = zero_zero(nat) ).
fof(fact_1630_bin__less__0__simps_I1_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),zero_zero(int))) ).
fof(fact_1631_zero__is__num__zero,axiom,
zero_zero(int) = hAPP(int,int,number_number_of(int),pls) ).
fof(fact_1632_rel__simps_I6_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),pls)) ).
fof(fact_1633_rel__simps_I3_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),min)) ).
fof(fact_1634_tan__pi,axiom,
hAPP(real,real,tan,pi) = zero_zero(real) ).
fof(fact_1635_rel__simps_I23_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),pls)) ).
fof(fact_1636_rel__simps_I20_J,axiom,
~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),min)) ).
fof(fact_1637_eq__number__of__Pls__Min,axiom,
hAPP(int,int,number_number_of(int),pls) != hAPP(int,int,number_number_of(int),min) ).
fof(fact_1638_tan__periodic__pi,axiom,
! [X] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),pi)) = hAPP(real,real,tan,X) ).
fof(fact_1639_not__neg__number__of__Pls,axiom,
~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),pls))) ).
fof(fact_1640_succ__Min,axiom,
hAPP(int,int,succ,min) = pls ).
fof(fact_1641_divide__Numeral0,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& number_ring(X_a) )
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),hAPP(int,X_a,number_number_of(X_a),pls)) = zero_zero(X_a) ) ).
fof(fact_1642_eq__0__number__of,axiom,
! [V_1] :
( zero_zero(nat) = hAPP(int,nat,number_number_of(nat),V_1)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_1),pls)) ) ).
fof(fact_1643_eq__number__of__0,axiom,
! [V_1] :
( hAPP(int,nat,number_number_of(nat),V_1) = zero_zero(nat)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_1),pls)) ) ).
fof(fact_1644_less__nat__number__of,axiom,
! [V_1,V_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,number_number_of(nat),V_1)),hAPP(int,nat,number_number_of(nat),V_2)))
<=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),V_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_2)) )
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_1),V_2)) ) ) ).
fof(fact_1645_number__of2,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),pls))) ).
fof(fact_1646_le__nat__number__of,axiom,
! [V_1,V_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),V_1)),hAPP(int,nat,number_number_of(nat),V_2)))
<=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_1),V_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V_1),pls)) ) ) ).
fof(fact_1647_less__special_I1_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),Y_1)) ) ) ).
fof(fact_1648_less__special_I3_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),X_1)),zero_zero(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),pls)) ) ) ).
fof(fact_1649_le__special_I1_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),Y_1)) ) ) ).
fof(fact_1650_le__special_I3_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),X_1)),zero_zero(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),pls)) ) ) ).
fof(fact_1651_semiring__mult__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_3,V] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_3))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),V_3)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),V_3)) ) ) ) ).
fof(fact_1652_semiring__add__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V_3,V] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V_3))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),hAPP(int,X_a,number_number_of(X_a),V_3)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),V_3)) ) ) ) ).
fof(fact_1653_less__0__number__of,axiom,
! [V_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,number_number_of(nat),V_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V_1)) ) ).
fof(fact_1654_add__nat__number__of,axiom,
! [V_3,V] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_3)) = hAPP(int,nat,number_number_of(nat),V_3) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_3),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_3)) = hAPP(int,nat,number_number_of(nat),V) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_3),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_3)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),V_3)) ) ) ) ) ).
fof(fact_1655_tan__npi,axiom,
! [N_1] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),pi)) = zero_zero(real) ).
fof(fact_1656_tan__periodic__n,axiom,
! [X,N_1] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),N_1)),pi))) = hAPP(real,real,tan,X) ).
fof(fact_1657_tan__periodic__nat,axiom,
! [X,N_1] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),pi))) = hAPP(real,real,tan,X) ).
fof(fact_1658_tan__periodic__int,axiom,
! [X,I_1] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,real_1(int),I_1)),pi))) = hAPP(real,real,tan,X) ).
fof(fact_1659_nat__number__of__mult__left,axiom,
! [V_3,K_4,V] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V_3)),K_4)) = zero_zero(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V_3)),K_4)) = 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),V_3))),K_4) ) ) ).
fof(fact_1660_mult__nat__number__of,axiom,
! [V_3,V] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_3)) = zero_zero(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),V)),hAPP(int,nat,number_number_of(nat),V_3)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),V),V_3)) ) ) ).
fof(fact_1661_semiring__1__add__number__of,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [V] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),V))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),V)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,succ,V)) ) ) ).
fof(fact_1662_eq__special_I1_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [Y_1] :
( zero_zero(X_a) = hAPP(int,X_a,number_number_of(X_a),Y_1)
<=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),pls),hAPP(int,int,uminus_uminus(int),Y_1))))) ) ) ).
fof(fact_1663_eq__special_I3_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [X_1] :
( hAPP(int,X_a,number_number_of(X_a),X_1) = zero_zero(X_a)
<=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),hAPP(int,int,uminus_uminus(int),pls))))) ) ) ).
fof(fact_1664_nat__number__of__diff__1,axiom,
! [V] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat)) = zero_zero(nat) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),V),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V)),one_one(nat)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,pred,V)) ) ) ).
fof(fact_1665_Suc__diff__number__of,axiom,
! [M_1,V] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),V))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M_1)),hAPP(int,nat,number_number_of(nat),V)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,pred,V))) ) ).
fof(fact_1666_diff__special_I2_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))) ) ).
fof(fact_1667_diff__special_I1_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),W))) ) ).
fof(fact_1668_iszero__number__of__Bit1,axiom,
! [X_a] :
( ( number_ring(X_a)
& ring_char_0(X_a) )
=> ! [W] : ~ hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,W)))) ) ).
fof(fact_1669_rel__simps_I51_J,axiom,
! [K,L] :
( hAPP(int,int,bit1,K) = hAPP(int,int,bit1,L)
<=> K = L ) ).
fof(fact_1670_minus__Bit1,axiom,
! [K_4] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,K_4)) = hAPP(int,int,bit1,hAPP(int,int,pred,hAPP(int,int,uminus_uminus(int),K_4))) ).
fof(fact_1671_rel__simps_I46_J,axiom,
! [K_4] : hAPP(int,int,bit1,K_4) != pls ).
fof(fact_1672_rel__simps_I39_J,axiom,
! [L_4] : pls != hAPP(int,int,bit1,L_4) ).
fof(fact_1673_rel__simps_I17_J,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit1,L)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) ) ).
fof(fact_1674_less__int__code_I16_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit1,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_1675_rel__simps_I34_J,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit1,L)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).
fof(fact_1676_less__eq__int__code_I16_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit1,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_1677_rel__simps_I47_J,axiom,
! [K] :
( hAPP(int,int,bit1,K) = min
<=> K = min ) ).
fof(fact_1678_rel__simps_I43_J,axiom,
! [L] :
( min = hAPP(int,int,bit1,L)
<=> min = L ) ).
fof(fact_1679_Bit1__Min,axiom,
hAPP(int,int,bit1,min) = min ).
fof(fact_1680_not__iszero__Numeral1,axiom,
! [X_a] :
( number_ring(X_a)
=> ~ hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)))) ) ).
fof(fact_1681_succ__pred,axiom,
! [X] : hAPP(int,int,succ,hAPP(int,int,pred,X)) = X ).
fof(fact_1682_iszero__def,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [Z_1] :
( hBOOL(hAPP(X_a,bool,iszero(X_a),Z_1))
<=> ti(X_a,Z_1) = zero_zero(X_a) ) ) ).
fof(fact_1683_iszero__0,axiom,
! [X_a] :
( semiring_1(X_a)
=> hBOOL(hAPP(X_a,bool,iszero(X_a),zero_zero(X_a))) ) ).
fof(fact_1684_not__iszero__1,axiom,
! [X_a] :
( semiring_1(X_a)
=> ~ hBOOL(hAPP(X_a,bool,iszero(X_a),one_one(X_a))) ) ).
fof(fact_1685_rel__simps_I12_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) ) ).
fof(fact_1686_bin__less__0__simps_I4_J,axiom,
! [W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,W_1)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),zero_zero(int))) ) ).
fof(fact_1687_rel__simps_I22_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),hAPP(int,int,bit1,K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) ) ).
fof(fact_1688_Bit1__def,axiom,
! [K_4] : hAPP(int,int,bit1,K_4) = 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_4)),K_4) ).
fof(fact_1689_rel__simps_I13_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),min))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),min)) ) ).
fof(fact_1690_rel__simps_I9_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),hAPP(int,int,bit1,K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) ) ).
fof(fact_1691_rel__simps_I30_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K)),min))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) ) ).
fof(fact_1692_rel__simps_I26_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),hAPP(int,int,bit1,K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),K)) ) ).
fof(fact_1693_neg__number__of__Bit1,axiom,
! [W_1] :
( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,W_1))))
<=> hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),W_1))) ) ).
fof(fact_1694_succ__Pls,axiom,
hAPP(int,int,succ,pls) = hAPP(int,int,bit1,pls) ).
fof(fact_1695_le__iff__pred__less,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,pred,K)),L)) ) ).
fof(fact_1696_pred__Pls,axiom,
hAPP(int,int,pred,pls) = min ).
fof(fact_1697_pred__def,axiom,
! [K_4] : hAPP(int,int,pred,K_4) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),one_one(int)) ).
fof(fact_1698_add__Min,axiom,
! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),min),K_4) = hAPP(int,int,pred,K_4) ).
fof(fact_1699_add__Min__right,axiom,
! [K_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),min) = hAPP(int,int,pred,K_4) ).
fof(fact_1700_number__of__Bit1,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),W))),hAPP(int,X_a,number_number_of(X_a),W)) ) ).
fof(fact_1701_mult__numeral__1__right,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))) = ti(X_a,A_1) ) ).
fof(fact_1702_mult__numeral__1,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))),A_1) = ti(X_a,A_1) ) ).
fof(fact_1703_semiring__norm_I110_J,axiom,
! [X_a] :
( number_ring(X_a)
=> one_one(X_a) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)) ) ).
fof(fact_1704_numeral__1__eq__1,axiom,
! [X_a] :
( number_ring(X_a)
=> hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)) = one_one(X_a) ) ).
fof(fact_1705_semiring__numeral__1__eq__1,axiom,
! [X_a] :
( number_semiring(X_a)
=> hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)) = one_one(X_a) ) ).
fof(fact_1706_eq__special_I4_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [X_1] :
( hAPP(int,X_a,number_number_of(X_a),X_1) = one_one(X_a)
<=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit1,pls)))))) ) ) ).
fof(fact_1707_eq__special_I2_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [Y_1] :
( one_one(X_a) = hAPP(int,X_a,number_number_of(X_a),Y_1)
<=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),hAPP(int,int,uminus_uminus(int),Y_1))))) ) ) ).
fof(fact_1708_divide__numeral__1,axiom,
! [X_a] :
( ( field(X_a)
& number_ring(X_a) )
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))) = ti(X_a,A_1) ) ).
fof(fact_1709_divide__Numeral1,axiom,
! [X_a] :
( ( field(X_a)
& number_ring(X_a) )
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))) = ti(X_a,X) ) ).
fof(fact_1710_rel__simps_I5_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),hAPP(int,int,bit1,K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) ) ).
fof(fact_1711_rel__simps_I29_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) ) ).
fof(fact_1712_one__is__num__one,axiom,
one_one(int) = hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,pls)) ).
fof(fact_1713_Numeral1__eq1__nat,axiom,
one_one(nat) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,pls)) ).
fof(fact_1714_nat__numeral__1__eq__1,axiom,
hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,pls)) = one_one(nat) ).
fof(fact_1715_diff__bin__simps_I4_J,axiom,
! [L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),pls),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),L_4)) ).
fof(fact_1716_minus__Min,axiom,
hAPP(int,int,uminus_uminus(int),min) = hAPP(int,int,bit1,pls) ).
fof(fact_1717_iszero__Numeral0,axiom,
! [X_a] :
( number_ring(X_a)
=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),pls))) ) ).
fof(fact_1718_nonzero__number__of__Min,axiom,
! [X_a] :
( number_ring(X_a)
=> ~ hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),min))) ) ).
fof(fact_1719_power3__eq__cube,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1)),A_1) ) ).
fof(fact_1720_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_1721_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_1722_Suc3__eq__add__3,axiom,
! [N_1] : hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,N_1))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),N_1) ).
fof(fact_1723_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_1724_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_1,hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) ).
fof(fact_1725_power3__minus,axiom,
! [A_1] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,uminus_uminus(int),A_1)),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_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).
fof(fact_1726_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_1727_number__of__pred,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,pred,W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),hAPP(int,X_a,number_number_of(X_a),W)) ) ).
fof(fact_1728_less__special_I4_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,number_number_of(X_a),X_1)),one_one(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),hAPP(int,int,bit1,pls))) ) ) ).
fof(fact_1729_less__special_I2_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,pls)),Y_1)) ) ) ).
fof(fact_1730_le__special_I4_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),X_1)),one_one(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),hAPP(int,int,bit1,pls))) ) ) ).
fof(fact_1731_le__special_I2_J,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,pls)),Y_1)) ) ) ).
fof(fact_1732_add__special_I2_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,pls)),W)) ) ).
fof(fact_1733_add__special_I3_J,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,number_number_of(X_a),V)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V),hAPP(int,int,bit1,pls))) ) ).
fof(fact_1734_Suc__diff__eq__diff__pred,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,number_number_of(nat),pls)),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,pls)))) ) ).
fof(fact_1735_Suc__div__eq__add3__div__number__of,axiom,
! [M_1,V] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M_1)))),hAPP(int,nat,number_number_of(nat),V)) = 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_1)),hAPP(int,nat,number_number_of(nat),V)) ).
fof(fact_1736_Suc__div__eq__add3__div,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M_1)))),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),M_1)),N_1) ).
fof(fact_1737_div__Suc__eq__div__add3,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,N_1)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),N_1)) ).
fof(fact_1738_Suc__mod__eq__add3__mod__number__of,axiom,
! [M_1,V] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M_1)))),hAPP(int,nat,number_number_of(nat),V)) = hAPP(nat,nat,hAPP(nat,fun(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_1)),hAPP(int,nat,number_number_of(nat),V)) ).
fof(fact_1739_Suc__mod__eq__add3__mod,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,M_1)))),N_1) = hAPP(nat,nat,hAPP(nat,fun(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_1)),N_1) ).
fof(fact_1740_mod__Suc__eq__mod__add3,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,suc,N_1)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),N_1)) ).
fof(fact_1741_neg__number__of__pred__iff__0,axiom,
! [V_1] :
( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
<=> hAPP(int,nat,number_number_of(nat),V_1) = zero_zero(nat) ) ).
fof(fact_1742_eq__number__of__eq,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [X_1,Y_1] :
( hAPP(int,X_a,number_number_of(X_a),X_1) = hAPP(int,X_a,number_number_of(X_a),Y_1)
<=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),hAPP(int,int,uminus_uminus(int),Y_1))))) ) ) ).
fof(fact_1743_tan__double,axiom,
! [X] :
( hAPP(real,real,cos,X) != zero_zero(real)
=> ( hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)) != zero_zero(real)
=> hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,tan,X))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,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_1744_cos__pi__eq__zero,axiom,
! [M_1] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),pi),hAPP(nat,real,real_1(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_1))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real) ).
fof(fact_1745_Euler_Oaux____2,axiom,
! [P_4] : 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_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_1746_neg__zmod__mult__2,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),zero_zero(int)))
=> hAPP(int,int,hAPP(int,fun(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))),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_1)) = 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,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),one_one(int))),A_1))),one_one(int)) ) ).
fof(fact_1747_zmod__number__of__Bit1,axiom,
! [V,W] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_mod(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,V))),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,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W)))),one_one(int)) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_mod(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,V))),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,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,number_number_of(int),V)),one_one(int))),hAPP(int,int,number_number_of(int),W)))),one_one(int)) ) ) ).
fof(fact_1748_arctan,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arctan,Y)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
& hAPP(real,real,tan,hAPP(real,real,arctan,Y)) = Y ) ).
fof(fact_1749_rel__simps_I48_J,axiom,
! [K,L] :
( hAPP(int,int,bit0,K) = hAPP(int,int,bit0,L)
<=> K = L ) ).
fof(fact_1750_rel__simps_I49_J,axiom,
! [K_4,L_4] : hAPP(int,int,bit0,K_4) != hAPP(int,int,bit1,L_4) ).
fof(fact_1751_rel__simps_I50_J,axiom,
! [K_4,L_4] : hAPP(int,int,bit1,K_4) != hAPP(int,int,bit0,L_4) ).
fof(fact_1752_Bit0__Pls,axiom,
hAPP(int,int,bit0,pls) = pls ).
fof(fact_1753_rel__simps_I38_J,axiom,
! [L] :
( pls = hAPP(int,int,bit0,L)
<=> pls = L ) ).
fof(fact_1754_rel__simps_I44_J,axiom,
! [K] :
( hAPP(int,int,bit0,K) = pls
<=> K = pls ) ).
fof(fact_1755_less__int__code_I13_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit0,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_1756_rel__simps_I14_J,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit0,L)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) ) ).
fof(fact_1757_less__eq__int__code_I13_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit0,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_1758_rel__simps_I31_J,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit0,L)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).
fof(fact_1759_mult__Bit0,axiom,
! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,bit0,K_4)),L_4) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),L_4)) ).
fof(fact_1760_Bit0__def,axiom,
! [K_4] : hAPP(int,int,bit0,K_4) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),K_4) ).
fof(fact_1761_add__Bit0__Bit0,axiom,
! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),L_4)) ).
fof(fact_1762_diff__bin__simps_I7_J,axiom,
! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),L_4)) ).
fof(fact_1763_rel__simps_I42_J,axiom,
! [L_4] : min != hAPP(int,int,bit0,L_4) ).
fof(fact_1764_rel__simps_I45_J,axiom,
! [K_4] : hAPP(int,int,bit0,K_4) != min ).
fof(fact_1765_minus__Bit0,axiom,
! [K_4] : hAPP(int,int,uminus_uminus(int),hAPP(int,int,bit0,K_4)) = hAPP(int,int,bit0,hAPP(int,int,uminus_uminus(int),K_4)) ).
fof(fact_1766_less__int__code_I15_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit0,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_1767_rel__simps_I16_J,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit0,L)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) ) ).
fof(fact_1768_rel__simps_I10_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),pls)) ) ).
fof(fact_1769_rel__simps_I4_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),hAPP(int,int,bit0,K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),pls),K)) ) ).
fof(fact_1770_bin__less__0__simps_I3_J,axiom,
! [W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,W_1)),zero_zero(int)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),zero_zero(int))) ) ).
fof(fact_1771_rel__simps_I32_J,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit1,L)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).
fof(fact_1772_less__eq__int__code_I14_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit1,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_1773_rel__simps_I27_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),pls))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),pls)) ) ).
fof(fact_1774_rel__simps_I21_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),hAPP(int,int,bit0,K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),pls),K)) ) ).
fof(fact_1775_add__Bit1__Bit0,axiom,
! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),L_4)) ).
fof(fact_1776_add__Bit0__Bit1,axiom,
! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),L_4)) ).
fof(fact_1777_diff__bin__simps_I9_J,axiom,
! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),L_4)) ).
fof(fact_1778_diff__bin__simps_I10_J,axiom,
! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),K_4),L_4)) ).
fof(fact_1779_diff__bin__simps_I3_J,axiom,
! [L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),pls),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),pls),L_4)) ).
fof(fact_1780_rel__simps_I8_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),hAPP(int,int,bit0,K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) ) ).
fof(fact_1781_rel__simps_I28_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit0,K)),min))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) ) ).
fof(fact_1782_zdiv__number__of__Bit0,axiom,
! [V,W] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,V))),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)),hAPP(int,int,number_number_of(int),W)) ).
fof(fact_1783_neg__number__of__Bit0,axiom,
! [W_1] :
( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,W_1))))
<=> hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),W_1))) ) ).
fof(fact_1784_pred__Bit1,axiom,
! [K_4] : hAPP(int,int,pred,hAPP(int,int,bit1,K_4)) = hAPP(int,int,bit0,K_4) ).
fof(fact_1785_pred__Bit0,axiom,
! [K_4] : hAPP(int,int,pred,hAPP(int,int,bit0,K_4)) = hAPP(int,int,bit1,hAPP(int,int,pred,K_4)) ).
fof(fact_1786_pred__Min,axiom,
hAPP(int,int,pred,min) = hAPP(int,int,bit0,min) ).
fof(fact_1787_iszero__number__of__Bit0,axiom,
! [X_a] :
( ( number_ring(X_a)
& ring_char_0(X_a) )
=> ! [W_1] :
( hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,W_1))))
<=> hBOOL(hAPP(X_a,bool,iszero(X_a),hAPP(int,X_a,number_number_of(X_a),W_1))) ) ) ).
fof(fact_1788_succ__Bit1,axiom,
! [K_4] : hAPP(int,int,succ,hAPP(int,int,bit1,K_4)) = hAPP(int,int,bit0,hAPP(int,int,succ,K_4)) ).
fof(fact_1789_succ__Bit0,axiom,
! [K_4] : hAPP(int,int,succ,hAPP(int,int,bit0,K_4)) = hAPP(int,int,bit1,K_4) ).
fof(fact_1790_number__of__Bit0,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,W)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),zero_zero(X_a)),hAPP(int,X_a,number_number_of(X_a),W))),hAPP(int,X_a,number_number_of(X_a),W)) ) ).
fof(fact_1791_rel__simps_I15_J,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),hAPP(int,int,bit1,L)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),L)) ) ).
fof(fact_1792_less__int__code_I14_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K1)),hAPP(int,int,bit1,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K1),K2)) ) ).
fof(fact_1793_rel__simps_I33_J,axiom,
! [K,L] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K)),hAPP(int,int,bit0,L)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),L)) ) ).
fof(fact_1794_less__eq__int__code_I15_J,axiom,
! [K1,K2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,bit1,K1)),hAPP(int,int,bit0,K2)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K1),K2)) ) ).
fof(fact_1795_mult__Bit1,axiom,
! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,bit1,K_4)),L_4) = 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_4),L_4))),L_4) ).
fof(fact_1796_rel__simps_I11_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,bit0,K)),min))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),K),min)) ) ).
fof(fact_1797_rel__simps_I25_J,axiom,
! [K] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),min),hAPP(int,int,bit0,K)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),min),K)) ) ).
fof(fact_1798_diff__bin__simps_I6_J,axiom,
! [L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),L_4)) ).
fof(fact_1799_diff__bin__simps_I5_J,axiom,
! [L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),hAPP(int,int,bit0,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),min),L_4)) ).
fof(fact_1800_diff__bin__simps_I8_J,axiom,
! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,bit0,K_4)),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,pred,K_4)),L_4)) ).
fof(fact_1801_add__Bit1__Bit1,axiom,
! [K_4,L_4] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,bit1,K_4)),hAPP(int,int,bit1,L_4)) = hAPP(int,int,bit0,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),K_4),hAPP(int,int,succ,L_4))) ).
fof(fact_1802_double__number__of__Bit0,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [W] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,W)) ) ).
fof(fact_1803_number__of1,axiom,
! [N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),N_1)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,N_1))))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,N_1)))) ) ) ).
fof(fact_1804_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_1805_semiring__mult__2,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),Z) ) ).
fof(fact_1806_mult__2,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),Z) ) ).
fof(fact_1807_semiring__mult__2__right,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),Z) ) ).
fof(fact_1808_mult__2__right,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),Z) ) ).
fof(fact_1809_semiring__one__add__one__is__two,axiom,
! [X_a] :
( number_semiring(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ) ).
fof(fact_1810_one__add__one__is__two,axiom,
! [X_a] :
( number_ring(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a)) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ) ).
fof(fact_1811_zero__eq__power2,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [Aa] :
( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(X_a)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_1812_zero__power2,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(X_a) ) ).
fof(fact_1813_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).
fof(fact_1814_power2__eq__square,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),A_1) ) ).
fof(fact_1815_one__power2,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),one_one(X_a)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(X_a) ) ).
fof(fact_1816_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_1817_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_1818_pos2,axiom,
hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_1819_power2__minus,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).
fof(fact_1820_power__even__eq,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_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)))),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).
fof(fact_1821_add__2__eq__Suc_H,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,N_1)) ).
fof(fact_1822_add__2__eq__Suc,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1) = hAPP(nat,nat,suc,hAPP(nat,nat,suc,N_1)) ).
fof(fact_1823_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_1824_power2__abs,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).
fof(fact_1825_abs__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).
fof(fact_1826_nat__mult__2,axiom,
! [Z] : 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) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Z),Z) ).
fof(fact_1827_nat__mult__2__right,axiom,
! [Z] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Z),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),Z) ).
fof(fact_1828_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_1829_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_1,hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_1830_eq__divide__2__times__iff,axiom,
! [X_1,Y_1,Z_1] :
( X_1 = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Y_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Z_1))
<=> hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Y_1),Z_1) ) ).
fof(fact_1831_cos__two__neq__zero,axiom,
hAPP(real,real,cos,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(real) ).
fof(fact_1832_div2__Suc__Suc,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,M_1))),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_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_1833_power2__ge__self,axiom,
! [X] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_1834_mod2__Suc__Suc,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,M_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),div_mod(nat),M_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_1835_pi__ge__two,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi)) ).
fof(fact_1836_pi__less__4,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_1837_cube__square,axiom,
! [A_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(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))))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) ).
fof(fact_1838_real__sum__of__halves,axiom,
! [X] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = X ).
fof(fact_1839_add__self__div__2,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),M_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = M_1 ).
fof(fact_1840_zmod__number__of__Bit0,axiom,
! [V,W] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,V))),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,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),V)),hAPP(int,int,number_number_of(int),W))) ).
fof(fact_1841_pi__half__neq__two,axiom,
hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ).
fof(fact_1842_power2__eq__iff__abs__eq,axiom,
! [Aa,Ba] :
( hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Aa),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),Ba),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))
<=> hAPP(int,int,abs_abs(int),Aa) = hAPP(int,int,abs_abs(int),Ba) ) ).
fof(fact_1843_abs__power2__distrib,axiom,
! [A_1] : hAPP(int,int,abs_abs(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))))) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,abs_abs(int),A_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_1844_zEvenI,axiom,
! [X_1,K] :
( X_1 = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K)
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven)) ) ).
fof(fact_1845_half__gt__zero__iff,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number_ring(X_a) )
=> ! [R_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),R_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),R_1)) ) ) ).
fof(fact_1846_half__gt__zero,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number_ring(X_a) )
=> ! [R_4] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),R_4))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),R_4),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).
fof(fact_1847_power2__eq__imp__eq,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ) ).
fof(fact_1848_power2__le__imp__le,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y)) ) ) ) ).
fof(fact_1849_zero__le__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_1850_power2__less__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),zero_zero(X_a))) ) ).
fof(fact_1851_zero__less__power2,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
<=> ti(X_a,Aa) != zero_zero(X_a) ) ) ).
fof(fact_1852_sum__power2__eq__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,Y_1] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(X_a)
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ).
fof(fact_1853_power2__eq__square__number__of,axiom,
! [X_b] :
( ( monoid_mult(X_b)
& number(X_b) )
=> ! [W] : hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),W)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),W)),hAPP(int,X_b,number_number_of(X_b),W)) ) ).
fof(fact_1854_power2__eq__1__iff,axiom,
! [X_a] :
( ring_11004092258visors(X_a)
=> ! [Aa] :
( hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(X_a)
<=> ( ti(X_a,Aa) = one_one(X_a)
| ti(X_a,Aa) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ) ).
fof(fact_1855_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),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(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)) ) ).
fof(fact_1856_less__2__cases,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
=> ( N_1 = zero_zero(nat)
| N_1 = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).
fof(fact_1857_ring__1__class_Opower__minus__even,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [A_1,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_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)))),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_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)))),N_1)) ) ).
fof(fact_1858_nat__2,axiom,
hAPP(int,nat,nat_1,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_1859_realpow__two__sum__zero__iff,axiom,
! [X_1,Y_1] :
( hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real)
<=> ( X_1 = zero_zero(real)
& Y_1 = zero_zero(real) ) ) ).
fof(fact_1860_cos__two__le__zero,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,cos,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),zero_zero(real))) ).
fof(fact_1861_cos__two__less__zero,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),zero_zero(real))) ).
fof(fact_1862_four__x__squared,axiom,
! [X] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(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_1863_two__realpow__ge__one,axiom,
! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1))) ).
fof(fact_1864_neq__one__mod__two,axiom,
! [X_1] :
( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(int)
<=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(int) ) ).
fof(fact_1865_two__realpow__gt,axiom,
! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1))) ).
fof(fact_1866_real__gt__half__sum,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Y)) ) ).
fof(fact_1867_real__less__half__sum,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_1868_realpow__square__minus__le,axiom,
! [U,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),U),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_1869_mod__exhaust__less__4,axiom,
! [M_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),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,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),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,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))
| hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))) ) ).
fof(fact_1870_pi__half__le__two,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_1871_pi__half__neq__zero,axiom,
hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(real) ).
fof(fact_1872_pi__half__less__two,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_1873_real__average__minus__second,axiom,
! [B,A_1] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),A_1)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),A_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A_1)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_1874_real__average__minus__first,axiom,
! [A_1,B] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),A_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),A_1)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_1875_power2__eq1__iff,axiom,
! [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)))) = one_one(int)
=> hAPP(int,int,abs_abs(int),A_1) = one_one(int) ) ).
fof(fact_1876_even__div__2__prop1,axiom,
! [X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(int) ) ).
fof(fact_1877_even__div__2__l,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zEven))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_1),Y_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Y_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).
fof(fact_1878_even__div__2__prop2,axiom,
! [X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = X_1 ) ).
fof(fact_1879_even__prod__div__2,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X_1),Y_1)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Y_1) ) ).
fof(fact_1880_even__sum__div__2,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),zEven))
=> hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Y_1)),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),X_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Y_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ) ).
fof(fact_1881_power2__less__imp__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),Y)) ) ) ) ).
fof(fact_1882_half,axiom,
! [X_a] :
( ( linord1117847801e_zero(X_a)
& number_ring(X_a) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_1883_sum__power2__le__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),zero_zero(X_a)))
<=> ( ti(X_a,X_1) = zero_zero(X_a)
& ti(X_a,Y_1) = zero_zero(X_a) ) ) ) ).
fof(fact_1884_sum__power2__ge__zero,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).
fof(fact_1885_not__sum__power2__lt__zero,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),zero_zero(X_a))) ) ).
fof(fact_1886_sum__power2__gt__zero__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))
<=> ( ti(X_a,X_1) != zero_zero(X_a)
| ti(X_a,Y_1) != zero_zero(X_a) ) ) ) ).
fof(fact_1887_power2__sum,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [X,Y] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),Y)) ) ).
fof(fact_1888_zero__le__even__power_H,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_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)))),N_1)))) ) ).
fof(fact_1889_even__power__le__0__imp__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,K_4] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_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)))),K_4))),zero_zero(X_a)))
=> ti(X_a,A_1) = zero_zero(X_a) ) ) ).
fof(fact_1890_comm__semiring__1__class_Onormalizing__semiring__rules_I37_J,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1))) ) ).
fof(fact_1891_power__odd__eq,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [A_1,N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ).
fof(fact_1892_power__minus1__even,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),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(X_a) ) ).
fof(fact_1893_int__pos__lt__two__imp__zero__or__one,axiom,
! [X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_1894_Suc__Suc__mult__two__diff__two,axiom,
! [N_1] :
( N_1 != zero_zero(nat)
=> hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1) ) ).
fof(fact_1895_lemma__Suc__Suc__4n__diff__2,axiom,
! [N_1] :
( N_1 != zero_zero(nat)
=> hAPP(nat,nat,suc,hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N_1) ) ).
fof(fact_1896_one__less__floor,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X_1)) ) ) ).
fof(fact_1897_floor__le__one,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)),one_one(int)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ) ).
fof(fact_1898_div__2__gt__zero,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_1899_mod2__gr__0,axiom,
! [M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M),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_mod(nat),M),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = one_one(nat) ) ).
fof(fact_1900_fact__lemma,axiom,
! [N_1] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N_1)) ).
fof(fact_1901_pi__half__ge__zero,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_1902_pi__half__gt__zero,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_1903_zdiv__number__of__Bit1,axiom,
! [V,W] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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))),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)),hAPP(int,int,number_number_of(int),W)) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,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))),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)),one_one(int))),hAPP(int,int,number_number_of(int),W)) ) ) ).
fof(fact_1904_less__one__imp__sqr__less__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),X)),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,hAPP(real,fun(nat,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_1905_zadd__power2,axiom,
! [A_1,B] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),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_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),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)),B))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_1906_zadd__power3,axiom,
! [A_1,B] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),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_1),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_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),B))),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_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).
fof(fact_1907_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_1908_zspecial__product,axiom,
! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(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))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_1909_m2pi__less__pi,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))),pi)) ).
fof(fact_1910_cos__pi__half,axiom,
hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = zero_zero(real) ).
fof(fact_1911_cos__two__pi,axiom,
hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi)) = one_one(real) ).
fof(fact_1912_cos__periodic,axiom,
! [X] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))) = hAPP(real,real,cos,X) ).
fof(fact_1913_cos__60,axiom,
hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_1914_zOddI,axiom,
! [X_1,K] :
( X_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K)),one_one(int))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd)) ) ).
fof(fact_1915_tan__periodic,axiom,
! [X] : hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))) = hAPP(real,real,tan,X) ).
fof(fact_1916_tan__45,axiom,
hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = one_one(real) ).
fof(fact_1917_arctan__ubound,axiom,
! [Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_1918_arctan1__eq__pi4,axiom,
hAPP(real,real,arctan,one_one(real)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_1919_power2__diff,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [X,Y] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),Y)) ) ).
fof(fact_1920_odd__0__le__power__imp__0__le,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,N_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)))))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),A_1)) ) ) ).
fof(fact_1921_odd__power__less__zero,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [N_1,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)))),zero_zero(X_a))) ) ) ).
fof(fact_1922_power__minus1__odd,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1))) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ).
fof(fact_1923_power__m1__even,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),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_1)) = one_one(X_a) ) ).
fof(fact_1924_power__m1__odd,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [N_1] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1))) = hAPP(int,X_a,number_number_of(X_a),min) ) ).
fof(fact_1925_Suc__mult__two__diff__one,axiom,
! [N_1] :
( N_1 != zero_zero(nat)
=> hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)),one_one(nat))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1) ) ).
fof(fact_1926_Int2_Oaux__2,axiom,
! [P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).
fof(fact_1927_Suc__n__div__2__gt__zero,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_1928_cos__double__less__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))),one_one(real))) ) ) ).
fof(fact_1929_Int2_Oaux__1,axiom,
! [P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,P_4)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ).
fof(fact_1930_cos__gt__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,cos,X))) ) ) ).
fof(fact_1931_zdiff__power3,axiom,
! [A_1,B] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B)),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_1),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_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),B))),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_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).
fof(fact_1932_zdiff__power2,axiom,
! [A_1,B] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),B)),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_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),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),A_1)),B))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ).
fof(fact_1933_minus__pi__half__less__zero,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),zero_zero(real))) ).
fof(fact_1934_cos__3over2__pi,axiom,
hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),pi)) = zero_zero(real) ).
fof(fact_1935_cos__2npi,axiom,
! [N_1] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,real_1(nat),N_1))),pi)) = one_one(real) ).
fof(fact_1936_tan__gt__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,tan,X))) ) ) ).
fof(fact_1937_tan__monotone,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,tan,Y)),hAPP(real,real,tan,X))) ) ) ) ).
fof(fact_1938_tan__monotone_H,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),X_1))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,tan,Y_1)),hAPP(real,real,tan,X_1))) ) ) ) ) ) ).
fof(fact_1939_tan__inverse,axiom,
! [Y] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,tan,Y)) = hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Y)) ).
fof(fact_1940_arctan__bounded,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arctan,Y)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arctan,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_1941_arctan__lbound,axiom,
! [Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arctan,Y))) ).
fof(fact_1942_neg__zdiv__mult__2,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),A_1),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))),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_1)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),B),one_one(int))),A_1) ) ).
fof(fact_1943_pos__zdiv__mult__2,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> 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))),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_1)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),A_1) ) ).
fof(fact_1944_pos__zmod__mult__2,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> hAPP(int,int,hAPP(int,fun(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))),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_1)) = 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,hAPP(int,fun(int,int),div_mod(int),B),A_1))) ) ).
fof(fact_1945_cos__ge__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,cos,X))) ) ) ).
fof(fact_1946_cos__gt__zero__pi,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,cos,X))) ) ) ).
fof(fact_1947_Euler_Oaux__2,axiom,
! [Pa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Pa),zOdd))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),Pa),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).
fof(fact_1948_tan__less__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,tan,X)),zero_zero(real))) ) ) ).
fof(fact_1949_machin,axiom,
hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(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_1950_arctan__tan,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hAPP(real,real,arctan,hAPP(real,real,tan,X)) = X ) ) ).
fof(fact_1951_tan__total__pi4,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),X)),one_one(real)))
=> ? [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),Z_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Z_2),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))
& hAPP(real,real,tan,Z_2) = X ) ) ).
fof(fact_1952_best__odd__division__abs,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X_1))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> ? [N_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Y_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N_2),X_1))))),X_1)) ) ) ).
fof(fact_1953_tan__total__pos,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
=> ? [X_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
& hAPP(real,real,tan,X_2) = Y ) ) ).
fof(fact_1954_best__division__abs,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ? [N_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_2),X))))),X)) ) ).
fof(fact_1955_inv__inv__aux,axiom,
! [P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_4))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),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_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))))) ) ).
fof(fact_1956_inv__is__inv__aux,axiom,
! [M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M_1))
=> hAPP(nat,nat,suc,hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M_1),one_one(int))) ) ).
fof(fact_1957_tan__total,axiom,
! [Y] :
? [X_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
& hAPP(real,real,tan,X_2) = Y
& ! [Y_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_2),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
& hAPP(real,real,tan,Y_2) = Y )
=> Y_2 = X_2 ) ) ).
fof(fact_1958_lemma__tan__total1,axiom,
! [Y] :
? [X_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
& hAPP(real,real,tan,X_2) = Y ) ).
fof(fact_1959_lemma__tan__total,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> ? [X_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),hAPP(real,real,tan,X_2))) ) ) ).
fof(fact_1960_zOddE,axiom,
! [X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> ~ ! [K_1] : X_1 != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K_1)),one_one(int)) ) ).
fof(fact_1961_real__sum__squared__expand,axiom,
! [X,Y] : hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,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(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)),Y)) ).
fof(fact_1962_cos__is__zero,axiom,
? [X_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
& hAPP(real,real,cos,X_2) = zero_zero(real)
& ! [Y_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_2),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
& hAPP(real,real,cos,Y_2) = zero_zero(real) )
=> Y_2 = X_2 ) ) ).
fof(fact_1963_zEvenE,axiom,
! [X_1] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zEven))
=> ~ ! [K_1] : X_1 != hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),K_1) ) ).
fof(fact_1964_negDivAlg__eqn__1__number__of,axiom,
! [W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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))))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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(int,fun(int,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))))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),W)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),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_1965_negDivAlg__eqn__number__of,axiom,
! [W,V] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V)))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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))))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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))) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,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))))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),V)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),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)))) ) ) ) ).
fof(fact_1966_negDivAlg__eqn,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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)) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),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))) ) ) ) ).
fof(fact_1967_posDivAlg__eqn,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A_1) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),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))) ) ) ) ).
fof(fact_1968_posDivAlg__eqn__number__of,axiom,
! [W,V] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V)))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,hAPP(int,int,number_number_of(int),W)),hAPP(int,int,number_number_of(int),V)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),V)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),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)))) ) ) ) ).
fof(fact_1969_posDivAlg_Osimps,axiom,
! [A_1,B] :
( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A_1) )
& ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),B))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),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))) ) ) ).
fof(fact_1970_posDivAlg__eqn__1__number__of,axiom,
! [W] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,number_number_of(int),W)))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,number_number_of(int),W)))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,one_one(int)),hAPP(int,int,number_number_of(int),W)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,hAPP(int,int,number_number_of(int),W)),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),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_1971_negDivAlg_Osimps,axiom,
! [A_1,B] :
( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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)) )
& ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_1),B)))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B),zero_zero(int))) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,A_1),B) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,B),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),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))) ) ) ).
fof(fact_1972_arcsin__lt__bounded,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),one_one(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ) ).
fof(fact_1973_arcsin__bounded,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ) ).
fof(fact_1974_arcsin__lbound,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y))) ) ) ).
fof(fact_1975_arcsin__ubound,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).
fof(fact_1976_lemma__STAR__cos,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N_1)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),N_1)) = zero_zero(real) ) ).
fof(fact_1977_real__of__nat__fact__not__zero,axiom,
! [N_1] : hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N_1)) != zero_zero(real) ).
fof(fact_1978_real__of__nat__fact__ge__zero,axiom,
! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N_1)))) ).
fof(fact_1979_real__of__nat__fact__gt__zero,axiom,
! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N_1)))) ).
fof(fact_1980_fact__diff__Suc,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),hAPP(nat,nat,suc,M_1)))
=> hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M_1)),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,M_1)),N_1)),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1))) ) ).
fof(fact_1981_fact__add__num__eq__if2__nat,axiom,
! [N_1,M_1] :
( ( M_1 = zero_zero(nat)
=> hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) = hAPP(nat,nat,fact_fact(nat),N_1) )
& ( M_1 != zero_zero(nat)
=> hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),one_one(nat))),N_1))) ) ) ).
fof(fact_1982_fact__add__num__eq__if__nat,axiom,
! [M_1,N_1] :
( ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) = zero_zero(nat)
=> hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) = one_one(nat) )
& ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1) != zero_zero(nat)
=> hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)),one_one(nat)))) ) ) ).
fof(fact_1983_fact__reduce__nat,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(nat,nat,fact_fact(nat),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N_1),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat)))) ) ).
fof(fact_1984_fact__num__eq__if__nat,axiom,
! [M_1] :
( ( M_1 = zero_zero(nat)
=> hAPP(nat,nat,fact_fact(nat),M_1) = one_one(nat) )
& ( M_1 != zero_zero(nat)
=> hAPP(nat,nat,fact_fact(nat),M_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),one_one(nat)))) ) ) ).
fof(fact_1985_lemma__STAR__sin,axiom,
! [N] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),hAPP(bool,fun(real,fun(real,real)),if(real),hAPP(nat,bool,even_odd_even(nat),N)),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),N)) = zero_zero(real) ).
fof(fact_1986_odd__1__nat,axiom,
~ hBOOL(hAPP(nat,bool,even_odd_even(nat),one_one(nat))) ).
fof(fact_1987_even__product__nat,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X_1),Y_1)))
<=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
| hBOOL(hAPP(nat,bool,even_odd_even(nat),Y_1)) ) ) ).
fof(fact_1988_even__sum__nat,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X_1),Y_1)))
<=> ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
& hBOOL(hAPP(nat,bool,even_odd_even(nat),Y_1)) )
| ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
& ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Y_1)) ) ) ) ).
fof(fact_1989_even__add,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)))
<=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),M))
<=> hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) ) ) ).
fof(fact_1990_odd__add,axiom,
! [M,N] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N)))
<=> ~ ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),M))
<=> ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) ) ) ).
fof(fact_1991_even__Suc,axiom,
! [X_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,suc,X_1)))
<=> ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1)) ) ).
fof(fact_1992_even__zero__nat,axiom,
hBOOL(hAPP(nat,bool,even_odd_even(nat),zero_zero(nat))) ).
fof(fact_1993_power__mono__odd,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y,N_1] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N_1))) ) ) ) ).
fof(fact_1994_odd__pos,axiom,
! [N_1] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1)) ) ).
fof(fact_1995_neg__power__if,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X,N_1] :
( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N_1) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1) )
& ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N_1) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)) ) ) ) ).
fof(fact_1996_Parity_Opower__minus__even,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X,N_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N_1) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1) ) ) ).
fof(fact_1997_power__minus__odd,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X,N_1] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),N_1) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)) ) ) ).
fof(fact_1998_power__even__abs,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,N_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),N_1) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1) ) ) ).
fof(fact_1999_even__difference__nat,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),X_1),Y_1)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_1))
| ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
& hBOOL(hAPP(nat,bool,even_odd_even(nat),Y_1)) )
| ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
& ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),Y_1)) ) ) ) ).
fof(fact_2000_zero__le__odd__power,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,N] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),N)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1)) ) ) ) ).
fof(fact_2001_zero__le__even__power,axiom,
! [X_a] :
( ( monoid_mult(X_a)
& linordered_ring(X_a) )
=> ! [X,N_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1))) ) ) ).
fof(fact_2002_zero__le__power__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),N)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa))
| hBOOL(hAPP(nat,bool,even_odd_even(nat),N)) ) ) ) ).
fof(fact_2003_zero__le__power__eq,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),N)))
<=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
| ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1)) ) ) ) ) ).
fof(fact_2004_power__less__zero__eq,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),N)),zero_zero(X_a)))
<=> ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),zero_zero(X_a))) ) ) ) ).
fof(fact_2005_minus__one__odd__power,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ).
fof(fact_2006_minus__one__even__power,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = one_one(X_a) ) ) ).
fof(fact_2007_minus__one__even__odd__power,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X] :
( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = one_one(X_a) )
& ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))),X) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ) ).
fof(fact_2008_power__mono__even,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,Y,N_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),hAPP(X_a,X_a,abs_abs(X_a),Y)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N_1))) ) ) ) ).
fof(fact_2009_even__nat__equiv__def2,axiom,
! [X_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
<=> ? [Y_2] : X_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),Y_2) ) ).
fof(fact_2010_odd__nat__equiv__def2,axiom,
! [X_1] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
<=> ? [Y_2] : X_1 = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),Y_2)) ) ).
fof(fact_2011_Parity_Oneg__one__odd__power,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [X] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = hAPP(int,X_a,number_number_of(X_a),min) ) ) ).
fof(fact_2012_power__even__abs__number__of,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X,W] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),W)))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,abs_abs(X_a),X)),hAPP(int,nat,number_number_of(nat),W)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),hAPP(int,nat,number_number_of(nat),W)) ) ) ).
fof(fact_2013_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_2014_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_2015_odd__nat__mod__two__eq__one,axiom,
! [X] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
=> hAPP(nat,nat,hAPP(nat,fun(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_2016_even__nat__mod__two__eq__zero,axiom,
! [X] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = zero_zero(nat) ) ).
fof(fact_2017_odd__nat__equiv__def,axiom,
! [X_1] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
<=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X_1),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = hAPP(nat,nat,suc,zero_zero(nat)) ) ).
fof(fact_2018_even__nat__equiv__def,axiom,
! [X_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
<=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X_1),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))) = zero_zero(nat) ) ).
fof(fact_2019_even__power__nat,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X_1),Y_1)))
<=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y_1)) ) ) ).
fof(fact_2020_power__le__zero__eq,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),N)),zero_zero(X_a)))
<=> ( N != zero_zero(nat)
& ( ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),zero_zero(X_a))) )
| ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
& ti(X_a,X_1) = zero_zero(X_a) ) ) ) ) ) ).
fof(fact_2021_zero__less__power__eq,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),N)))
<=> ( N = zero_zero(nat)
| ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
& ti(X_a,X_1) != zero_zero(X_a) )
| ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X_1)) ) ) ) ) ).
fof(fact_2022_zero__le__power__eq__number__of,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,W_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),hAPP(int,nat,number_number_of(nat),W_1))))
<=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),W_1)))
| ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),W_1)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),X_1)) ) ) ) ) ).
fof(fact_2023_power__less__zero__eq__number__of,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,W_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),hAPP(int,nat,number_number_of(nat),W_1))),zero_zero(X_a)))
<=> ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),W_1)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),zero_zero(X_a))) ) ) ) ).
fof(fact_2024_neg__one__even__odd__power,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [X] :
( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = one_one(X_a) )
& ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = hAPP(int,X_a,number_number_of(X_a),min) ) ) ) ).
fof(fact_2025_Parity_Oneg__one__even__power,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [X] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),X))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),min)),X) = one_one(X_a) ) ) ).
fof(fact_2026_even__num__iff,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
<=> ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ).
fof(fact_2027_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_2028_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_2029_power__le__zero__eq__number__of,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,W_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),hAPP(int,nat,number_number_of(nat),W_1))),zero_zero(X_a)))
<=> ( hAPP(int,nat,number_number_of(nat),W_1) != zero_zero(nat)
& ( ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),W_1)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),zero_zero(X_a))) )
| ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),W_1)))
& ti(X_a,X_1) = zero_zero(X_a) ) ) ) ) ) ).
fof(fact_2030_zero__less__power__eq__number__of,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X_1,W_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),hAPP(int,nat,number_number_of(nat),W_1))))
<=> ( hAPP(int,nat,number_number_of(nat),W_1) = zero_zero(nat)
| ( hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),W_1)))
& ti(X_a,X_1) != zero_zero(X_a) )
| ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,number_number_of(nat),W_1)))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X_1)) ) ) ) ) ).
fof(fact_2031_even__mult__two__ex,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
<=> ? [M_3] : N = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M_3) ) ).
fof(fact_2032_even__even__mod__4__iff,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
<=> hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).
fof(fact_2033_odd__Suc__mult__two__ex,axiom,
! [N] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
<=> ? [M_3] : N = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M_3)) ) ).
fof(fact_2034_lemma__even__div2,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ).
fof(fact_2035_fact__nonzero__nat,axiom,
! [N_1] : hAPP(nat,nat,fact_fact(nat),N_1) != zero_zero(nat) ).
fof(fact_2036_fact__mono__nat,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,fact_fact(nat),M_1)),hAPP(nat,nat,fact_fact(nat),N_1))) ) ).
fof(fact_2037_fact__1__nat,axiom,
hAPP(nat,nat,fact_fact(nat),one_one(nat)) = one_one(nat) ).
fof(fact_2038_lemma__not__even__div2,axiom,
! [N_1] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ).
fof(fact_2039_lemma__odd__mod__4__div__2,axiom,
! [N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))
=> ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_2040_lemma__even__mod__4__div__2,axiom,
! [N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = one_one(nat)
=> hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_2041_Fact_Ofact__0__nat,axiom,
hAPP(nat,nat,fact_fact(nat),zero_zero(nat)) = hAPP(nat,nat,suc,zero_zero(nat)) ).
fof(fact_2042_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_2043_fact__less__mono__nat,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,fact_fact(nat),M_1)),hAPP(nat,nat,fact_fact(nat),N_1))) ) ) ).
fof(fact_2044_fact__gt__zero__nat,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,fact_fact(nat),N_1))) ).
fof(fact_2045_cos__zero__iff,axiom,
! [X_1] :
( hAPP(real,real,cos,X_1) = zero_zero(real)
<=> ( ? [N_2] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_2))
& X_1 = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_2)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) )
| ? [N_2] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_2))
& X_1 = hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_2)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ) ).
fof(fact_2046_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_2047_fact__ge__one__nat,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),hAPP(nat,nat,fact_fact(nat),N_1))) ).
fof(fact_2048_fact__ge__Suc__0__nat,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,zero_zero(nat))),hAPP(nat,nat,fact_fact(nat),N_1))) ).
fof(fact_2049_fact__plus__one__nat,axiom,
! [N_1] : hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),one_one(nat))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),one_one(nat))),hAPP(nat,nat,fact_fact(nat),N_1)) ).
fof(fact_2050_fact__mod,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,fact_fact(nat),N_1)),hAPP(nat,nat,fact_fact(nat),M_1)) = zero_zero(nat) ) ).
fof(fact_2051_cos__zero__lemma,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hAPP(real,real,cos,X) = zero_zero(real)
=> ? [N_2] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_2))
& X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_2)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ) ) ).
fof(fact_2052_odd__square,axiom,
! [N_1] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> ? [X_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X_2)),one_one(nat)) ) ).
fof(fact_2053_sin__coeff__def,axiom,
! [X_2] :
( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_2))
=> hAPP(nat,real,sin_coeff,X_2) = zero_zero(real) )
& ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X_2))
=> hAPP(nat,real,sin_coeff,X_2) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(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_2),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_1(nat),hAPP(nat,nat,fact_fact(nat),X_2))) ) ) ).
fof(fact_2054_even__square,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> ? [X_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X_2) ) ).
fof(fact_2055_cos__coeff__def,axiom,
! [X_2] :
( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_2))
=> hAPP(nat,real,cos_coeff,X_2) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X_2),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),X_2))) )
& ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),X_2))
=> hAPP(nat,real,cos_coeff,X_2) = zero_zero(real) ) ) ).
fof(fact_2056_WilsonRuss_Oinv__def,axiom,
! [P_4,A_1] : hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_4) ).
fof(fact_2057_arctan__inverse,axiom,
! [X] :
( X != zero_zero(real)
=> hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),X)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sgn_sgn(real),X)),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(real,real,arctan,X)) ) ).
fof(fact_2058_tan__half,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hAPP(real,real,tan,X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X))),one_one(real))) ) ) ).
fof(fact_2059_sin__cos__npi,axiom,
! [N_1] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)))),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)),N_1) ).
fof(fact_2060_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ) ).
fof(fact_2061_sin__zero,axiom,
hAPP(real,real,sin,zero_zero(real)) = zero_zero(real) ).
fof(fact_2062_sgn__sgn,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),A_1)) = hAPP(X_a,X_a,sgn_sgn(X_a),A_1) ) ).
fof(fact_2063_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_2064_sgn__zero__iff,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1] :
( hAPP(X_a,X_a,sgn_sgn(X_a),X_1) = zero_zero(X_a)
<=> ti(X_a,X_1) = zero_zero(X_a) ) ) ).
fof(fact_2065_sgn__0__0,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa] :
( hAPP(X_a,X_a,sgn_sgn(X_a),Aa) = zero_zero(X_a)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_2066_sgn__zero,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> hAPP(X_a,X_a,sgn_sgn(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_2067_sgn0,axiom,
! [X_a] :
( sgn_if(X_a)
=> hAPP(X_a,X_a,sgn_sgn(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_2068_sgn__mult,axiom,
! [X_a] :
( real_n1866405975lgebra(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)),hAPP(X_a,X_a,sgn_sgn(X_a),Y)) ) ).
fof(fact_2069_sgn__times,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),A_1)),hAPP(X_a,X_a,sgn_sgn(X_a),B)) ) ).
fof(fact_2070_sgn__one,axiom,
! [X_a] :
( real_n2089651433ebra_1(X_a)
=> hAPP(X_a,X_a,sgn_sgn(X_a),one_one(X_a)) = one_one(X_a) ) ).
fof(fact_2071_sgn__minus,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)) ) ).
fof(fact_2072_sin__le__one,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sin,X)),one_one(real))) ).
fof(fact_2073_sin__pi,axiom,
hAPP(real,real,sin,pi) = zero_zero(real) ).
fof(fact_2074_ln__less__self,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,X)),X)) ) ).
fof(fact_2075_ln__inj__iff,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1))
=> ( hAPP(real,real,ln,X_1) = hAPP(real,real,ln,Y_1)
<=> X_1 = Y_1 ) ) ) ).
fof(fact_2076_ln__less__cancel__iff,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,X_1)),hAPP(real,real,ln,Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1)) ) ) ) ).
fof(fact_2077_ln__one,axiom,
hAPP(real,real,ln,one_one(real)) = zero_zero(real) ).
fof(fact_2078_sgn__less,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),Aa)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_2079_sgn__greater,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,sgn_sgn(X_a),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Aa)) ) ) ).
fof(fact_2080_real__sgn__eq,axiom,
! [X] : hAPP(real,real,sgn_sgn(real),X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,abs_abs(real),X)) ).
fof(fact_2081_mult__sgn__abs,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)),hAPP(X_a,X_a,abs_abs(X_a),X)) = ti(X_a,X) ) ).
fof(fact_2082_abs__sgn,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [K_4] : hAPP(X_a,X_a,abs_abs(X_a),K_4) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),K_4),hAPP(X_a,X_a,sgn_sgn(X_a),K_4)) ) ).
fof(fact_2083_ln__eq__minus__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hAPP(real,real,ln,X) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),one_one(real))
=> X = one_one(real) ) ) ).
fof(fact_2084_sin__ge__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),pi))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sin,X))) ) ) ).
fof(fact_2085_sin__gt__zero__pi,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),pi))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sin,X))) ) ) ).
fof(fact_2086_sin__ge__minus__one,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),hAPP(real,real,sin,X))) ).
fof(fact_2087_cos__one__sin__zero,axiom,
! [X] :
( hAPP(real,real,cos,X) = one_one(real)
=> hAPP(real,real,sin,X) = zero_zero(real) ) ).
fof(fact_2088_abs__sin__le__one,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,sin,X))),one_one(real))) ).
fof(fact_2089_sin__add,axiom,
! [X,Y] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,sin,Y))) ).
fof(fact_2090_sin__diff,axiom,
! [X,Y] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,sin,Y))) ).
fof(fact_2091_sin__diff2,axiom,
! [X,Y] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Y)),hAPP(real,real,sin,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,Y)),hAPP(real,real,cos,X))) ).
fof(fact_2092_sin__periodic__pi2,axiom,
! [X] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),pi),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X)) ).
fof(fact_2093_sin__periodic__pi,axiom,
! [X] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),pi)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X)) ).
fof(fact_2094_ln__le__cancel__iff,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,X_1)),hAPP(real,real,ln,Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1)) ) ) ) ).
fof(fact_2095_tan__def,axiom,
! [X] : hAPP(real,real,tan,X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,X)) ).
fof(fact_2096_ln__ge__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,ln,X))) ) ).
fof(fact_2097_ln__gt__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,ln,X))) ) ).
fof(fact_2098_ln__gt__zero__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,ln,X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X_1)) ) ) ).
fof(fact_2099_ln__eq__zero__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( hAPP(real,real,ln,X_1) = zero_zero(real)
<=> X_1 = one_one(real) ) ) ).
fof(fact_2100_ln__less__zero__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,X_1)),zero_zero(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),one_one(real))) ) ) ).
fof(fact_2101_ln__less__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,ln,X)),zero_zero(real))) ) ) ).
fof(fact_2102_ln__gt__zero__imp__gt__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,ln,X)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ).
fof(fact_2103_real__sgn__pos,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,sgn_sgn(real),X) = one_one(real) ) ).
fof(fact_2104_sgn__pos,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hAPP(X_a,X_a,sgn_sgn(X_a),A_1) = one_one(X_a) ) ) ).
fof(fact_2105_sgn__1__pos,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa] :
( hAPP(X_a,X_a,sgn_sgn(X_a),Aa) = one_one(X_a)
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Aa)) ) ) ).
fof(fact_2106_ln__le__minus__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,X)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),one_one(real)))) ) ).
fof(fact_2107_sin__npi2,axiom,
! [N_1] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),pi),hAPP(nat,real,real_1(nat),N_1))) = zero_zero(real) ).
fof(fact_2108_sin__npi,axiom,
! [N_1] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),pi)) = zero_zero(real) ).
fof(fact_2109_sin__cos__squared__add3,axiom,
! [X] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,X))) = one_one(real) ).
fof(fact_2110_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_2111_cos__diff2,axiom,
! [X,Y] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,Y)),hAPP(real,real,cos,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,Y)),hAPP(real,real,sin,X))) ).
fof(fact_2112_cos__diff,axiom,
! [X,Y] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,Y))) ).
fof(fact_2113_cos__add,axiom,
! [X,Y] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,sin,Y))) ).
fof(fact_2114_ln__ge__zero__imp__ge__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,ln,X)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ).
fof(fact_2115_ln__ge__zero__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,ln,X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X_1)) ) ) ).
fof(fact_2116_ln__mult,axiom,
! [Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,ln,X)),hAPP(real,real,ln,Y)) ) ) ).
fof(fact_2117_ln__add__one__self__le__self,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).
fof(fact_2118_ln__div,axiom,
! [Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,X)),hAPP(real,real,ln,Y)) ) ) ).
fof(fact_2119_ln__add__one__self__le__self2,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).
fof(fact_2120_ln__one__minus__pos__upper__bound,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X))),hAPP(real,real,uminus_uminus(real),X))) ) ) ).
fof(fact_2121_aux5,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),one_one(real)))
=> hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),X))))) ) ).
fof(fact_2122_real__sgn__neg,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
=> hAPP(real,real,sgn_sgn(real),X) = hAPP(int,real,number_number_of(real),min) ) ).
fof(fact_2123_sgn__real__def,axiom,
! [A_1] :
( ( A_1 = zero_zero(real)
=> hAPP(real,real,sgn_sgn(real),A_1) = zero_zero(real) )
& ( A_1 != zero_zero(real)
=> ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> hAPP(real,real,sgn_sgn(real),A_1) = one_one(real) )
& ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> hAPP(real,real,sgn_sgn(real),A_1) = hAPP(real,real,uminus_uminus(real),one_one(real)) ) ) ) ) ).
fof(fact_2124_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(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,sgn_sgn(real),X) = one_one(real) )
& ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,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_2125_sgn__neg,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> hAPP(X_a,X_a,sgn_sgn(X_a),A_1) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ).
fof(fact_2126_sgn__1__neg,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Aa] :
( hAPP(X_a,X_a,sgn_sgn(X_a),Aa) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_2127_sgn__if,axiom,
! [X_a] :
( sgn_if(X_a)
=> ! [X] :
( ( ti(X_a,X) = zero_zero(X_a)
=> hAPP(X_a,X_a,sgn_sgn(X_a),X) = zero_zero(X_a) )
& ( ti(X_a,X) != zero_zero(X_a)
=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
=> hAPP(X_a,X_a,sgn_sgn(X_a),X) = one_one(X_a) )
& ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
=> hAPP(X_a,X_a,sgn_sgn(X_a),X) = hAPP(X_a,X_a,uminus_uminus(X_a),one_one(X_a)) ) ) ) ) ) ).
fof(fact_2128_sin__arcsin,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> hAPP(real,real,sin,hAPP(real,real,arcsin,Y)) = Y ) ) ).
fof(fact_2129_ln__realpow,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,ln,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N_1)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,ln,X)) ) ).
fof(fact_2130_sin__gt__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sin,X))) ) ) ).
fof(fact_2131_sin__double,axiom,
! [X] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,sin,X))),hAPP(real,real,cos,X)) ).
fof(fact_2132_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(real,fun(real,real),plus_plus(real),hAPP(real,real,tan,X)),hAPP(real,real,tan,Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))) ) ) ).
fof(fact_2133_sin__two__pi,axiom,
hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi)) = zero_zero(real) ).
fof(fact_2134_sin__periodic,axiom,
! [X] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),pi))) = hAPP(real,real,sin,X) ).
fof(fact_2135_sin__30,axiom,
hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_2136_sin__pi__half,axiom,
hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = one_one(real) ).
fof(fact_2137_sin__gt__zero2,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sin,X))) ) ) ).
fof(fact_2138_ln__one__plus__pos__lower__bound,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(real,real),plus_plus(real),one_one(real)),X)))) ) ) ).
fof(fact_2139_sin__cos__squared__add2__mult,axiom,
! [R_4,A_1] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),hAPP(real,real,cos,A_1))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),hAPP(real,real,sin,A_1))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_2140_sin__2npi,axiom,
! [N_1] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,real_1(nat),N_1))),pi)) = zero_zero(real) ).
fof(fact_2141_sin__cos__squared__add,axiom,
! [X] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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_2142_sin__cos__squared__add2,axiom,
! [X] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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_2143_sin__squared__eq,axiom,
! [X] : hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,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_2144_cos__squared__eq,axiom,
! [X] : hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,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_2145_sin__monotone__2pi_H,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sin,Y)),hAPP(real,real,sin,X))) ) ) ) ).
fof(fact_2146_cos__sin__eq,axiom,
! [X] : hAPP(real,real,cos,X) = hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X)) ).
fof(fact_2147_sin__cos__eq,axiom,
! [X] : hAPP(real,real,sin,X) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X)) ).
fof(fact_2148_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).
fof(fact_2149_sin__less__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sin,X)),zero_zero(real))) ) ) ).
fof(fact_2150_cos__double,axiom,
! [X] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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_2151_sin__3over2__pi,axiom,
hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),pi)) = hAPP(real,real,uminus_uminus(real),one_one(real)) ).
fof(fact_2152_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(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_2153_arcsin__sin,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hAPP(real,real,arcsin,hAPP(real,real,sin,X)) = X ) ) ).
fof(fact_2154_sin__cos__add,axiom,
! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sin,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,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(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X)),hAPP(real,real,cos,Y))),hAPP(real,real,hAPP(real,fun(real,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_2155_sin__cos__minus,axiom,
! [X] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,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(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,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_2156_ln__one__minus__pos__lower__bound,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,uminus_uminus(real),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(real,real),minus_minus(real),one_one(real)),X)))) ) ) ).
fof(fact_2157_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).
fof(fact_2158_sin__expansion__lemma,axiom,
! [X,M_1] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,M_1))),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),M_1)),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_2159_sin__zero__iff,axiom,
! [X_1] :
( hAPP(real,real,sin,X_1) = zero_zero(real)
<=> ( ? [N_2] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_2))
& X_1 = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_2)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) )
| ? [N_2] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_2))
& X_1 = hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_2)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ) ).
fof(fact_2160_arcsin__pi,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),pi))
& hAPP(real,real,sin,hAPP(real,real,arcsin,Y)) = Y ) ) ) ).
fof(fact_2161_arcsin,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,real,arcsin,Y)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,arcsin,Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
& hAPP(real,real,sin,hAPP(real,real,arcsin,Y)) = Y ) ) ) ).
fof(fact_2162_cos__expansion__lemma,axiom,
! [X,M_1] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,M_1))),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),M_1)),pi)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).
fof(fact_2163_sin__zero__lemma,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hAPP(real,real,sin,X) = zero_zero(real)
=> ? [N_2] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N_2))
& X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_2)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ) ) ).
fof(fact_2164_sin__total,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),Y))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
=> ? [X_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
& hAPP(real,real,sin,X_2) = Y
& ! [Y_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_2),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
& hAPP(real,real,sin,Y_2) = Y )
=> Y_2 = X_2 ) ) ) ) ).
fof(fact_2165_Ln_Oaux1,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),N_1)))) ) ) ).
fof(fact_2166_cos__arcsin,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,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(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).
fof(fact_2167_sin__arccos,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(int,real,number_number_of(real),min)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,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(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ) ).
fof(fact_2168_abs__inverse,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [A_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)) ) ).
fof(fact_2169_INVERSE__ZERO,axiom,
hAPP(real,real,inverse_inverse(real),zero_zero(real)) = zero_zero(real) ).
fof(fact_2170_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_2171_nonzero__inverse__eq__imp__eq,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1,B] :
( hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,inverse_inverse(X_a),B)
=> ( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B) != zero_zero(X_a)
=> ti(X_a,A_1) = ti(X_a,B) ) ) ) ) ).
fof(fact_2172_inverse__zero__imp__zero,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] :
( hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = zero_zero(X_a)
=> ti(X_a,A_1) = zero_zero(X_a) ) ) ).
fof(fact_2173_nonzero__inverse__inverse__eq,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = ti(X_a,A_1) ) ) ).
fof(fact_2174_nonzero__imp__inverse__nonzero,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),A_1) != zero_zero(X_a) ) ) ).
fof(fact_2175_inverse__nonzero__iff__nonzero,axiom,
! [X_a] :
( divisi14063676e_zero(X_a)
=> ! [Aa] :
( hAPP(X_a,X_a,inverse_inverse(X_a),Aa) = zero_zero(X_a)
<=> ti(X_a,Aa) = zero_zero(X_a) ) ) ).
fof(fact_2176_inverse__zero,axiom,
! [X_a] :
( divisi14063676e_zero(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_2177_field__inverse__zero,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_2178_inverse__mult__distrib,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ).
fof(fact_2179_real__sqrt__le__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,X_1)),hAPP(real,real,sqrt,Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1)) ) ).
fof(fact_2180_real__sqrt__le__mono,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y))) ) ).
fof(fact_2181_real__sqrt__eq__0__iff,axiom,
! [X_1] :
( hAPP(real,real,sqrt,X_1) = zero_zero(real)
<=> X_1 = zero_zero(real) ) ).
fof(fact_2182_real__sqrt__zero,axiom,
hAPP(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).
fof(fact_2183_inverse__inverse__eq,axiom,
! [X_a] :
( divisi14063676e_zero(X_a)
=> ! [A_1] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = ti(X_a,A_1) ) ).
fof(fact_2184_inverse__eq__iff__eq,axiom,
! [X_a] :
( divisi14063676e_zero(X_a)
=> ! [Aa,Ba] :
( hAPP(X_a,X_a,inverse_inverse(X_a),Aa) = hAPP(X_a,X_a,inverse_inverse(X_a),Ba)
<=> ti(X_a,Aa) = ti(X_a,Ba) ) ) ).
fof(fact_2185_inverse__eq__imp__eq,axiom,
! [X_a] :
( divisi14063676e_zero(X_a)
=> ! [A_1,B] :
( hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,inverse_inverse(X_a),B)
=> ti(X_a,A_1) = ti(X_a,B) ) ) ).
fof(fact_2186_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_2187_real__sqrt__eq__iff,axiom,
! [X_1,Y_1] :
( hAPP(real,real,sqrt,X_1) = hAPP(real,real,sqrt,Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_2188_inverse__divide,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),B),A_1) ) ).
fof(fact_2189_real__sqrt__divide,axiom,
! [X,Y] : hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y)) ).
fof(fact_2190_real__sqrt__one,axiom,
hAPP(real,real,sqrt,one_one(real)) = one_one(real) ).
fof(fact_2191_real__sqrt__eq__1__iff,axiom,
! [X_1] :
( hAPP(real,real,sqrt,X_1) = one_one(real)
<=> X_1 = one_one(real) ) ).
fof(fact_2192_real__sqrt__power,axiom,
! [X,K_4] : hAPP(real,real,sqrt,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),K_4)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,X)),K_4) ).
fof(fact_2193_inverse__minus__eq,axiom,
! [X_a] :
( divisi14063676e_zero(X_a)
=> ! [A_1] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) ) ).
fof(fact_2194_power__inverse,axiom,
! [X_a] :
( divisi14063676e_zero(X_a)
=> ! [A_1,N_1] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),N_1) ) ).
fof(fact_2195_real__sqrt__mult,axiom,
! [X,Y] : hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y)) ).
fof(fact_2196_inverse__1,axiom,
! [X_a] :
( division_ring(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),one_one(X_a)) = one_one(X_a) ) ).
fof(fact_2197_inverse__eq__1__iff,axiom,
! [X_a] :
( field_inverse_zero(X_a)
=> ! [X_1] :
( hAPP(X_a,X_a,inverse_inverse(X_a),X_1) = one_one(X_a)
<=> ti(X_a,X_1) = one_one(X_a) ) ) ).
fof(fact_2198_real__sqrt__less__mono,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,X)),hAPP(real,real,sqrt,Y))) ) ).
fof(fact_2199_real__sqrt__less__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,X_1)),hAPP(real,real,sqrt,Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1)) ) ).
fof(fact_2200_sqrt__divide__self__eq,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,X)),X) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,X)) ) ).
fof(fact_2201_inverse__nonnegative__iff__nonnegative,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),Aa)) ) ) ).
fof(fact_2202_inverse__nonpositive__iff__nonpositive,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Aa)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_2203_inverse__less__imp__less__neg,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).
fof(fact_2204_inverse__less__imp__less,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),A_1)) ) ) ) ).
fof(fact_2205_inverse__negative__imp__negative,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),zero_zero(X_a)))
=> ( ti(X_a,A_1) != zero_zero(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a))) ) ) ) ).
fof(fact_2206_less__imp__inverse__less__neg,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).
fof(fact_2207_less__imp__inverse__less,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).
fof(fact_2208_negative__imp__inverse__negative,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),zero_zero(X_a))) ) ) ).
fof(fact_2209_inverse__positive__imp__positive,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)))
=> ( ti(X_a,A_1) != zero_zero(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1)) ) ) ) ).
fof(fact_2210_positive__imp__inverse__positive,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ).
fof(fact_2211_inverse__negative__iff__negative,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Aa)),zero_zero(X_a)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),zero_zero(X_a))) ) ) ).
fof(fact_2212_inverse__positive__iff__positive,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),Aa)) ) ) ).
fof(fact_2213_nonzero__inverse__mult__distrib,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [B,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) ) ) ) ).
fof(fact_2214_inverse__unique,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1,B] :
( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B) = one_one(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = ti(X_a,B) ) ) ).
fof(fact_2215_nonzero__power__inverse,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [N_1,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),N_1) ) ) ).
fof(fact_2216_real__sqrt__ge__0__iff,axiom,
! [Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_1)) ) ).
fof(fact_2217_real__sqrt__le__0__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,X_1)),zero_zero(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),zero_zero(real))) ) ).
fof(fact_2218_real__sqrt__ge__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,X))) ) ).
fof(fact_2219_real__sqrt__eq__zero__cancel,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hAPP(real,real,sqrt,X) = zero_zero(real)
=> X = zero_zero(real) ) ) ).
fof(fact_2220_real__sqrt__gt__0__iff,axiom,
! [Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sqrt,Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1)) ) ).
fof(fact_2221_real__sqrt__lt__0__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,X_1)),zero_zero(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),zero_zero(real))) ) ).
fof(fact_2222_real__sqrt__not__eq__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,sqrt,X) != zero_zero(real) ) ).
fof(fact_2223_real__sqrt__gt__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sqrt,X))) ) ).
fof(fact_2224_nonzero__inverse__minus__eq,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),A_1)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) ) ) ).
fof(fact_2225_divide__inverse,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ).
fof(fact_2226_field__divide__inverse,axiom,
! [X_a] :
( field(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ).
fof(fact_2227_nonzero__abs__inverse,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,abs_abs(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,abs_abs(X_a),A_1)) ) ) ).
fof(fact_2228_real__sqrt__ge__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,sqrt,X))) ) ).
fof(fact_2229_real__sqrt__le__1__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,X_1)),one_one(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),one_one(real))) ) ).
fof(fact_2230_real__sqrt__ge__1__iff,axiom,
! [Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,sqrt,Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y_1)) ) ).
fof(fact_2231_real__sqrt__gt__1__iff,axiom,
! [Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,sqrt,Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Y_1)) ) ).
fof(fact_2232_real__sqrt__lt__1__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,X_1)),one_one(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),one_one(real))) ) ).
fof(fact_2233_field__class_Onormalizing__field__rules_I2_J,axiom,
! [X_a] :
( field(X_a)
=> ! [A_1] : hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1) ) ).
fof(fact_2234_inverse__eq__divide,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] : hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1) ) ).
fof(fact_2235_real__sqrt__abs2,axiom,
! [X] : hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X)) = hAPP(real,real,abs_abs(real),X) ).
fof(fact_2236_real__divide__def,axiom,
! [R,S_6] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),R),S_6) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R),hAPP(real,real,inverse_inverse(real),S_6)) ).
fof(fact_2237_divide__real__def,axiom,
! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),hAPP(real,real,inverse_inverse(real),Y)) ).
fof(fact_2238_inverse__le__imp__le__neg,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) ) ) ) ).
fof(fact_2239_inverse__le__imp__le,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),A_1)) ) ) ) ).
fof(fact_2240_le__imp__inverse__le__neg,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).
fof(fact_2241_le__imp__inverse__le,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).
fof(fact_2242_inverse__le__1__iff,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),X_1)),one_one(X_a)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),zero_zero(X_a)))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),X_1)) ) ) ) ).
fof(fact_2243_one__less__inverse,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).
fof(fact_2244_one__less__inverse__iff,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),X_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),one_one(X_a))) ) ) ) ).
fof(fact_2245_division__ring__inverse__add,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [B,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B))),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ) ) ).
fof(fact_2246_inverse__add,axiom,
! [X_a] :
( field(X_a)
=> ! [B,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ) ) ).
fof(fact_2247_right__inverse,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = one_one(X_a) ) ) ).
fof(fact_2248_field__inverse,axiom,
! [X_a] :
( field(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),A_1) = one_one(X_a) ) ) ).
fof(fact_2249_left__inverse,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),A_1) = one_one(X_a) ) ) ).
fof(fact_2250_division__ring__inverse__diff,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [B,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1))),hAPP(X_a,X_a,inverse_inverse(X_a),B)) ) ) ) ).
fof(fact_2251_nonzero__inverse__eq__divide,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),A_1) ) ) ).
fof(fact_2252_inverse__numeral__1,axiom,
! [X_a] :
( ( field(X_a)
& number_ring(X_a) )
=> hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls))) = hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit1,pls)) ) ).
fof(fact_2253_inverse__eq__divide__number__of,axiom,
! [X_b] :
( ( division_ring(X_b)
& number(X_b) )
=> ! [W] : hAPP(X_b,X_b,inverse_inverse(X_b),hAPP(int,X_b,number_number_of(X_b),W)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),one_one(X_b)),hAPP(int,X_b,number_number_of(X_b),W)) ) ).
fof(fact_2254_le__real__sqrt__sumsq,axiom,
! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),Y))))) ).
fof(fact_2255_real__mult__inverse__cancel2,axiom,
! [Y,U,X1_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X1_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X1_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),U)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),hAPP(real,real,inverse_inverse(real),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),U),hAPP(real,real,inverse_inverse(real),X1_1)))) ) ) ) ).
fof(fact_2256_real__mult__inverse__cancel,axiom,
! [Y,U,X1_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X1_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X1_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),U)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),X)),Y)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),X1_1)),U))) ) ) ) ).
fof(fact_2257_real__mult__inverse__left,axiom,
! [X] :
( X != zero_zero(real)
=> hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),X)),X) = one_one(real) ) ).
fof(fact_2258_real__inv__sqrt__pow2,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(nat,real,hAPP(real,fun(nat,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_2259_inverse__less__iff,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Aa)),hAPP(X_a,X_a,inverse_inverse(X_a),Ba)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Ba)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),Aa)) )
& ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Ba)),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba)) ) ) ) ) ).
fof(fact_2260_inverse__le__iff,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Aa)),hAPP(X_a,X_a,inverse_inverse(X_a),Ba)))
<=> ( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Ba)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),Aa)) )
& ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Ba)),zero_zero(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba)) ) ) ) ) ).
fof(fact_2261_one__le__inverse,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),one_one(X_a)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),A_1))) ) ) ) ).
fof(fact_2262_inverse__less__1__iff,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),X_1)),one_one(X_a)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),zero_zero(X_a)))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),one_one(X_a)),X_1)) ) ) ) ).
fof(fact_2263_one__le__inverse__iff,axiom,
! [X_a] :
( linord1117847801e_zero(X_a)
=> ! [X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),one_one(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),X_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),one_one(X_a))) ) ) ) ).
fof(fact_2264_inverse__diff__inverse,axiom,
! [X_a] :
( division_ring(X_a)
=> ! [B,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))),hAPP(X_a,X_a,inverse_inverse(X_a),B))) ) ) ) ).
fof(fact_2265_real__sqrt__mult__self__sum__ge__zero,axiom,
! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),Y))))) ).
fof(fact_2266_inv__real__of__nat__fact__ge__zero,axiom,
! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N_1))))) ).
fof(fact_2267_inv__real__of__nat__fact__gt__zero,axiom,
! [N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N_1))))) ).
fof(fact_2268_ln__inverse,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,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_2269_DERIV__inverse__lemma,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [H_2,B,A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> ( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),B))),H_2) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)),H_2))),hAPP(X_a,X_a,inverse_inverse(X_a),B))) ) ) ) ).
fof(fact_2270_zsgn__def,axiom,
! [I_1] :
( ( I_1 = zero_zero(int)
=> hAPP(int,int,sgn_sgn(int),I_1) = zero_zero(int) )
& ( I_1 != zero_zero(int)
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),I_1))
=> hAPP(int,int,sgn_sgn(int),I_1) = one_one(int) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),I_1))
=> hAPP(int,int,sgn_sgn(int),I_1) = hAPP(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).
fof(fact_2271_real__sqrt__two__ge__zero,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_2272_real__sqrt__two__gt__zero,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_2273_real__sqrt__two__gt__one,axiom,
hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_2274_real__sqrt__abs,axiom,
! [X] : hAPP(real,real,sqrt,hAPP(nat,real,hAPP(real,fun(nat,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_2275_tan__60,axiom,
hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) = hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) ).
fof(fact_2276_real__sqrt__pow2__iff,axiom,
! [X_1] :
( hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,X_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = X_1
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1)) ) ).
fof(fact_2277_real__sqrt__pow2,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(nat,real,hAPP(real,fun(nat,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_2278_real__sqrt__unique,axiom,
! [Y,X] :
( hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
=> hAPP(real,real,sqrt,X) = Y ) ) ).
fof(fact_2279_real__sqrt__pow2__gt__zero,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,hAPP(real,fun(nat,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_2280_lemma__real__divide__sqrt__less,axiom,
! [U] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),U))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),U)) ) ).
fof(fact_2281_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A_1,C,B,D] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),C)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),D)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),C),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),D),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) ).
fof(fact_2282_real__sqrt__sum__squares__ge2,axiom,
! [Y,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).
fof(fact_2283_real__sqrt__sum__squares__ge1,axiom,
! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).
fof(fact_2284_real__sqrt__sum__squares__eq__cancel,axiom,
! [X,Y] :
( hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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_2285_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X,Y] :
( hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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_2286_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X,Y,Xa_1,Ya] : hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_2287_cos__45,axiom,
hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_2288_cos__30,axiom,
hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_2289_sin__60,axiom,
hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_2290_sin__45,axiom,
hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_2291_real__sqrt__sum__squares__ge__zero,axiom,
! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).
fof(fact_2292_real__sqrt__ge__abs2,axiom,
! [Y,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),Y)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).
fof(fact_2293_real__sqrt__ge__abs1,axiom,
! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).
fof(fact_2294_tan__30,axiom,
hAPP(real,real,tan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit1,hAPP(int,int,bit1,pls))))) ).
fof(fact_2295_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X,Y,Xa_1,Ya] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Xa_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Ya),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) ).
fof(fact_2296_cos__arccos__lemma1,axiom,
! [X_5,Y_4] : hAPP(real,real,cos,hAPP(real,real,arccos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_5),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_5),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_5),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_5),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).
fof(fact_2297_cos__x__y__le__one,axiom,
! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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_2298_real__sqrt__sum__squares__less,axiom,
! [Y,X,U] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),X)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U),hAPP(real,real,sqrt,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),U)) ) ) ).
fof(fact_2299_tan__sec,axiom,
! [X] :
( hAPP(real,real,cos,X) != zero_zero(real)
=> hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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_2300_lemma__sqrt__hcomplex__capprox,axiom,
! [Y,X,U] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),U))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),U),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),U)) ) ) ) ) ) ).
fof(fact_2301_arctan__half,axiom,
! [X] : hAPP(real,real,arctan,X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,arctan,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) ).
fof(fact_2302_sin__arccos__abs,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,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(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_2303_sin__arccos__lemma1,axiom,
! [X_5,Y_4] : hAPP(real,real,sin,hAPP(real,real,arccos,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_5),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_5),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X_5),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_5),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y_4),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_2304_reals__Archimedean,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ? [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_2)))),X)) ) ).
fof(fact_2305_DERIV__arccos,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),arccos),X_1),hAPP(real,real,inverse_inverse(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))))) ) ) ).
fof(fact_2306_DERIV__arcsin,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),arcsin),X_1),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) ) ) ).
fof(fact_2307_Bolzano__bisect__diff,axiom,
! [P_1,N,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N))),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Ba),Aa)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) ) ).
fof(fact_2308_DERIV__cmult__Id,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [C_1,X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1)),X_1),C_1)) ) ).
fof(fact_2309_DERIV__unique,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [E_3,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),E_3))
=> ti(X_a,D_2) = ti(X_a,E_3) ) ) ) ).
fof(fact_2310_DERIV__cong,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_3,F,X_1,X_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),X_3))
=> ( ti(X_a,X_3) = ti(X_a,Y_3)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),Y_3)) ) ) ) ).
fof(fact_2311_Deriv_Olemma__DERIV__subst,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [E_3,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> ( ti(X_a,D_2) = ti(X_a,E_3)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),E_3)) ) ) ) ).
fof(fact_2312_Bolzano__bisect_Osimps_I1_J,axiom,
! [P_1,Aa,Ba] : hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),zero_zero(nat)) = hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),Aa),Ba) ).
fof(fact_2313_Bolzano__bisect__le,axiom,
! [P_1,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N_2))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N_2)))) ) ).
fof(fact_2314_DERIV__sin,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),sin),X_1),hAPP(real,real,cos,X_1))) ).
fof(fact_2315_DERIV__cos,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),cos),X_1),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X_1)))) ).
fof(fact_2316_Bolzano__bisect__fst__le__Suc,axiom,
! [P_1,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N_2))),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),hAPP(nat,nat,suc,N_2))))) ) ).
fof(fact_2317_Bolzano__bisect__Suc__le__snd,axiom,
! [P_1,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),hAPP(nat,nat,suc,N_2)))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N_2)))) ) ).
fof(fact_2318_DERIV__ln,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),ln),X_1),hAPP(real,real,inverse_inverse(real),X_1))) ) ).
fof(fact_2319_DERIV__ln__divide,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),ln),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),X_1))) ) ).
fof(fact_2320_DERIV__real__sqrt,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),sqrt),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,X_1))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ) ).
fof(fact_2321_DERIV__tan,axiom,
! [X_1] :
( hAPP(real,real,cos,X_1) != zero_zero(real)
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),tan),X_1),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).
fof(fact_2322_DERIV__arctan,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),arctan),X_1),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))) ).
fof(fact_2323_DERIV__real__sqrt__generic,axiom,
! [D_2,X_1] :
( X_1 != zero_zero(real)
=> ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> D_2 = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,X_1))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) )
=> ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),zero_zero(real)))
=> D_2 = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,sqrt,X_1)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),sqrt),X_1),D_2)) ) ) ) ).
fof(fact_2324_DERIV__const__average,axiom,
! [V_1,K,Aa,Ba] :
( Aa != Ba
=> ( ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),V_1),X_2),K))
=> hAPP(real,real,V_1,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Aa),Ba)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,V_1,Aa)),hAPP(real,real,V_1,Ba))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ) ).
fof(fact_2325_DERIV__local__max,axiom,
! [D_1,F,X_1,L] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),L))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
=> ( ! [Y_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X_1),Y_2))),D_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,Y_2)),hAPP(real,real,F,X_1))) )
=> L = zero_zero(real) ) ) ) ).
fof(fact_2326_DERIV__local__min,axiom,
! [D_1,F,X_1,L] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),L))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
=> ( ! [Y_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X_1),Y_2))),D_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,X_1)),hAPP(real,real,F,Y_2))) )
=> L = zero_zero(real) ) ) ) ).
fof(fact_2327_DERIV__local__const,axiom,
! [D_1,F,X_1,L] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),L))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
=> ( ! [Y_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X_1),Y_2))),D_1))
=> hAPP(real,real,F,X_1) = hAPP(real,real,F,Y_2) )
=> L = zero_zero(real) ) ) ) ).
fof(fact_2328_not__P__Bolzano__bisect,axiom,
! [N,Aa,Ba,P_1] :
( ! [A_3,B_3,C_4] :
( hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_3)))
=> ( hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_3),C_4)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_3))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_3),C_4))
=> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),C_4))) ) ) ) )
=> ( ~ hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),Aa),Ba)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> ~ hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N))))) ) ) ) ).
fof(fact_2329_not__P__Bolzano__bisect_H,axiom,
! [Aa,Ba,P_1] :
( ! [A_3,B_3,C_4] :
( ( hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_3)))
& hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_3),C_4)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_3))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_3),C_4)) )
=> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),C_4))) )
=> ( ~ hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),Aa),Ba)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> ! [N_2] : ~ hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N_2))),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N_2))))) ) ) ) ).
fof(fact_2330_DERIV__pos__inc__left,axiom,
! [F,X_1,L] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),L))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
=> ? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [H_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X_1),H_1))),hAPP(real,real,F,X_1))) ) ) ) ) ) ).
fof(fact_2331_DERIV__neg__dec__left,axiom,
! [F,X_1,L] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),L))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
=> ? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [H_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,X_1)),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X_1),H_1)))) ) ) ) ) ) ).
fof(fact_2332_DERIV__pos__inc__right,axiom,
! [F,X_1,L] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),L))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
=> ? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [H_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,X_1)),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X_1),H_1)))) ) ) ) ) ) ).
fof(fact_2333_DERIV__neg__dec__right,axiom,
! [F,X_1,L] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_1),L))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
=> ? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [H_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H_1),D_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X_1),H_1))),hAPP(real,real,F,X_1))) ) ) ) ) ) ).
fof(fact_2334_DERIV__const__ratio__const2,axiom,
! [F,K,Aa,Ba] :
( Aa != Ba
=> ( ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),K))
=> hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,Ba)),hAPP(real,real,F,Aa))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Ba),Aa)) = K ) ) ).
fof(fact_2335_DERIV__const__ratio__const,axiom,
! [F,K,Aa,Ba] :
( Aa != Ba
=> ( ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),K))
=> hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,Ba)),hAPP(real,real,F,Aa)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Ba),Aa)),K) ) ) ).
fof(fact_2336_DERIV__real__root__generic,axiom,
! [D_2,X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( X_1 != zero_zero(real)
=> ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> D_2 = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))) ) )
=> ( ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),zero_zero(real)))
=> D_2 = hAPP(real,real,uminus_uminus(real),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat))))))) ) )
=> ( ( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
=> D_2 = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,N)),X_1),D_2)) ) ) ) ) ) ).
fof(fact_2337_DERIV__even__real__root,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),zero_zero(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,N)),X_1),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(nat,real,real_1(nat),N))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))))) ) ) ) ).
fof(fact_2338_ln__bound,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,X)),X)) ) ).
fof(fact_2339_DERIV__real__root,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,N)),X_1),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))))) ) ) ).
fof(fact_2340_real__root__zero,axiom,
! [N_1] : hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),zero_zero(real)) = zero_zero(real) ).
fof(fact_2341_real__root__Suc__0,axiom,
! [X] : hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,suc,zero_zero(nat))),X) = X ).
fof(fact_2342_real__root__commute,axiom,
! [X,N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,M_1),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)) = hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(real,real,hAPP(nat,fun(real,real),root,M_1),X)) ) ) ).
fof(fact_2343_real__root__eq__iff,axiom,
! [X_1,Y_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1) = hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y_1)
<=> X_1 = Y_1 ) ) ).
fof(fact_2344_odd__real__root__power__cancel,axiom,
! [X,N_1] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N_1)) = X ) ).
fof(fact_2345_odd__real__root__pow,axiom,
! [X,N_1] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),N_1) = X ) ).
fof(fact_2346_odd__real__root__unique,axiom,
! [Y,X,N_1] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N_1))
=> ( hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),N_1) = X
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X) = Y ) ) ).
fof(fact_2347_real__root__le__mono,axiom,
! [X,Y,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),Y))) ) ) ).
fof(fact_2348_real__root__le__iff,axiom,
! [X_1,Y_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1)) ) ) ).
fof(fact_2349_real__root__eq__0__iff,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1) = zero_zero(real)
<=> X_1 = zero_zero(real) ) ) ).
fof(fact_2350_real__root__less__mono,axiom,
! [X,Y,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),Y))) ) ) ).
fof(fact_2351_real__root__less__iff,axiom,
! [X_1,Y_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1)) ) ) ).
fof(fact_2352_real__root__mult,axiom,
! [X,Y,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),Y)) ) ).
fof(fact_2353_real__root__mult__exp,axiom,
! [X,N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)),X) = hAPP(real,real,hAPP(nat,fun(real,real),root,M_1),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)) ) ) ).
fof(fact_2354_real__root__power,axiom,
! [X,K_4,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),K_4)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),K_4) ) ).
fof(fact_2355_real__root__eq__1__iff,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1) = one_one(real)
<=> X_1 = one_one(real) ) ) ).
fof(fact_2356_real__root__one,axiom,
! [N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),one_one(real)) = one_one(real) ) ).
fof(fact_2357_real__root__divide,axiom,
! [X,Y,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),Y)) ) ).
fof(fact_2358_real__root__minus,axiom,
! [X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(real,real,uminus_uminus(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)) ) ).
fof(fact_2359_real__root__abs,axiom,
! [X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(real,real,abs_abs(real),X)) = hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)) ) ).
fof(fact_2360_real__root__inverse,axiom,
! [X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)) ) ).
fof(fact_2361_real__root__ge__zero,axiom,
! [X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X))) ) ) ).
fof(fact_2362_real__root__le__0__iff,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),zero_zero(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),zero_zero(real))) ) ) ).
fof(fact_2363_real__root__ge__0__iff,axiom,
! [Y_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_1)) ) ) ).
fof(fact_2364_real__root__gt__0__iff,axiom,
! [Y_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1)) ) ) ).
fof(fact_2365_real__root__lt__0__iff,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),zero_zero(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),zero_zero(real))) ) ) ).
fof(fact_2366_real__root__gt__zero,axiom,
! [X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X))) ) ) ).
fof(fact_2367_real__root__decreasing,axiom,
! [X,N_6,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_6))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_6),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X))) ) ) ) ).
fof(fact_2368_real__root__le__1__iff,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),one_one(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),one_one(real))) ) ) ).
fof(fact_2369_real__root__ge__1__iff,axiom,
! [Y_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y_1)) ) ) ).
fof(fact_2370_real__root__strict__decreasing,axiom,
! [X,N_6,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_6))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_6),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X))) ) ) ) ).
fof(fact_2371_real__root__lt__1__iff,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),one_one(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),one_one(real))) ) ) ).
fof(fact_2372_real__root__gt__1__iff,axiom,
! [Y_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Y_1)) ) ) ).
fof(fact_2373_real__root__pos,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,suc,N_1)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(nat,nat,suc,N_1))) = X ) ).
fof(fact_2374_real__root__less__mono__lemma,axiom,
! [Y,X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),Y))) ) ) ) ).
fof(fact_2375_real__root__pos__pos,axiom,
! [X_5,N_7] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_7))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_5))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_7),X_5))) ) ) ).
fof(fact_2376_real__root__mult__lemma,axiom,
! [Y,X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),Y)) ) ) ) ).
fof(fact_2377_real__root__pos__mult__exp,axiom,
! [X,N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)),X) = hAPP(real,real,hAPP(nat,fun(real,real),root,M_1),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)) ) ) ) ).
fof(fact_2378_real__root__pos__unique,axiom,
! [X,Y,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
=> ( hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),Y),N_1) = X
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X) = Y ) ) ) ).
fof(fact_2379_real__root__pow__pos2,axiom,
! [X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),N_1) = X ) ) ).
fof(fact_2380_real__root__pos2,axiom,
! [X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N_1)) = X ) ) ).
fof(fact_2381_real__root__pow__pos,axiom,
! [X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),N_1) = X ) ) ).
fof(fact_2382_real__root__increasing,axiom,
! [X,N_6,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_6))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_6),X))) ) ) ) ) ).
fof(fact_2383_real__root__strict__increasing,axiom,
! [X,N_6,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),N_6))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)),hAPP(real,real,hAPP(nat,fun(real,real),root,N_6),X))) ) ) ) ) ).
fof(fact_2384_sqrt__def,axiom,
sqrt = hAPP(nat,fun(real,real),root,hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_2385_real__root__inverse__lemma,axiom,
! [X,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N_1),X)) ) ) ).
fof(fact_2386_DERIV__odd__real__root,axiom,
! [X_1,N] :
( ~ hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
=> ( X_1 != zero_zero(real)
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),root,N)),X_1),hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))))) ) ) ).
fof(fact_2387_negDivAlg_Opsimps,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),negDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Aa),Ba)))
=> ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aa),Ba)))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),zero_zero(int))) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,Aa),Ba) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),min)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aa),Ba)) )
& ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aa),Ba)))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),zero_zero(int))) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,Aa),Ba) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,Ba),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),negDivAlg,Aa),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)))),Ba))) ) ) ) ).
fof(fact_2388_aux4,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(real,real),plus_plus(real),one_one(real)),X))) ) ) ).
fof(fact_2389_MVT2,axiom,
! [F,F_2,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),Ba))
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Ba)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),hAPP(real,real,F_2,X_2))) )
=> ? [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),Z_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Z_2),Ba))
& hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,F,Ba)),hAPP(real,real,F,Aa)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Ba),Aa)),hAPP(real,real,F_2,Z_2)) ) ) ) ).
fof(fact_2390_polar__ex1,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> ? [R_2,A_3] :
( X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_2),hAPP(real,real,cos,A_3))
& Y = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_2),hAPP(real,real,sin,A_3)) ) ) ).
fof(fact_2391_exp__less__mono,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),X)),hAPP(real,real,exp(real),Y))) ) ).
fof(fact_2392_exp__le__cancel__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),X_1)),hAPP(real,real,exp(real),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1)) ) ).
fof(fact_2393_exp__less__cancel__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),X_1)),hAPP(real,real,exp(real),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1)) ) ).
fof(fact_2394_exp__less__cancel,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),X)),hAPP(real,real,exp(real),Y)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y)) ) ).
fof(fact_2395_exp__inj__iff,axiom,
! [X_1,Y_1] :
( hAPP(real,real,exp(real),X_1) = hAPP(real,real,exp(real),Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_2396_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_2397_ln__unique,axiom,
! [Y,X] :
( hAPP(real,real,exp(real),Y) = X
=> hAPP(real,real,ln,X) = Y ) ).
fof(fact_2398_exp__ln__eq,axiom,
! [U,X] :
( hAPP(real,real,exp(real),U) = X
=> hAPP(real,real,ln,X) = U ) ).
fof(fact_2399_ln__exp,axiom,
! [X] : hAPP(real,real,ln,hAPP(real,real,exp(real),X)) = X ).
fof(fact_2400_exp__not__eq__zero,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X] : hAPP(X_a,X_a,exp(X_a),X) != zero_zero(X_a) ) ).
fof(fact_2401_DERIV__exp,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),exp(X_a)),X_1),hAPP(X_a,X_a,exp(X_a),X_1))) ) ).
fof(fact_2402_not__exp__le__zero,axiom,
! [X] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),X)),zero_zero(real))) ).
fof(fact_2403_exp__ge__zero,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,exp(real),X))) ).
fof(fact_2404_exp__gt__zero,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,exp(real),X))) ).
fof(fact_2405_not__exp__less__zero,axiom,
! [X] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),X)),zero_zero(real))) ).
fof(fact_2406_exp__eq__one__iff,axiom,
! [X_1] :
( hAPP(real,real,exp(real),X_1) = one_one(real)
<=> X_1 = zero_zero(real) ) ).
fof(fact_2407_exp__zero,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> hAPP(X_a,X_a,exp(X_a),zero_zero(X_a)) = one_one(X_a) ) ).
fof(fact_2408_exp__add,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X,Y] : hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,exp(X_a),X)),hAPP(X_a,X_a,exp(X_a),Y)) ) ).
fof(fact_2409_mult__exp__exp,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,exp(X_a),X)),hAPP(X_a,X_a,exp(X_a),Y)) = hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) ) ).
fof(fact_2410_exp__diff,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X,Y] : hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,exp(X_a),X)),hAPP(X_a,X_a,exp(X_a),Y)) ) ).
fof(fact_2411_exp__minus,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X] : hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,exp(X_a),X)) ) ).
fof(fact_2412_one__le__exp__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,exp(real),X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1)) ) ).
fof(fact_2413_exp__le__one__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),X_1)),one_one(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),zero_zero(real))) ) ).
fof(fact_2414_exp__gt__one,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,exp(real),X))) ) ).
fof(fact_2415_exp__less__one__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,exp(real),X_1)),one_one(real)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),zero_zero(real))) ) ).
fof(fact_2416_one__less__exp__iff,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),hAPP(real,real,exp(real),X_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1)) ) ).
fof(fact_2417_exp__ge__add__one__self,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)),hAPP(real,real,exp(real),X))) ).
fof(fact_2418_exp__real__of__nat__mult,axiom,
! [N_1,X] : hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),X)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,exp(real),X)),N_1) ).
fof(fact_2419_exp__ln__iff,axiom,
! [X_1] :
( hAPP(real,real,exp(real),hAPP(real,real,ln,X_1)) = X_1
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1)) ) ).
fof(fact_2420_exp__ln,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,exp(real),hAPP(real,real,ln,X)) = X ) ).
fof(fact_2421_exp__ge__add__one__self__aux,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)),hAPP(real,real,exp(real),X))) ) ).
fof(fact_2422_ln__x__over__x__mono,axiom,
! [Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),one_one(real))),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,Y)),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,X)),X))) ) ) ).
fof(fact_2423_exp__bound,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,exp(real),X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ) ).
fof(fact_2424_negDivAlg_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),negDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
=> ( ! [A_3,B_3] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),negDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_3),B_3)))
=> ( ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),A_3),B_3)))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_3),zero_zero(int))) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,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_3))) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_3),B_3)) ) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1)) ) ) ).
fof(fact_2425_posDivAlg_Opsimps,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),posDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Aa),Ba)))
=> ( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),Ba))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),zero_zero(int))) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,Aa),Ba) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),Aa) )
& ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),Ba))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),zero_zero(int))) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,Aa),Ba) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,Ba),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),posDivAlg,Aa),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)))),Ba))) ) ) ) ).
fof(fact_2426_lemma__exp__total,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y))
=> ? [X_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Y),one_one(real))))
& hAPP(real,real,exp(real),X_2) = Y ) ) ).
fof(fact_2427_posDivAlg_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),posDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
=> ( ! [A_3,B_3] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),posDivAlg_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_3),B_3)))
=> ( ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_3),B_3))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),B_3),zero_zero(int))) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,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_3))) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_3),B_3)) ) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1)) ) ) ).
fof(fact_2428_log__base__10__eq1,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,hAPP(real,real,exp(real),one_one(real)))),hAPP(real,real,ln,hAPP(int,real,number_number_of(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_2429_log__base__10__eq2,axiom,
! [X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(int,real,number_number_of(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_2430_exp__total,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> ? [X_2] : hAPP(real,real,exp(real),X_2) = Y ) ).
fof(fact_2431_polar__ex2,axiom,
! [X,Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),zero_zero(real)))
=> ? [R_2,A_3] :
( X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_2),hAPP(real,real,cos,A_3))
& Y = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_2),hAPP(real,real,sin,A_3)) ) ) ).
fof(fact_2432_log__one,axiom,
! [A_1] : hAPP(real,real,hAPP(real,fun(real,real),log,A_1),one_one(real)) = zero_zero(real) ).
fof(fact_2433_Log_Olog__def,axiom,
! [A_1,X] : hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,X)),hAPP(real,real,ln,A_1)) ).
fof(fact_2434_log__less__cancel__iff,axiom,
! [Y_1,X_1,Aa] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Aa))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),log,Aa),X_1)),hAPP(real,real,hAPP(real,fun(real,real),log,Aa),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1)) ) ) ) ) ).
fof(fact_2435_log__eq__one,axiom,
! [A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ( A_1 != one_one(real)
=> hAPP(real,real,hAPP(real,fun(real,real),log,A_1),A_1) = one_one(real) ) ) ).
fof(fact_2436_log__ln,axiom,
! [X] : hAPP(real,real,ln,X) = hAPP(real,real,hAPP(real,fun(real,real),log,hAPP(real,real,exp(real),one_one(real))),X) ).
fof(fact_2437_log__le__cancel__iff,axiom,
! [Y_1,X_1,Aa] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),Aa))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),log,Aa),X_1)),hAPP(real,real,hAPP(real,fun(real,real),log,Aa),Y_1)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1)) ) ) ) ) ).
fof(fact_2438_log__nat__power,axiom,
! [B,N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(real,fun(real,real),log,B),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N_1)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(real,real,hAPP(real,fun(real,real),log,B),X)) ) ).
fof(fact_2439_log__mult,axiom,
! [Y,X,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ( A_1 != one_one(real)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),Y)) ) ) ) ) ).
fof(fact_2440_log__divide,axiom,
! [Y,X,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ( A_1 != one_one(real)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),Y)) ) ) ) ) ).
fof(fact_2441_log__inverse,axiom,
! [X,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ( A_1 != one_one(real)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,inverse_inverse(real),X)) = hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)) ) ) ) ).
fof(fact_2442_log__eq__div__ln__mult__log,axiom,
! [X,B,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ( A_1 != one_one(real)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),B))
=> ( B != one_one(real)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,ln,B)),hAPP(real,real,ln,A_1))),hAPP(real,real,hAPP(real,fun(real,real),log,B),X)) ) ) ) ) ) ).
fof(fact_2443_two__times__odd__div__two__plus__one,axiom,
! [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_2444_DERIV__isconst__all,axiom,
! [X_1,Y_1,F] :
( ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),zero_zero(real)))
=> hAPP(real,real,F,X_1) = hAPP(real,real,F,Y_1) ) ).
fof(fact_2445_Code__Numeral_Oint__of__code,axiom,
! [K_4] :
( ( K_4 = zero_zero(code_code_numeral)
=> hAPP(code_code_numeral,int,code_int_of,K_4) = zero_zero(int) )
& ( K_4 != zero_zero(code_code_numeral)
=> ( ( hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(code_code_numeral)
=> hAPP(code_code_numeral,int,code_int_of,K_4) = 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(code_code_numeral,int,code_int_of,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) )
& ( hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(code_code_numeral)
=> hAPP(code_code_numeral,int,code_int_of,K_4) = 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(code_code_numeral,int,code_int_of,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),K_4),hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),one_one(int)) ) ) ) ) ).
fof(fact_2446_inv__inv,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_4))
=> hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_1)) = A_1 ) ) ) ) ).
fof(fact_2447_even__neg,axiom,
! [X_1] :
( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,uminus_uminus(int),X_1)))
<=> hBOOL(hAPP(int,bool,even_odd_even(int),X_1)) ) ).
fof(fact_2448_code__numeral__zero__minus__one,axiom,
hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),minus_minus(code_code_numeral),zero_zero(code_code_numeral)),one_one(code_code_numeral)) = zero_zero(code_code_numeral) ).
fof(fact_2449_even__difference,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),Y_1)))
<=> ( ( hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
& hBOOL(hAPP(int,bool,even_odd_even(int),Y_1)) )
| ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
& ~ hBOOL(hAPP(int,bool,even_odd_even(int),Y_1)) ) ) ) ).
fof(fact_2450_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_2451_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_2452_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_2453_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_2454_even__sum,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),Y_1)))
<=> ( ( hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
& hBOOL(hAPP(int,bool,even_odd_even(int),Y_1)) )
| ( ~ hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
& ~ hBOOL(hAPP(int,bool,even_odd_even(int),Y_1)) ) ) ) ).
fof(fact_2455_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_2456_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_2457_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_2458_Parity_Oeven__product,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X_1),Y_1)))
<=> ( hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
| hBOOL(hAPP(int,bool,even_odd_even(int),Y_1)) ) ) ).
fof(fact_2459_odd__one__int,axiom,
~ hBOOL(hAPP(int,bool,even_odd_even(int),one_one(int))) ).
fof(fact_2460_even__zero__int,axiom,
hBOOL(hAPP(int,bool,even_odd_even(int),zero_zero(int))) ).
fof(fact_2461_odd__pow,axiom,
! [N_1,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_1))) ) ).
fof(fact_2462_even__power,axiom,
! [X_1,N] :
( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X_1),N)))
<=> ( hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
& N != zero_zero(nat) ) ) ).
fof(fact_2463_zero__code__numeral__code,axiom,
zero_zero(code_code_numeral) = hAPP(int,code_code_numeral,number_number_of(code_code_numeral),pls) ).
fof(fact_2464_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_2465_inv__less__p__minus__1,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_1)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))) ) ) ) ).
fof(fact_2466_inv__g__1,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_1))) ) ) ) ).
fof(fact_2467_inv__not__p__minus__1,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
=> hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_1) != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)) ) ) ) ).
fof(fact_2468_inv__not__1,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
=> hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_1) != one_one(int) ) ) ) ).
fof(fact_2469_inv__distinct,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
=> A_1 != hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_1) ) ) ) ).
fof(fact_2470_pos__int__even__equiv__nat__even,axiom,
! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
<=> hBOOL(hAPP(nat,bool,even_odd_even(nat),hAPP(int,nat,nat_1,X_1))) ) ) ).
fof(fact_2471_prime__g__5,axiom,
! [P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( P_4 != hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))
=> ( P_4 != hAPP(int,int,number_number_of(int),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_4)) ) ) ) ).
fof(fact_2472_inv__not__0,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))
=> hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_1) != zero_zero(int) ) ) ) ).
fof(fact_2473_even__equiv__def,axiom,
! [X_1] :
( hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
<=> ? [Y_2] : X_1 = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Y_2) ) ).
fof(fact_2474_zprime__zOdd__eq__grt__2,axiom,
! [Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Pa),zOdd))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa)) ) ) ).
fof(fact_2475_even__def,axiom,
! [X_1] :
( hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
<=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X_1),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(int) ) ).
fof(fact_2476_two__times__even__div__two,axiom,
! [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_2477_one__code__numeral__code,axiom,
one_one(code_code_numeral) = hAPP(int,code_code_numeral,number_number_of(code_code_numeral),hAPP(int,int,bit1,pls)) ).
fof(fact_2478_odd__equiv__def,axiom,
! [X_1] :
( ~ hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
<=> ? [Y_2] : X_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Y_2)),one_one(int)) ) ).
fof(fact_2479_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_2480_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_2481_div__mod__code__numeral__def,axiom,
! [N_1,M_1] : hAPP(code_code_numeral,product_prod(code_code_numeral,code_code_numeral),hAPP(code_code_numeral,fun(code_code_numeral,product_prod(code_code_numeral,code_code_numeral)),code_d418564891umeral,N_1),M_1) = hAPP(code_code_numeral,product_prod(code_code_numeral,code_code_numeral),hAPP(code_code_numeral,fun(code_code_numeral,product_prod(code_code_numeral,code_code_numeral)),product_Pair(code_code_numeral,code_code_numeral),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_div(code_code_numeral),N_1),M_1)),hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),div_mod(code_code_numeral),N_1),M_1)) ).
fof(fact_2482_wset__mem__inv__mem,axiom,
! [Ba,Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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))))),Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa)))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),inv,Pa),Ba)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa))) ) ) ) ) ).
fof(fact_2483_wset__inv__mem__mem,axiom,
! [Ba,Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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))))),Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ba))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ba),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),inv,Pa),Ba)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa)))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa))) ) ) ) ) ) ) ).
fof(fact_2484_Euler__part2,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,zero_zero(int)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_4)) ) ) ) ).
fof(fact_2485_zcong__id,axiom,
! [M_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,M_1),zero_zero(int)),M_1)) ).
fof(fact_2486_IntPrimes_Ozcong__zero,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),zero_zero(int)))
<=> Aa = Ba ) ).
fof(fact_2487_zcong__zmod,axiom,
! [Aa,Ba,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Aa),M)),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Ba),M)),M)) ) ).
fof(fact_2488_zcong__1,axiom,
! [A_1,B] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),one_one(int))) ).
fof(fact_2489_zcong__zmult,axiom,
! [C,D,A_1,B,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),D),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,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),D)),M_1)) ) ) ).
fof(fact_2490_zcong__scalar2,axiom,
! [K_4,A_1,B,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),A_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),B)),M_1)) ) ).
fof(fact_2491_zcong__scalar,axiom,
! [K_4,A_1,B,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),K_4)),M_1)) ) ).
fof(fact_2492_zcong__zmult__prop2,axiom,
! [C_1,D_1,Aa,Ba,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_1),Aa)),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),D_1),Ba)),M)) ) ) ).
fof(fact_2493_zcong__zmult__prop1,axiom,
! [C_1,D_1,Aa,Ba,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Aa),D_1)),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ba),D_1)),M)) ) ) ).
fof(fact_2494_zcong__zmult__self,axiom,
! [A_1,M_1,B] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),M_1)),M_1)) ).
fof(fact_2495_zcong__zadd,axiom,
! [C,D,A_1,B,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),D),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,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),D)),M_1)) ) ) ).
fof(fact_2496_zcong__shift,axiom,
! [C,A_1,B,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,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),C)),M_1)) ) ).
fof(fact_2497_zcong__zdiff,axiom,
! [C,D,A_1,B,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),D),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),A_1),C)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),B),D)),M_1)) ) ) ).
fof(fact_2498_zcong__zminus,axiom,
! [Aa,Ba,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),hAPP(int,int,uminus_uminus(int),M)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M)) ) ).
fof(fact_2499_zcong__zpower,axiom,
! [Z,X,Y,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Z)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y),Z)),M_1)) ) ).
fof(fact_2500_zcong__sym,axiom,
! [Aa,Ba,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Ba),Aa),M)) ) ).
fof(fact_2501_zcong__refl,axiom,
! [K_4,M_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),K_4),M_1)) ).
fof(fact_2502_zcong__eq__trans,axiom,
! [D,C,A_1,B,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> ( B = C
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,C),D),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),D),M_1)) ) ) ) ).
fof(fact_2503_zcong__trans,axiom,
! [C,A_1,B,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B),C),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),C),M_1)) ) ) ).
fof(fact_2504_zcong__not,axiom,
! [B,M_1,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),A_1))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1)) ) ) ) ) ).
fof(fact_2505_zcong__not__zero,axiom,
! [M_1,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M_1))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),M_1)) ) ) ).
fof(fact_2506_zcong__less__eq,axiom,
! [M_1,Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y),M_1))
=> X = Y ) ) ) ) ) ) ).
fof(fact_2507_zcong__iff__lin,axiom,
! [Aa,Ba,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M))
<=> ? [K_1] : Ba = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),K_1)) ) ).
fof(fact_2508_Int2_Ozcong__zero,axiom,
! [M_1,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),M_1))
=> X = zero_zero(int) ) ) ) ).
fof(fact_2509_zcong__zless__0,axiom,
! [M_1,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),M_1))
=> A_1 = zero_zero(int) ) ) ) ).
fof(fact_2510_zcong__zless__imp__eq,axiom,
! [B,M_1,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> A_1 = B ) ) ) ) ) ).
fof(fact_2511_zcong__zmod__eq,axiom,
! [Aa,Ba,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M))
<=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Aa),M) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Ba),M) ) ) ).
fof(fact_2512_inv__not__p__minus__1__aux,axiom,
! [Aa,Pa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int)))),one_one(int)),Pa))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))),Pa)) ) ).
fof(fact_2513_zcong__zpower__zmult,axiom,
! [Z,X,Y,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),Y)),one_one(int)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,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))),one_one(int)),P_4)) ) ).
fof(fact_2514_wset__mem__mem,axiom,
! [Pa,Aa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Aa))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa))) ) ).
fof(fact_2515_zcong__zmult__prop3,axiom,
! [Y,X,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Y),zero_zero(int)),P_4))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)),zero_zero(int)),P_4)) ) ) ) ).
fof(fact_2516_zcong__zprime__prod__zero__contra,axiom,
! [B,A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_4))
& ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B),zero_zero(int)),P_4)) )
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),zero_zero(int)),P_4)) ) ) ) ).
fof(fact_2517_zcong__zprime__prod__zero,axiom,
! [B,A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),zero_zero(int)),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_4))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,B),zero_zero(int)),P_4)) ) ) ) ) ).
fof(fact_2518_wset__subset,axiom,
! [Ba,Pa,Aa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Aa))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Aa),one_one(int))),Pa)))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa))) ) ) ).
fof(fact_2519_zcong__square,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),A_1)),one_one(int)),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),one_one(int)),P_4))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),P_4)) ) ) ) ) ).
fof(fact_2520_zcong__square__zless,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),A_1)),one_one(int)),P_4))
=> ( A_1 = one_one(int)
| A_1 = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)) ) ) ) ) ) ).
fof(fact_2521_wset__g__1,axiom,
! [Ba,Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ba)) ) ) ) ).
fof(fact_2522_wset__less,axiom,
! [Ba,Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ba),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int)))) ) ) ) ).
fof(fact_2523_inv__is__inv,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),A_1),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),inv,P_4),A_1))),one_one(int)),P_4)) ) ) ) ).
fof(fact_2524_wset__mem__imp__or,axiom,
! [Ba,Pa,Aa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Aa))
=> ( ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Aa),one_one(int))),Pa)))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa)))
=> ( Ba = Aa
| Ba = hAPP(int,int,hAPP(int,fun(int,int),inv,Pa),Aa) ) ) ) ) ).
fof(fact_2525_Wilson__Russ,axiom,
! [P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))),hAPP(int,int,number_number_of(int),min)),P_4)) ) ).
fof(fact_2526_one__not__neg__one__mod__m,axiom,
! [M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M_1))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,one_one(int)),hAPP(int,int,number_number_of(int),min)),M_1)) ) ).
fof(fact_2527_zcong__neg__1__impl__ne__1,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),hAPP(int,int,number_number_of(int),min)),P_4))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),one_one(int)),P_4)) ) ) ).
fof(fact_2528_neg__one__power__eq__mod__m,axiom,
! [J_2,K_4,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,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_4)),M_1))
=> 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_4) ) ) ).
fof(fact_2529_wset__mem,axiom,
! [Ba,Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Ba))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),Aa))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa))) ) ) ) ) ).
fof(fact_2530_d22set__eq__wset,axiom,
! [Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> hAPP(int,fun(int,bool),d22set,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),Pa) ) ).
fof(fact_2531_Euler__Criterion,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_4)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_4)) ) ) ).
fof(fact_2532_zfact__prop,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),P_4)) ) ) ) ) ).
fof(fact_2533_Euler__part1,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),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_4)) ) ) ) ) ).
fof(fact_2534_Euler__part3,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),one_one(int)),P_4)) ) ) ) ) ).
fof(fact_2535_Legendre__def,axiom,
! [A_1,P_4] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_4))
=> hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_4) = zero_zero(int) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_4))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_1))
=> hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_4) = one_one(int) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_1))
=> hAPP(int,int,hAPP(int,fun(int,int),legendre,A_1),P_4) = hAPP(int,int,number_number_of(int),min) ) ) ) ) ).
fof(fact_2536_QuadRes__def,axiom,
! [M,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,M),X_1))
<=> ? [Y_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y_2),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X_1),M)) ) ).
fof(fact_2537_zcong__zless__unique,axiom,
! [A_1,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M_1))
=> ? [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),M_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),X_2),M_1))
& ! [Y_2] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_2))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_2),M_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),Y_2),M_1)) )
=> Y_2 = X_2 ) ) ) ).
fof(fact_2538_nat__of__aux__code,axiom,
! [N_1,I_1] :
( ( I_1 = zero_zero(code_code_numeral)
=> hAPP(nat,nat,hAPP(code_code_numeral,fun(nat,nat),code_nat_of_aux,I_1),N_1) = N_1 )
& ( I_1 != zero_zero(code_code_numeral)
=> hAPP(nat,nat,hAPP(code_code_numeral,fun(nat,nat),code_nat_of_aux,I_1),N_1) = hAPP(nat,nat,hAPP(code_code_numeral,fun(nat,nat),code_nat_of_aux,hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),minus_minus(code_code_numeral),I_1),one_one(code_code_numeral))),hAPP(nat,nat,suc,N_1)) ) ) ).
fof(fact_2539_mod__mod__is__mod,axiom,
! [X,M_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M_1)),M_1)) ).
fof(fact_2540_Residues_Oaux,axiom,
! [X,M_1,Y] :
( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Y),M_1)
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),M_1)) ) ).
fof(fact_2541_MultInvPair__distinct,axiom,
! [J_2,A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,P_4),A_1))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),J_2))),P_4)) ) ) ) ) ) ).
fof(fact_2542_MultInv__prop2a,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),X)),X)),one_one(int)),P_4)) ) ) ) ).
fof(fact_2543_MultInv__prop2,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),X))),one_one(int)),P_4)) ) ) ) ).
fof(fact_2544_phi__prime,axiom,
! [P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> hAPP(int,nat,phi,P_4) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))) ) ).
fof(fact_2545_aux______1,axiom,
! [J_2,A_1,P_4,K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),K_4))),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),K_4))),K_4)),P_4)) ) ).
fof(fact_2546_aux______3,axiom,
! [J_2,K_4,A_1,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),A_1),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,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,hAPP(int,fun(int,int),multInv,P_4),J_2)),J_2)),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),J_2)),A_1)),P_4)) ) ).
fof(fact_2547_MultInv__prop1,axiom,
! [X,Y,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),X)),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),Y)),P_4)) ) ) ).
fof(fact_2548_MultInv__zcong__prop1,axiom,
! [A_1,J_2,K_4,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),K_4),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),J_2))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),K_4))),P_4)) ) ) ).
fof(fact_2549_MultInv__def,axiom,
! [P_4,X] : hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),X) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_2550_MultInv__prop5,axiom,
! [Y,X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Y),zero_zero(int)),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),X)),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),Y)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),Y),P_4)) ) ) ) ) ) ).
fof(fact_2551_MultInv__prop4,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),X))),X),P_4)) ) ) ) ).
fof(fact_2552_MultInv__prop3,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),X)),zero_zero(int)),P_4)) ) ) ) ).
fof(fact_2553_Int2_Oaux____1,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),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,hAPP(int,fun(int,int),multInv,P_4),X))),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),X)))),P_4)) ) ) ) ).
fof(fact_2554_Int2_Oaux____2,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,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,hAPP(int,fun(int,int),multInv,P_4),X))),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),X)))),X),P_4)) ) ) ) ).
fof(fact_2555_MultInv__zcong__prop3,axiom,
! [J_2,K_4,A_1,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),zero_zero(int)),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),zero_zero(int)),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),J_2))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),K_4))),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),K_4),P_4)) ) ) ) ) ) ) ).
fof(fact_2556_MultInv__zcong__prop2,axiom,
! [A_1,J_2,K_4,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),zero_zero(int)),P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),K_4))),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),J_2))),P_4)) ) ) ) ) ) ).
fof(fact_2557_aux______2,axiom,
! [J_2,A_1,K_4,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),zero_zero(int)),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),K_4))),K_4)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),J_2),K_4)),A_1),P_4)) ) ) ) ) ).
fof(fact_2558_aux______4,axiom,
! [K_4,A_1,J_2,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J_2),zero_zero(int)),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,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,hAPP(int,fun(int,int),multInv,P_4),J_2)),J_2)),K_4)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),J_2)),A_1)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),multInv,P_4),J_2))),P_4)) ) ) ) ) ).
fof(fact_2559_SR__def,axiom,
! [Pa] : hAPP(int,fun(int,bool),sr,Pa) = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Pa))) ).
fof(fact_2560_StandardRes__prop4,axiom,
! [X,Y,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,M_1),X)),hAPP(int,int,hAPP(int,fun(int,int),standardRes,M_1),Y))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)),M_1)) ) ).
fof(fact_2561_of__nat__double,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_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)))),X))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,X))),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,X))) ) ) ).
fof(fact_2562_complex__inverse,axiom,
! [A_1,B] : hAPP(complex,complex,inverse_inverse(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),A_1),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),B)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),A_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),B),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).
fof(fact_2563_negative__zle,axiom,
! [N_1,M_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N_1))),hAPP(nat,int,semiring_1_of_nat(int),M_1))) ).
fof(fact_2564_negative__zless,axiom,
! [N_1,M_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1)))),hAPP(nat,int,semiring_1_of_nat(int),M_1))) ).
fof(fact_2565_complex__zero__def,axiom,
zero_zero(complex) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),zero_zero(real)) ).
fof(fact_2566_Complex__eq__0,axiom,
! [Aa,Ba] :
( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Aa),Ba) = zero_zero(complex)
<=> ( Aa = zero_zero(real)
& Ba = zero_zero(real) ) ) ).
fof(fact_2567_number__of__int,axiom,
! [X_a] :
( number_semiring(X_a)
=> ! [N_1] : hAPP(int,X_a,number_number_of(X_a),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1) ) ).
fof(fact_2568_complex__divide__def,axiom,
! [X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),X),Y) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),hAPP(complex,complex,inverse_inverse(complex),Y)) ).
fof(fact_2569_abs__int__eq,axiom,
! [M_1] : hAPP(int,int,abs_abs(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)) = hAPP(nat,int,semiring_1_of_nat(int),M_1) ).
fof(fact_2570_real__of__nat__def,axiom,
real_1(nat) = semiring_1_of_nat(real) ).
fof(fact_2571_real__eq__of__nat,axiom,
real_1(nat) = semiring_1_of_nat(real) ).
fof(fact_2572_abs__of__nat,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [N_1] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1) ) ).
fof(fact_2573_nat__int,axiom,
! [N_1] : hAPP(int,nat,nat_1,hAPP(nat,int,semiring_1_of_nat(int),N_1)) = N_1 ).
fof(fact_2574_ceiling__of__nat,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [N_1] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),N_1) ) ).
fof(fact_2575_of__nat__eq__iff,axiom,
! [X_a] :
( semiring_char_0(X_a)
=> ! [M,N] :
( hAPP(nat,X_a,semiring_1_of_nat(X_a),M) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N)
<=> M = N ) ) ).
fof(fact_2576_floor__of__nat,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [N_1] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),N_1) ) ).
fof(fact_2577_complex__add,axiom,
! [A_1,B,C,D] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,C),D)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),C)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),B),D)) ).
fof(fact_2578_complex_Oinject,axiom,
! [Real1,Real2,Real1_2,Real2_2] :
( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1),Real2) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1_2),Real2_2)
<=> ( Real1 = Real1_2
& Real2 = Real2_2 ) ) ).
fof(fact_2579_complex__minus,axiom,
! [A_1,B] : hAPP(complex,complex,uminus_uminus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),A_1)),hAPP(real,real,uminus_uminus(real),B)) ).
fof(fact_2580_Ints__of__nat,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [N] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),ring_1_Ints(X_a))) ) ).
fof(fact_2581_not__neg__int,axiom,
! [N_1] : ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(nat,int,semiring_1_of_nat(int),N_1))) ).
fof(fact_2582_complex__diff,axiom,
! [A_1,B,C,D] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,C),D)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),C)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B),D)) ).
fof(fact_2583_StandardRes__SR__prop,axiom,
! [X_1,Pa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),sr,Pa)))
=> hAPP(int,int,hAPP(int,fun(int,int),standardRes,Pa),X_1) = X_1 ) ).
fof(fact_2584_StandardRes__prop1,axiom,
! [X,M_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),hAPP(int,int,hAPP(int,fun(int,int),standardRes,M_1),X)),M_1)) ).
fof(fact_2585_StandardRes__def,axiom,
! [M_1,X] : hAPP(int,int,hAPP(int,fun(int,int),standardRes,M_1),X) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X),M_1) ).
fof(fact_2586_zero__le__imp__of__nat,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1))) ) ).
fof(fact_2587_of__nat__0__le__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1))) ) ).
fof(fact_2588_of__nat__less__0__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1] : ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),zero_zero(X_a))) ) ).
fof(fact_2589_of__nat__0,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),zero_zero(nat)) = zero_zero(X_a) ) ).
fof(fact_2590_of__nat__less__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
fof(fact_2591_less__imp__of__nat__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1))) ) ) ).
fof(fact_2592_of__nat__less__imp__less,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M_1,N_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ) ).
fof(fact_2593_of__nat__le__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [M,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
fof(fact_2594_of__nat__add,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M_1,N_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) ) ).
fof(fact_2595_of__nat__mult,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M_1,N_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) ) ).
fof(fact_2596_of__nat__1,axiom,
! [X_a] :
( semiring_1(X_a)
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),one_one(nat)) = one_one(X_a) ) ).
fof(fact_2597_of__nat__fact__not__zero,axiom,
! [X_a] :
( semiring_char_0(X_a)
=> ! [N_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N_1)) != zero_zero(X_a) ) ).
fof(fact_2598_int__eq__0__conv,axiom,
! [N] :
( hAPP(nat,int,semiring_1_of_nat(int),N) = zero_zero(int)
<=> N = zero_zero(nat) ) ).
fof(fact_2599_int__0,axiom,
hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).
fof(fact_2600_transfer__int__nat__numerals_I1_J,axiom,
zero_zero(int) = hAPP(nat,int,semiring_1_of_nat(int),zero_zero(nat)) ).
fof(fact_2601_of__nat__power,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M_1,N_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M_1),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),N_1) ) ).
fof(fact_2602_int__less__0__conv,axiom,
! [K_4] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),K_4)),zero_zero(int))) ).
fof(fact_2603_zero__zle__int,axiom,
! [N_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),N_1))) ).
fof(fact_2604_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),Z))) ).
fof(fact_2605_transfer__int__nat__quantifiers_I2_J,axiom,
! [P_1] :
( ? [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
& hBOOL(hAPP(int,bool,P_1,X_2)) )
<=> ? [X_2] : hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_2))) ) ).
fof(fact_2606_transfer__int__nat__quantifiers_I1_J,axiom,
! [P_1] :
( ! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
=> hBOOL(hAPP(int,bool,P_1,X_2)) )
<=> ! [X_2] : hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),X_2))) ) ).
fof(fact_2607_zless__int,axiom,
! [M,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
fof(fact_2608_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)),hAPP(nat,int,semiring_1_of_nat(int),Y_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_1)) ) ).
fof(fact_2609_zle__int,axiom,
! [M,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
fof(fact_2610_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)),hAPP(nat,int,semiring_1_of_nat(int),Y_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_1),Y_1)) ) ).
fof(fact_2611_zmult__int,axiom,
! [M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)) ).
fof(fact_2612_int__mult,axiom,
! [M_1,N_1] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) ).
fof(fact_2613_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,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y)) ).
fof(fact_2614_zle__iff__zadd,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z_1))
<=> ? [N_2] : Z_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_1),hAPP(nat,int,semiring_1_of_nat(int),N_2)) ) ).
fof(fact_2615_zadd__int__left,axiom,
! [M_1,N_1,Z] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),N_1)),Z)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1))),Z) ).
fof(fact_2616_zadd__int,axiom,
! [M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) ).
fof(fact_2617_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,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),X),Y)) ).
fof(fact_2618_int__1,axiom,
hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).
fof(fact_2619_transfer__int__nat__numerals_I2_J,axiom,
one_one(int) = hAPP(nat,int,semiring_1_of_nat(int),one_one(nat)) ).
fof(fact_2620_negative__eq__positive,axiom,
! [N,M] :
( hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N)) = hAPP(nat,int,semiring_1_of_nat(int),M)
<=> ( N = zero_zero(nat)
& M = zero_zero(nat) ) ) ).
fof(fact_2621_not__int__zless__negative,axiom,
! [N_1,M_1] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),N_1)),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)))) ).
fof(fact_2622_Complex__eq__number__of,axiom,
! [Aa,Ba,W_1] :
( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Aa),Ba) = hAPP(int,complex,number_number_of(complex),W_1)
<=> ( Aa = hAPP(int,real,number_number_of(real),W_1)
& Ba = zero_zero(real) ) ) ).
fof(fact_2623_Complex__eq__1,axiom,
! [Aa,Ba] :
( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Aa),Ba) = one_one(complex)
<=> ( Aa = one_one(real)
& Ba = zero_zero(real) ) ) ).
fof(fact_2624_complex__one__def,axiom,
one_one(complex) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,one_one(real)),zero_zero(real)) ).
fof(fact_2625_real__of__int__of__nat__eq,axiom,
! [N_1] : hAPP(int,real,real_1(int),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,real,real_1(nat),N_1) ).
fof(fact_2626_Divides_Otransfer__int__nat__functions_I1_J,axiom,
! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),Y)) ).
fof(fact_2627_zdiv__int,axiom,
! [A_1,B] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(nat,int,semiring_1_of_nat(int),A_1)),hAPP(nat,int,semiring_1_of_nat(int),B)) ).
fof(fact_2628_Divides_Otransfer__int__nat__functions_I2_J,axiom,
! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) ).
fof(fact_2629_zmod__int,axiom,
! [A_1,B] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(nat,int,semiring_1_of_nat(int),A_1)),hAPP(nat,int,semiring_1_of_nat(int),B)) ).
fof(fact_2630_zpower__int,axiom,
! [M_1,N_1] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),N_1) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M_1),N_1)) ).
fof(fact_2631_int__power,axiom,
! [M_1,N_1] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M_1),N_1)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),N_1) ).
fof(fact_2632_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X,N_1] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(nat,int,semiring_1_of_nat(int),X)),N_1) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N_1)) ).
fof(fact_2633_not__neg__nat,axiom,
! [Z] :
( ~ hBOOL(hAPP(int,bool,nat_neg,Z))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z)) = Z ) ).
fof(fact_2634_floor__real__of__nat,axiom,
! [N_1] : hAPP(real,int,archim2085082626_floor(real),hAPP(nat,real,real_1(nat),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),N_1) ).
fof(fact_2635_Parity_Otransfer__int__nat__relations,axiom,
! [X_1] :
( hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)))
<=> hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1)) ) ).
fof(fact_2636_even__nat__def,axiom,
! [X_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
<=> hBOOL(hAPP(int,bool,even_odd_even(int),hAPP(nat,int,semiring_1_of_nat(int),X_1))) ) ).
fof(fact_2637_ceiling__real__of__nat,axiom,
! [N_1] : hAPP(real,int,archimedean_ceiling(real),hAPP(nat,real,real_1(nat),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),N_1) ).
fof(fact_2638_Nat__Transfer_Otransfer__int__nat__functions_I3_J,axiom,
! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),nat_tsub,hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),X),Y)) ).
fof(fact_2639_of__nat__Suc,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [M_1] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,suc,M_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)) ) ).
fof(fact_2640_StandardRes__ubound,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),X)),P_4)) ) ).
fof(fact_2641_of__nat__fact__ge__zero,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N_1)))) ) ).
fof(fact_2642_of__nat__fact__gt__zero,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N_1)))) ) ).
fof(fact_2643_of__nat__diff,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),M_1)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) ) ) ).
fof(fact_2644_StandardRes__prop3,axiom,
! [X_1,Pa] :
( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X_1),zero_zero(int)),Pa))
<=> hAPP(int,int,hAPP(int,fun(int,int),standardRes,Pa),X_1) != zero_zero(int) ) ).
fof(fact_2645_StandardRes__eq__zcong,axiom,
! [M,X_1] :
( hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),X_1) = zero_zero(int)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X_1),zero_zero(int)),M)) ) ).
fof(fact_2646_int__le__0__conv,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),N)),zero_zero(int)))
<=> N = zero_zero(nat) ) ).
fof(fact_2647_int__Suc0__eq__1,axiom,
hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,zero_zero(nat))) = one_one(int) ).
fof(fact_2648_zless__iff__Suc__zadd,axiom,
! [W_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_1))
<=> ? [N_2] : Z_1 = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W_1),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_2))) ) ).
fof(fact_2649_int__Suc,axiom,
! [M_1] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,M_1)) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),one_one(int)),hAPP(nat,int,semiring_1_of_nat(int),M_1)) ).
fof(fact_2650_nat__0__le,axiom,
! [Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z)) = Z ) ).
fof(fact_2651_int__eq__iff,axiom,
! [M,Z_1] :
( hAPP(nat,int,semiring_1_of_nat(int),M) = Z_1
<=> ( M = hAPP(int,nat,nat_1,Z_1)
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) ) ) ).
fof(fact_2652_int__nat__eq,axiom,
! [Z] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z)) = Z )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,Z)) = zero_zero(int) ) ) ).
fof(fact_2653_zless__nat__eq__int__zless,axiom,
! [M,Z_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),hAPP(int,nat,nat_1,Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(nat,int,semiring_1_of_nat(int),M)),Z_1)) ) ).
fof(fact_2654_int__zle__neg,axiom,
! [N,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(nat,int,semiring_1_of_nat(int),N)),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),M))))
<=> ( N = zero_zero(nat)
& M = zero_zero(nat) ) ) ).
fof(fact_2655_negative__zle__0,axiom,
! [N_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N_1))),zero_zero(int))) ).
fof(fact_2656_nat__zminus__int,axiom,
! [N_1] : hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N_1))) = zero_zero(nat) ).
fof(fact_2657_zdiff__int,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),M_1)),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)) ) ).
fof(fact_2658_complex__mult,axiom,
! [A_1,B,C,D] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,C),D)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),C)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),D))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),D)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B),C))) ).
fof(fact_2659_neg__zminus__int,axiom,
! [N_1] : hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1))))) ).
fof(fact_2660_of__nat__0__less__iff,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).
fof(fact_2661_inv__of__nat__fact__ge__zero,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N_1))))) ) ).
fof(fact_2662_inv__of__nat__fact__gt__zero,axiom,
! [X_a] :
( linordered_field(X_a)
=> ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,fact_fact(nat),N_1))))) ) ).
fof(fact_2663_zero__less__int__conv,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
fof(fact_2664_zmult__zless__mono2__lemma,axiom,
! [K_4,I_1,J_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I_1),J_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),K_4)),I_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),K_4)),J_2))) ) ) ).
fof(fact_2665_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,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,hAPP(int,int,bit1,pls)))) ).
fof(fact_2666_split__nat,axiom,
! [P_1,I] :
( hBOOL(hAPP(nat,bool,P_1,hAPP(int,nat,nat_1,I)))
<=> ( ! [N_2] :
( I = hAPP(nat,int,semiring_1_of_nat(int),N_2)
=> hBOOL(hAPP(nat,bool,P_1,N_2)) )
& ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
=> hBOOL(hAPP(nat,bool,P_1,zero_zero(nat))) ) ) ) ).
fof(fact_2667_nat__eq__iff,axiom,
! [W_1,M] :
( hAPP(int,nat,nat_1,W_1) = M
<=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W_1))
=> W_1 = hAPP(nat,int,semiring_1_of_nat(int),M) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W_1))
=> M = zero_zero(nat) ) ) ) ).
fof(fact_2668_nat__eq__iff2,axiom,
! [M,W_1] :
( M = hAPP(int,nat,nat_1,W_1)
<=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W_1))
=> W_1 = hAPP(nat,int,semiring_1_of_nat(int),M) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W_1))
=> M = zero_zero(nat) ) ) ) ).
fof(fact_2669_negative__zless__0,axiom,
! [N_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1)))),zero_zero(int))) ).
fof(fact_2670_not__zle__0__negative,axiom,
! [N_1] : ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_1))))) ).
fof(fact_2671_int__eq__iff__number__of,axiom,
! [M,V_1] :
( hAPP(nat,int,semiring_1_of_nat(int),M) = hAPP(int,int,number_number_of(int),V_1)
<=> ( M = hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),V_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V_1))) ) ) ).
fof(fact_2672_int__nat__number__of,axiom,
! [V] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),V)) = zero_zero(int) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,int,number_number_of(int),V) ) ) ).
fof(fact_2673_StandardRes__lbound,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),P_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),standardRes,P_4),X))) ) ).
fof(fact_2674_StandardRes__prop2,axiom,
! [X1_2,X2_2,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),X1_2) = hAPP(int,int,hAPP(int,fun(int,int),standardRes,M),X2_2)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X1_2),X2_2),M)) ) ) ).
fof(fact_2675_floor__minus__real__of__nat,axiom,
! [N_1] : hAPP(real,int,archim2085082626_floor(real),hAPP(real,real,uminus_uminus(real),hAPP(nat,real,real_1(nat),N_1))) = hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),N_1)) ).
fof(fact_2676_of__nat__number__of__eq,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V] :
( ( hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = zero_zero(X_a) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,X_a,number_number_of(X_a),V) ) ) ) ).
fof(fact_2677_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,semiring_1_of_nat(int),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_2678_nat__less__iff,axiom,
! [M,W_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,W_1)),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),hAPP(nat,int,semiring_1_of_nat(int),M))) ) ) ).
fof(fact_2679_zdiff__int__split,axiom,
! [P_1,X_1,Y_1] :
( hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),X_1),Y_1))))
<=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),Y_1),X_1))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)),hAPP(nat,int,semiring_1_of_nat(int),Y_1)))) )
& ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_1),Y_1))
=> hBOOL(hAPP(int,bool,P_1,zero_zero(int))) ) ) ) ).
fof(fact_2680_of__nat__less__two__power,axiom,
! [X_a] :
( ( number_ring(X_a)
& linordered_idom(X_a) )
=> ! [N_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,number_number_of(X_a),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1))) ) ).
fof(fact_2681_of__nat__number__of__lemma,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = hAPP(int,X_a,number_number_of(X_a),V) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),V)))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,number_number_of(nat),V)) = zero_zero(X_a) ) ) ) ).
fof(fact_2682_ex__inverse__of__nat__less,axiom,
! [X_a] :
( archim1804426504_field(X_a)
=> ! [X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
=> ? [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_2))),X)) ) ) ) ).
fof(fact_2683_ex__inverse__of__nat__Suc__less,axiom,
! [X_a] :
( archim1804426504_field(X_a)
=> ! [X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
=> ? [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,suc,N_2)))),X)) ) ) ).
fof(fact_2684_SRStar__mult__prop2,axiom,
! [X_1,Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),zero_zero(int)),Pa))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),sRStar,Pa)))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,Pa),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),multInv,Pa),X_1)))),hAPP(int,fun(int,bool),sRStar,Pa))) ) ) ) ) ).
fof(fact_2685_lemma__termdiff3,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [N_1,Z,K_5,H_2] :
( ti(X_a,H_2) != zero_zero(X_a)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),Z)),K_5))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),H_2))),K_5))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z),H_2)),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),N_1))),H_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(nat,nat,suc,zero_zero(nat)))))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,semiring_1_of_nat(real),N_1)),hAPP(nat,real,semiring_1_of_nat(real),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(nat,nat,suc,zero_zero(nat)))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),K_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(X_a,real,norm_norm(X_a),H_2)))) ) ) ) ) ).
fof(fact_2686_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [X_1,Y_1] :
( hAPP(nat,int,semiring_1_of_nat(int),X_1) = hAPP(nat,int,semiring_1_of_nat(int),Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_2687_int__if__cong,axiom,
! [X_1,Y_1,P_1] :
( ( hBOOL(P_1)
=> hAPP(nat,int,semiring_1_of_nat(int),X_1) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),P_1),X_1),Y_1)) )
& ( ~ hBOOL(P_1)
=> hAPP(nat,int,semiring_1_of_nat(int),Y_1) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),P_1),X_1),Y_1)) ) ) ).
fof(fact_2688_norm__of__nat,axiom,
! [X_a] :
( real_n2089651433ebra_1(X_a)
=> ! [N_1] : hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1)) = hAPP(nat,real,semiring_1_of_nat(real),N_1) ) ).
fof(fact_2689_int__int__eq,axiom,
! [M,N] :
( hAPP(nat,int,semiring_1_of_nat(int),M) = hAPP(nat,int,semiring_1_of_nat(int),N)
<=> M = N ) ).
fof(fact_2690_norm__minus__commute,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,B] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),A_1)) ) ).
fof(fact_2691_complex__diff__def,axiom,
! [X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),hAPP(complex,complex,uminus_uminus(complex),Y)) ).
fof(fact_2692_real__norm__def,axiom,
! [R_4] : hAPP(real,real,norm_norm(real),R_4) = hAPP(real,real,abs_abs(real),R_4) ).
fof(fact_2693_abs__norm__cancel,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1] : hAPP(real,real,abs_abs(real),hAPP(X_a,real,norm_norm(X_a),A_1)) = hAPP(X_a,real,norm_norm(X_a),A_1) ) ).
fof(fact_2694_norm__minus__cancel,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,real,norm_norm(X_a),X) ) ).
fof(fact_2695_norm__eq__zero,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1] :
( hAPP(X_a,real,norm_norm(X_a),X_1) = zero_zero(real)
<=> ti(X_a,X_1) = zero_zero(X_a) ) ) ).
fof(fact_2696_norm__zero,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> hAPP(X_a,real,norm_norm(X_a),zero_zero(X_a)) = zero_zero(real) ) ).
fof(fact_2697_StandardRes__SRStar__prop3,axiom,
! [X_1,Pa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),sRStar,Pa)))
=> hAPP(int,int,hAPP(int,fun(int,int),standardRes,Pa),X_1) = X_1 ) ).
fof(fact_2698_norm__mult,axiom,
! [X_a] :
( real_n1866405975lgebra(X_a)
=> ! [X,Y] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(X_a,real,norm_norm(X_a),Y)) ) ).
fof(fact_2699_norm__ge__zero,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(X_a,real,norm_norm(X_a),X))) ) ).
fof(fact_2700_norm__not__less__zero,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X)),zero_zero(real))) ) ).
fof(fact_2701_norm__one,axiom,
! [X_a] :
( real_n2089651433ebra_1(X_a)
=> hAPP(X_a,real,norm_norm(X_a),one_one(X_a)) = one_one(real) ) ).
fof(fact_2702_norm__power,axiom,
! [X_a] :
( real_n1866405975lgebra(X_a)
=> ! [X,N_1] : hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(X_a,real,norm_norm(X_a),X)),N_1) ) ).
fof(fact_2703_norm__divide,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& real_normed_field(X_a) )
=> ! [A_1,B] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)) ) ).
fof(fact_2704_complex__mod__triangle__ineq2,axiom,
! [B,A_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),B),A_1))),hAPP(complex,real,norm_norm(complex),B))),hAPP(complex,real,norm_norm(complex),A_1))) ).
fof(fact_2705_complex__mod__minus__le__complex__mod,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,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_2706_norm__inverse,axiom,
! [X_a] :
( ( divisi14063676e_zero(X_a)
& real_n1866405975lgebra(X_a) )
=> ! [A_1] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(real,real,inverse_inverse(real),hAPP(X_a,real,norm_norm(X_a),A_1)) ) ).
fof(fact_2707_SRStar__SR__prop,axiom,
! [X_1,Pa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),sRStar,Pa)))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),sr,Pa))) ) ).
fof(fact_2708_norm__le__zero__iff,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),X_1)),zero_zero(real)))
<=> ti(X_a,X_1) = zero_zero(X_a) ) ) ).
fof(fact_2709_zero__less__norm__iff,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,norm_norm(X_a),X_1)))
<=> ti(X_a,X_1) != zero_zero(X_a) ) ) ).
fof(fact_2710_norm__mult__ineq,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(X_a,real,norm_norm(X_a),Y)))) ) ).
fof(fact_2711_norm__mult__less,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y,S_4,X,R_4] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X)),R_4))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Y)),S_4))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),S_4))) ) ) ) ).
fof(fact_2712_nonzero__norm__divide,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [A_1,B] :
( ti(X_a,B) != zero_zero(X_a)
=> hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),A_1),B)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)) ) ) ).
fof(fact_2713_norm__triangle__ineq,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(X_a,real,norm_norm(X_a),Y)))) ) ).
fof(fact_2714_norm__add__less,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [Y,S_4,X,R_4] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X)),R_4))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Y)),S_4))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_4),S_4))) ) ) ) ).
fof(fact_2715_norm__triangle__ineq4,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,B] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)))) ) ).
fof(fact_2716_norm__power__ineq,axiom,
! [X_a] :
( real_n2089651433ebra_1(X_a)
=> ! [X,N_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(X_a,real,norm_norm(X_a),X)),N_1))) ) ).
fof(fact_2717_norm__diff__ineq,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,B] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)))) ) ).
fof(fact_2718_norm__triangle__ineq2,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,B] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) ) ).
fof(fact_2719_SRStar__def,axiom,
! [Pa] : hAPP(int,fun(int,bool),sRStar,Pa) = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Pa))) ).
fof(fact_2720_norm__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& real_n2089651433ebra_1(X_a) )
=> ! [W] : hAPP(X_a,real,norm_norm(X_a),hAPP(int,X_a,number_number_of(X_a),W)) = hAPP(real,real,abs_abs(real),hAPP(int,real,number_number_of(real),W)) ) ).
fof(fact_2721_nonzero__norm__inverse,axiom,
! [X_a] :
( real_n1866405975lgebra(X_a)
=> ! [A_1] :
( ti(X_a,A_1) != zero_zero(X_a)
=> hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),A_1)) = hAPP(real,real,inverse_inverse(real),hAPP(X_a,real,norm_norm(X_a),A_1)) ) ) ).
fof(fact_2722_StandardRes__SRStar__prop1a,axiom,
! [X_1,Pa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),sRStar,Pa)))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X_1),zero_zero(int)),Pa)) ) ).
fof(fact_2723_norm__ratiotest__lemma,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X,Y,C] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),C),zero_zero(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C),hAPP(X_a,real,norm_norm(X_a),Y))))
=> ti(X_a,X) = zero_zero(X_a) ) ) ) ).
fof(fact_2724_norm__diff__triangle__ineq,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,B,C,D] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),C),D)))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),C))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),D))))) ) ).
fof(fact_2725_norm__triangle__ineq3,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,B] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(X_a,real,norm_norm(X_a),A_1)),hAPP(X_a,real,norm_norm(X_a),B)))),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),A_1),B)))) ) ).
fof(fact_2726_norm__sgn,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X] :
( ( ti(X_a,X) = zero_zero(X_a)
=> hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)) = zero_zero(real) )
& ( ti(X_a,X) != zero_zero(X_a)
=> hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,sgn_sgn(X_a),X)) = one_one(real) ) ) ) ).
fof(fact_2727_cmod__unit__one,axiom,
! [A_1] : hAPP(complex,real,norm_norm(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_1)),hAPP(real,real,sin,A_1))) = one_one(real) ).
fof(fact_2728_lemma__NBseq__def,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [X_3] :
( ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_3,N_2))),K_2)) )
<=> ? [N_3] :
! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_3,N_2))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_3)))) ) ) ).
fof(fact_2729_lemma__NBseq__def2,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [X_3] :
( ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_3,N_2))),K_2)) )
<=> ? [N_3] :
! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_3,N_2))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_3)))) ) ) ).
fof(fact_2730_Bseq__inverse__lemma,axiom,
! [X_a] :
( real_n1866405975lgebra(X_a)
=> ! [R_4,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),R_4),hAPP(X_a,real,norm_norm(X_a),X)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_4))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),X))),hAPP(real,real,inverse_inverse(real),R_4))) ) ) ) ).
fof(fact_2731_complex__norm,axiom,
! [X,Y] : hAPP(complex,real,norm_norm(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_2732_StandardRes__SRStar__prop4,axiom,
! [X_1,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),sRStar,Pa)))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,Pa),X_1)),hAPP(int,fun(int,bool),sRStar,Pa))) ) ) ) ).
fof(fact_2733_StandardRes__SRStar__prop1,axiom,
! [X_1,Pa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,Pa),X_1)),hAPP(int,fun(int,bool),sRStar,Pa)))
<=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X_1),zero_zero(int)),Pa)) ) ) ).
fof(fact_2734_SRStar__mult__prop1,axiom,
! [Y_1,X_1,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),sRStar,Pa)))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_1),hAPP(int,fun(int,bool),sRStar,Pa)))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,Pa),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X_1),Y_1))),hAPP(int,fun(int,bool),sRStar,Pa))) ) ) ) ) ).
fof(fact_2735_StandardRes__SRStar__prop2,axiom,
! [X_1,Pa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),hAPP(int,fun(int,bool),sRStar,Pa)))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),standardRes,Pa),hAPP(int,int,hAPP(int,fun(int,int),multInv,Pa),X_1))),hAPP(int,fun(int,bool),sRStar,Pa))) ) ) ) ).
fof(fact_2736_bounded__bilinear_Opos__bounded,axiom,
! [X_c,X_b,X_a] :
( ( real_normed_vector(X_a)
& real_normed_vector(X_b)
& real_normed_vector(X_c) )
=> ! [Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [A_3,B_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_c,real,norm_norm(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,A_3),B_3))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_3)),hAPP(X_b,real,norm_norm(X_b),B_3))),K_2))) ) ) ) ).
fof(fact_2737_divide_Opos__bounded,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y] :
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X_2),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ) ).
fof(fact_2738_lemmaCauchy,axiom,
! [X_b,X_a] :
( ( ord(X_a)
& real_normed_vector(X_b) )
=> ! [X_3,M_2] :
( ! [N_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),M_2),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,X_3,M_2)),hAPP(X_a,X_b,X_3,N_2)))),one_one(real))) )
=> ! [N_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),M_2),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_3,N_2))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,X_3,M_2))))) ) ) ) ).
fof(fact_2739_mult_Opos__bounded,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [A_3,B_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_3))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_3)),hAPP(X_a,real,norm_norm(X_a),B_3))),K_2))) ) ) ).
fof(fact_2740_mult__left_Opos__bounded,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y] :
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_2),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ) ).
fof(fact_2741_mult__right_Opos__bounded,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X] :
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ) ).
fof(fact_2742_bounded__bilinear_Ononneg__bounded,axiom,
! [X_c,X_b,X_a] :
( ( real_normed_vector(X_a)
& real_normed_vector(X_b)
& real_normed_vector(X_c) )
=> ! [Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [A_3,B_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_c,real,norm_norm(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,A_3),B_3))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_3)),hAPP(X_b,real,norm_norm(X_b),B_3))),K_2))) ) ) ) ).
fof(fact_2743_negD,axiom,
! [X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X),zero_zero(int)))
=> ? [N_2] : X = hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_2))) ) ).
fof(fact_2744_divide_Ononneg__bounded,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y] :
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X_2),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ) ).
fof(fact_2745_zero__less__imp__eq__int,axiom,
! [K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),K_4))
=> ? [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_2))
& K_4 = hAPP(nat,int,semiring_1_of_nat(int),N_2) ) ) ).
fof(fact_2746_mult_Ononneg__bounded,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [A_3,B_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_3))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_3)),hAPP(X_a,real,norm_norm(X_a),B_3))),K_2))) ) ) ).
fof(fact_2747_mult__right_Ononneg__bounded,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X] :
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ) ).
fof(fact_2748_mult__left_Ononneg__bounded,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y] :
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_2),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ) ).
fof(fact_2749_bounded__bilinear_Obounded,axiom,
! [X_c,X_b,X_a] :
( ( real_normed_vector(X_a)
& real_normed_vector(X_b)
& real_normed_vector(X_c) )
=> ! [Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> ? [K_2] :
! [A_3,B_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_c,real,norm_norm(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,A_3),B_3))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_3)),hAPP(X_b,real,norm_norm(X_b),B_3))),K_2))) ) ) ).
fof(fact_2750_divide_Obounded,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y] :
? [K_2] :
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X_2),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ).
fof(fact_2751_mult__left_Obounded,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y] :
? [K_2] :
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_2),Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ).
fof(fact_2752_mult__right_Obounded,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X] :
? [K_2] :
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ).
fof(fact_2753_mult_Obounded,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ? [K_2] :
! [A_3,B_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_3),B_3))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),A_3)),hAPP(X_a,real,norm_norm(X_a),B_3))),K_2))) ) ).
fof(fact_2754_ex__less__of__nat__mult,axiom,
! [X_a] :
( archim1804426504_field(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),X))
=> ? [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_2)),X))) ) ) ).
fof(fact_2755_complex_Osize_I1_J,axiom,
! [Real1_1,Real2_1] : hAPP(complex,nat,complex_size,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1_1),Real2_1)) = zero_zero(nat) ).
fof(fact_2756_bezw_Osimps,axiom,
! [X,Y] :
( ( Y = zero_zero(nat)
=> hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) )
& ( Y != zero_zero(nat)
=> hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),Y))))) ) ) ).
fof(fact_2757_bezw__0,axiom,
! [X] : hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),zero_zero(nat)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).
fof(fact_2758_bezw__non__0,axiom,
! [X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
=> hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)))),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),X),Y))))) ) ).
fof(fact_2759_zero__le__imp__eq__int,axiom,
! [K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
=> ? [N_2] : K_4 = hAPP(nat,int,semiring_1_of_nat(int),N_2) ) ).
fof(fact_2760_nonneg__eq__int,axiom,
! [Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> ~ ! [M_3] : Z != hAPP(nat,int,semiring_1_of_nat(int),M_3) ) ).
fof(fact_2761_nonneg__int__cases,axiom,
! [K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
=> ~ ! [N_2] : K_4 != hAPP(nat,int,semiring_1_of_nat(int),N_2) ) ).
fof(fact_2762_cos__arg__i__mult__zero__neg,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y),zero_zero(real)))
=> hAPP(real,real,cos,hAPP(complex,real,arg,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),Y))) = zero_zero(real) ) ).
fof(fact_2763_cos__arg__i__mult__zero__pos,axiom,
! [Y] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> hAPP(real,real,cos,hAPP(complex,real,arg,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),Y))) = zero_zero(real) ) ).
fof(fact_2764_int__nat__abs,axiom,
! [X] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))) = hAPP(int,int,abs_abs(int),X) ).
fof(fact_2765_cos__arg__i__mult__zero,axiom,
! [Y] :
( Y != zero_zero(real)
=> hAPP(real,real,cos,hAPP(complex,real,arg,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),Y))) = zero_zero(real) ) ).
fof(fact_2766_complex_Osize_I2_J,axiom,
! [Real1_1,Real2_1] : hAPP(complex,nat,size_size(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1_1),Real2_1)) = zero_zero(nat) ).
fof(fact_2767_SRStar__card,axiom,
! [Pa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),sRStar,Pa))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int)) ) ).
fof(fact_2768_int__diff__cases,axiom,
! [Z] :
~ ! [M_3,N_2] : Z != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(nat,int,semiring_1_of_nat(int),M_3)),hAPP(nat,int,semiring_1_of_nat(int),N_2)) ).
fof(fact_2769_MultInvPair__card__two,axiom,
! [J,Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),zero_zero(int)),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,Pa),Aa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,J),zero_zero(int)),Pa))
=> hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),hAPP(int,fun(int,fun(int,fun(int,bool))),multInvPair,Aa),Pa),J)) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ) ) ) ) ) ).
fof(fact_2770_SetS__elems__card,axiom,
! [Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),zero_zero(int)),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,Pa),Aa))
=> ! [X_2] :
( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_2),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,Aa),Pa)))
=> hAPP(fun(int,bool),nat,finite_card(int),X_2) = hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))) ) ) ) ) ) ).
fof(fact_2771_power2__i,axiom,
hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),ii),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(int,complex,number_number_of(complex),min) ).
fof(fact_2772_complex__i__not__zero,axiom,
ii != zero_zero(complex) ).
fof(fact_2773_complex__i__not__one,axiom,
ii != one_one(complex) ).
fof(fact_2774_complex__i__not__number__of,axiom,
! [W] : ii != hAPP(int,complex,number_number_of(complex),W) ).
fof(fact_2775_complex__i__mult__minus,axiom,
! [X] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),X)) = hAPP(complex,complex,uminus_uminus(complex),X) ).
fof(fact_2776_inverse__i,axiom,
hAPP(complex,complex,inverse_inverse(complex),ii) = hAPP(complex,complex,uminus_uminus(complex),ii) ).
fof(fact_2777_i__def,axiom,
ii = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),one_one(real)) ).
fof(fact_2778_Complex__eq__i,axiom,
! [X_1,Y_1] :
( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X_1),Y_1) = ii
<=> ( X_1 = zero_zero(real)
& Y_1 = one_one(real) ) ) ).
fof(fact_2779_Complex__mult__i,axiom,
! [A_1,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)),ii) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),B)),A_1) ).
fof(fact_2780_i__mult__Complex,axiom,
! [A_1,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),B)),A_1) ).
fof(fact_2781_i__squared,axiom,
hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),ii) = hAPP(int,complex,number_number_of(complex),min) ).
fof(fact_2782_i__mult__eq2,axiom,
hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),ii) = hAPP(complex,complex,uminus_uminus(complex),one_one(complex)) ).
fof(fact_2783_SetS__card,axiom,
! [Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),zero_zero(int)),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,Pa),Aa))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(fun(int,bool),bool),nat,finite_card(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,Aa),Pa))) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ) ) ) ) ).
fof(fact_2784_complex__inverse__complex__split,axiom,
! [X,Y] : hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,of_real(complex),X)),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,of_real(complex),Y)))) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),Y),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,real),power_power(real),Y),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))) ).
fof(fact_2785_MultInvPair__prop2,axiom,
! [Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),zero_zero(int)),Pa))
=> hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,Aa),Pa)) = hAPP(int,fun(int,bool),sRStar,Pa) ) ) ) ).
fof(fact_2786_of__real__eq__iff,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [X_1,Y_1] :
( hAPP(real,X_a,of_real(X_a),X_1) = hAPP(real,X_a,of_real(X_a),Y_1)
<=> X_1 = Y_1 ) ) ).
fof(fact_2787_of__real__0,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> hAPP(real,X_a,of_real(X_a),zero_zero(real)) = zero_zero(X_a) ) ).
fof(fact_2788_of__real_Ozero,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> hAPP(real,X_a,of_real(X_a),zero_zero(real)) = zero_zero(X_a) ) ).
fof(fact_2789_of__real__eq__0__iff,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [X_1] :
( hAPP(real,X_a,of_real(X_a),X_1) = zero_zero(X_a)
<=> X_1 = zero_zero(real) ) ) ).
fof(fact_2790_of__real__mult,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).
fof(fact_2791_of__real__number__of__eq,axiom,
! [X_a] :
( ( number_ring(X_a)
& real_algebra_1(X_a) )
=> ! [W] : hAPP(real,X_a,of_real(X_a),hAPP(int,real,number_number_of(real),W)) = hAPP(int,X_a,number_number_of(X_a),W) ) ).
fof(fact_2792_of__real__add,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).
fof(fact_2793_of__real_Oadd,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).
fof(fact_2794_of__real__1,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> hAPP(real,X_a,of_real(X_a),one_one(real)) = one_one(X_a) ) ).
fof(fact_2795_of__real__power,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [X,N_1] : hAPP(real,X_a,of_real(X_a),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(real,X_a,of_real(X_a),X)),N_1) ) ).
fof(fact_2796_complex__eq__cancel__iff2,axiom,
! [X_1,Y_1,Xa_2] :
( hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X_1),Y_1) = hAPP(real,complex,of_real(complex),Xa_2)
<=> ( X_1 = Xa_2
& Y_1 = zero_zero(real) ) ) ).
fof(fact_2797_complex__of__real__def,axiom,
! [R_4] : hAPP(real,complex,of_real(complex),R_4) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,R_4),zero_zero(real)) ).
fof(fact_2798_of__real__divide,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& real_field(X_a) )
=> ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).
fof(fact_2799_of__real__diff,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).
fof(fact_2800_of__real_Odiff,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [X,Y] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ).
fof(fact_2801_of__real__minus,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [X] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,uminus_uminus(real),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(real,X_a,of_real(X_a),X)) ) ).
fof(fact_2802_of__real_Ominus,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [X] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,uminus_uminus(real),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(real,X_a,of_real(X_a),X)) ) ).
fof(fact_2803_norm__of__real,axiom,
! [X_a] :
( real_n2089651433ebra_1(X_a)
=> ! [R_4] : hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),R_4)) = hAPP(real,real,abs_abs(real),R_4) ) ).
fof(fact_2804_of__real__inverse,axiom,
! [X_a] :
( ( divisi14063676e_zero(X_a)
& real_div_algebra(X_a) )
=> ! [X] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,inverse_inverse(real),X)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(real,X_a,of_real(X_a),X)) ) ).
fof(fact_2805_exp__of__real,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X] : hAPP(X_a,X_a,exp(X_a),hAPP(real,X_a,of_real(X_a),X)) = hAPP(real,X_a,of_real(X_a),hAPP(real,real,exp(real),X)) ) ).
fof(fact_2806_of__real__of__nat__eq,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [N_1] : hAPP(real,X_a,of_real(X_a),hAPP(nat,real,semiring_1_of_nat(real),N_1)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1) ) ).
fof(fact_2807_sgn__of__real,axiom,
! [X_a] :
( real_n2089651433ebra_1(X_a)
=> ! [R_4] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(real,X_a,of_real(X_a),R_4)) = hAPP(real,X_a,of_real(X_a),hAPP(real,real,sgn_sgn(real),R_4)) ) ).
fof(fact_2808_sgn__eq,axiom,
! [Z] : hAPP(complex,complex,sgn_sgn(complex),Z) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),Z),hAPP(real,complex,of_real(complex),hAPP(complex,real,norm_norm(complex),Z))) ).
fof(fact_2809_nonzero__of__real__divide,axiom,
! [X_a] :
( real_field(X_a)
=> ! [X,Y] :
( Y != zero_zero(real)
=> hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(real,X_a,of_real(X_a),X)),hAPP(real,X_a,of_real(X_a),Y)) ) ) ).
fof(fact_2810_nonzero__of__real__inverse,axiom,
! [X_a] :
( real_div_algebra(X_a)
=> ! [X] :
( X != zero_zero(real)
=> hAPP(real,X_a,of_real(X_a),hAPP(real,real,inverse_inverse(real),X)) = hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(real,X_a,of_real(X_a),X)) ) ) ).
fof(fact_2811_Complex__mult__complex__of__real,axiom,
! [X,Y,R_4] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)),hAPP(real,complex,of_real(complex),R_4)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),R_4)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),R_4)) ).
fof(fact_2812_complex__of__real__mult__Complex,axiom,
! [R_4,X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_4)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),X)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),Y)) ).
fof(fact_2813_complex__of__real__add__Complex,axiom,
! [R_4,X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,of_real(complex),R_4)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),R_4),X)),Y) ).
fof(fact_2814_Complex__add__complex__of__real,axiom,
! [X,Y,R_4] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)),hAPP(real,complex,of_real(complex),R_4)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),R_4)),Y) ).
fof(fact_2815_i__mult__eq,axiom,
hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),ii) = hAPP(real,complex,of_real(complex),hAPP(int,real,number_number_of(real),min)) ).
fof(fact_2816_i__complex__of__real,axiom,
! [R_4] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),ii),hAPP(real,complex,of_real(complex),R_4)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),R_4) ).
fof(fact_2817_complex__of__real__i,axiom,
! [R_4] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_4)),ii) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,zero_zero(real)),R_4) ).
fof(fact_2818_complex__of__real__minus__one,axiom,
hAPP(real,complex,of_real(complex),hAPP(real,real,uminus_uminus(real),one_one(real))) = hAPP(complex,complex,uminus_uminus(complex),one_one(complex)) ).
fof(fact_2819_cmod__complex__polar,axiom,
! [R_4,A_1] : hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_4)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_1)),hAPP(real,real,sin,A_1)))) = hAPP(real,real,abs_abs(real),R_4) ).
fof(fact_2820_of__real_Opos__bounded,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_2)),K_2))) ) ) ).
fof(fact_2821_of__real_Ononneg__bounded,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_2)),K_2))) ) ) ).
fof(fact_2822_expi__two__pi__i,axiom,
hAPP(complex,complex,expi,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(int,complex,number_number_of(complex),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,complex,of_real(complex),pi))),ii)) = one_one(complex) ).
fof(fact_2823_expi__add,axiom,
! [A_1,B] : hAPP(complex,complex,expi,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),A_1),B)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,expi,A_1)),hAPP(complex,complex,expi,B)) ).
fof(fact_2824_expi__zero,axiom,
hAPP(complex,complex,expi,zero_zero(complex)) = one_one(complex) ).
fof(fact_2825_of__real_Obounded,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ? [K_2] :
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(real,X_a,of_real(X_a),X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_2)),K_2))) ) ).
fof(fact_2826_complex__split__polar,axiom,
! [Z] :
? [R_2,A_3] : Z = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_2)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_3)),hAPP(real,real,sin,A_3))) ).
fof(fact_2827_UnionI,axiom,
! [X_a,A_2,X_3,C_2] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_3),C_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),X_3))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),C_2))) ) ) ).
fof(fact_2828_Union__upper,axiom,
! [X_a,B_1,A_2] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),B_1),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2))) ) ).
fof(fact_2829_Union__mono,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),A_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),B_1))) ) ).
fof(fact_2830_Sup__le__iff,axiom,
! [X_a] :
( comple44059174attice(X_a)
=> ! [A_2,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2)),Ba))
<=> ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Ba)) ) ) ) ).
fof(fact_2831_complete__lattice__class_OSup__upper,axiom,
! [X_a] :
( comple44059174attice(X_a)
=> ! [X_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2))) ) ) ).
fof(fact_2832_Sup__subset__mono,axiom,
! [X_a] :
( comple44059174attice(X_a)
=> ! [A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2)),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),B_1))) ) ) ).
fof(fact_2833_complete__lattice__class_OSup__upper2,axiom,
! [X_a] :
( comple44059174attice(X_a)
=> ! [V_1,U_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),U_1),A_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),V_1),U_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),V_1),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2))) ) ) ) ).
fof(fact_2834_less__Sup__iff,axiom,
! [X_a] :
( comple1803745165norder(X_a)
=> ! [Aa,S] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),S)))
<=> ? [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),S))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),X_2)) ) ) ) ).
fof(fact_2835_UnionE,axiom,
! [X_a,A_2,C_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),C_2)))
=> ~ ! [X_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),A_2),X_4))
=> ~ hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_4),C_2)) ) ) ).
fof(fact_2836_Union__SetS__finite,axiom,
! [Aa,Pa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,Aa),Pa)))) ) ).
fof(fact_2837_complete__lattice__class_OSup__least,axiom,
! [X_a] :
( comple44059174attice(X_a)
=> ! [Z_1,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_2),Z_1)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(X_a,bool),X_a,complete_Sup_Sup(X_a),A_2)),Z_1)) ) ) ).
fof(fact_2838_wset__fin,axiom,
! [Aa,Pa] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa))) ).
fof(fact_2839_d22set__fin,axiom,
! [Aa] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),d22set,Aa))) ).
fof(fact_2840_SetS__elems__finite,axiom,
! [Aa,Pa,X_2] :
( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_2),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,Aa),Pa)))
=> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),X_2)) ) ).
fof(fact_2841_neg__one__special,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> 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,number_number_of(int),min)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,number_number_of(int),min)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))) = one_one(int) ) ).
fof(fact_2842_SRStar__finite,axiom,
! [Pa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),sRStar,Pa))) ) ).
fof(fact_2843_diff__card__le__card__Diff,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1))),hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))) ) ).
fof(fact_2844_card__Diff__subset,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1)) ) ) ).
fof(fact_2845_card__seteq,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)))
=> ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ) ) ).
fof(fact_2846_finite__Diff,axiom,
! [X_a,B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) ) ).
fof(fact_2847_finite__Diff2,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ) ).
fof(fact_2848_finite__code,axiom,
! [X_a] :
( finite_finite(X_a)
=> ! [A_2] : hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).
fof(fact_2849_finite,axiom,
! [X_a] :
( finite_finite(X_a)
=> ! [A_2] : hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).
fof(fact_2850_SetS__finite,axiom,
! [Aa,Pa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> hBOOL(hAPP(fun(fun(int,bool),bool),bool,finite_finite_1(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,Aa),Pa))) ) ).
fof(fact_2851_rev__finite__subset,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ) ).
fof(fact_2852_finite__subset,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ) ).
fof(fact_2853_finite__UnionD,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)))
=> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),A_2)) ) ).
fof(fact_2854_card__infinite,axiom,
! [X_a,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2) = zero_zero(nat) ) ).
fof(fact_2855_psubset__card__mono,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1))) ) ) ).
fof(fact_2856_card__ge__0__finite,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).
fof(fact_2857_card__psubset,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1)) ) ) ) ).
fof(fact_2858_card__mono,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_a,bool),nat,finite_card(X_a),B_1))) ) ) ).
fof(fact_2859_finite__Union,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),A_2))
=> ( ! [M_4] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),M_4),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),M_4)) )
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2))) ) ) ).
fof(fact_2860_card_Oneutral,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ( ? [X_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> one_one(nat) = zero_zero(nat) )
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2) = zero_zero(nat) ) ) ).
fof(fact_2861_DiffE,axiom,
! [X_a,C_1,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
=> ~ ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),B_1)) ) ) ).
fof(fact_2862_equalityI,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ) ).
fof(fact_2863_subsetD,axiom,
! [X_a,C_1,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),B_1)) ) ) ).
fof(fact_2864_DiffI,axiom,
! [X_a,B_1,C_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),A_2))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1))) ) ) ).
fof(fact_2865_psubsetD,axiom,
! [X_a,C_1,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),B_1)) ) ) ).
fof(fact_2866_psubset__trans,axiom,
! [X_a,C_2,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),B_1),C_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C_2)) ) ) ).
fof(fact_2867_equalityE,axiom,
! [X_a,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
=> ~ ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ) ).
fof(fact_2868_subset__trans,axiom,
! [X_a,C_2,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),C_2)) ) ) ).
fof(fact_2869_set__mp,axiom,
! [X_a,X_1,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_1)) ) ) ).
fof(fact_2870_set__rev__mp,axiom,
! [X_a,B_1,X_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_1)) ) ) ).
fof(fact_2871_in__mono,axiom,
! [X_a,X_1,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),B_1)) ) ) ).
fof(fact_2872_equalityD2,axiom,
! [X_a,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ).
fof(fact_2873_equalityD1,axiom,
! [X_a,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).
fof(fact_2874_set__eq__subset,axiom,
! [X_a,A_2,B_1] :
( ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1)
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
& hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ) ).
fof(fact_2875_subset__refl,axiom,
! [X_a,A_2] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),A_2)) ).
fof(fact_2876_psubset__eq,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
& ti(fun(X_a,bool),A_2) != ti(fun(X_a,bool),B_1) ) ) ).
fof(fact_2877_subset__iff__psubset__eq,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
| ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ) ).
fof(fact_2878_psubset__imp__subset,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).
fof(fact_2879_psubset__subset__trans,axiom,
! [X_a,C_2,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C_2)) ) ) ).
fof(fact_2880_subset__psubset__trans,axiom,
! [X_a,C_2,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),B_1),C_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),C_2)) ) ) ).
fof(fact_2881_Compl__anti__mono,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),B_1)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))) ) ).
fof(fact_2882_Compl__subset__Compl__iff,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),B_1)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2)) ) ).
fof(fact_2883_set__diff__eq,axiom,
! [X_a,A_2,B_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1)))) ).
fof(fact_2884_Diff__iff,axiom,
! [X_a,C_1,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),A_2))
& ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),B_1)) ) ) ).
fof(fact_2885_Diff__idemp,axiom,
! [X_a,A_2,B_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),B_1) = hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1) ).
fof(fact_2886_DiffD1,axiom,
! [X_a,C_1,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),A_2)) ) ).
fof(fact_2887_DiffD2,axiom,
! [X_a,C_1,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),B_1)) ) ).
fof(fact_2888_Diff__subset,axiom,
! [X_a,A_2,B_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),A_2)) ).
fof(fact_2889_Diff__mono,axiom,
! [X_a,D_2,B_1,A_2,C_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),C_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),D_2),B_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),C_2),D_2))) ) ) ).
fof(fact_2890_double__diff,axiom,
! [X_a,C_2,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),C_2))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),C_2),A_2)) = ti(fun(X_a,bool),A_2) ) ) ).
fof(fact_2891_subsetI,axiom,
! [X_a,B_1,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),B_1)) )
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).
fof(fact_2892_psubset__imp__ex__mem,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
=> ? [B_3] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),A_2))) ) ).
fof(fact_2893_Union__least,axiom,
! [X_a,C_2,A_2] :
( ! [X_4] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_4),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),X_4),C_2)) )
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)),C_2)) ) ).
fof(fact_2894_ComplI,axiom,
! [X_a,C_1,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2))) ) ).
fof(fact_2895_Compl__eq,axiom,
! [X_a,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))) ).
fof(fact_2896_double__complement,axiom,
! [X_a,A_2] : hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)) = ti(fun(X_a,bool),A_2) ).
fof(fact_2897_Compl__iff,axiom,
! [X_a,C_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)))
<=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),A_2)) ) ).
fof(fact_2898_Compl__eq__Compl__iff,axiom,
! [X_a,A_2,B_1] :
( hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2) = hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),B_1)
<=> ti(fun(X_a,bool),A_2) = ti(fun(X_a,bool),B_1) ) ).
fof(fact_2899_ComplD,axiom,
! [X_a,C_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),A_2)) ) ).
fof(fact_2900_ComplE,axiom,
! [X_a,C_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),C_1),A_2)) ) ).
fof(fact_2901_complex_Orecs,axiom,
! [X_a,F1,Real1,Real2] : hAPP(complex,X_a,hAPP(fun(real,fun(real,X_a)),fun(complex,X_a),complex_rec(X_a),F1),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1),Real2)) = hAPP(real,X_a,hAPP(real,fun(real,X_a),F1,Real1),Real2) ).
fof(fact_2902_complex_Osimps_I2_J,axiom,
! [X_a,F1,Real1,Real2] : hAPP(complex,X_a,hAPP(fun(real,fun(real,X_a)),fun(complex,X_a),complex_case(X_a),F1),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Real1),Real2)) = hAPP(real,X_a,hAPP(real,fun(real,X_a),F1,Real1),Real2) ).
fof(fact_2903_ex__less__of__nat,axiom,
! [X_a] :
( archim1804426504_field(X_a)
=> ! [X] :
? [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_2))) ) ).
fof(fact_2904_ex__le__of__nat,axiom,
! [X_a] :
( archim1804426504_field(X_a)
=> ! [X] :
? [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(nat,X_a,semiring_1_of_nat(X_a),N_2))) ) ).
fof(fact_2905_card__Pow,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(fun(X_a,bool),bool),nat,finite_card(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)) ) ).
fof(fact_2906_DERIV__neg__imp__decreasing,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),Ba))
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Ba)) )
=> ? [Y_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_2),zero_zero(real))) ) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Ba)),hAPP(real,real,F,Aa))) ) ) ).
fof(fact_2907_PowI,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))) ) ).
fof(fact_2908_Pow__top,axiom,
! [X_a,A_2] : hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2))) ).
fof(fact_2909_Pow__def,axiom,
! [X_a,A_2] : hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2) = hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool),collect(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),combc(fun(X_a,bool),fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool))),A_2)) ).
fof(fact_2910_Pow__iff,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).
fof(fact_2911_PowD,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1)) ) ).
fof(fact_2912_Union__Pow__eq,axiom,
! [X_a,A_2] : hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)) = ti(fun(X_a,bool),A_2) ).
fof(fact_2913_Pow__mono,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))) ) ).
fof(fact_2914_finite__Pow__iff,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).
fof(fact_2915_subset__Pow__Union,axiom,
! [X_a,A_2] : hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),A_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),A_2)))) ).
fof(fact_2916_DERIV__pos__imp__increasing,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),Ba))
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Ba)) )
=> ? [Y_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y_2)) ) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Aa)),hAPP(real,real,F,Ba))) ) ) ).
fof(fact_2917_coprime__sos,axiom,
! [X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_2918_accp__subset,axiom,
! [X_a,R1_1,R2_1] :
( hBOOL(hAPP(fun(X_a,fun(X_a,bool)),bool,hAPP(fun(X_a,fun(X_a,bool)),fun(fun(X_a,fun(X_a,bool)),bool),ord_less_eq(fun(X_a,fun(X_a,bool))),R1_1),R2_1))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),R2_1)),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),R1_1))) ) ).
fof(fact_2919_coprime__commute,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,Aa),Ba))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,Ba),Aa)) ) ).
fof(fact_2920_coprime__lmul2,axiom,
! [D,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),B)) ) ).
fof(fact_2921_coprime__rmul2,axiom,
! [D,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),A_1)) ) ).
fof(fact_2922_coprime__mul,axiom,
! [B,D,A_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),B))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B))) ) ) ).
fof(fact_2923_coprime__mul__eq,axiom,
! [D_1,Aa,Ba] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Aa),Ba)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),Aa))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),Ba)) ) ) ).
fof(fact_2924_coprime__1_H,axiom,
! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,one_one(nat)),A_1)) ).
fof(fact_2925_coprime__1,axiom,
! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),one_one(nat))) ).
fof(fact_2926_coprime__refl,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,N),N))
<=> N = one_one(nat) ) ).
fof(fact_2927_coprime__exp__imp,axiom,
! [N_1,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),N_1))) ) ).
fof(fact_2928_coprime__exp,axiom,
! [N_1,D,A_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),A_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N_1))) ) ).
fof(fact_2929_coprime__Suc0_H,axiom,
! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,suc,zero_zero(nat))),A_1)) ).
fof(fact_2930_coprime__Suc0,axiom,
! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),hAPP(nat,nat,suc,zero_zero(nat)))) ).
fof(fact_2931_coprime__0,axiom,
! [D_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D_1),zero_zero(nat)))
<=> D_1 = one_one(nat) ) ).
fof(fact_2932_coprime__0_H,axiom,
! [D_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,zero_zero(nat)),D_1))
<=> D_1 = one_one(nat) ) ).
fof(fact_2933_coprime__plus1,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),one_one(nat))),N_1)) ).
fof(fact_2934_coprime__exp2,axiom,
! [Aa,N,Ba] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Aa),hAPP(nat,nat,suc,N))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Ba),hAPP(nat,nat,suc,N))))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,Aa),Ba)) ) ).
fof(fact_2935_coprime__minus1,axiom,
! [N_1] :
( N_1 != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat))),N_1)) ) ).
fof(fact_2936_accp_Osimps,axiom,
! [X_a,R_1,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),R_1),Aa))
<=> ? [X_2] :
( ti(X_a,Aa) = ti(X_a,X_2)
& ! [Xa] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_1,Xa),X_2))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),R_1),Xa)) ) ) ) ).
fof(fact_2937_accp__downward,axiom,
! [X_a,Aa,R_1,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),R_1),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_1,Aa),Ba))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,bool),accp(X_a),R_1),Aa)) ) ) ).
fof(fact_2938_coprime__bezout,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,Aa),Ba))
<=> ? [X_2,Y_2] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Aa),X_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ba),Y_2)) = 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),Ba),X_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Aa),Y_2)) = one_one(nat) ) ) ).
fof(fact_2939_nat_Osize_I2_J,axiom,
! [Nat_1] : hAPP(nat,nat,nat_size,hAPP(nat,nat,suc,Nat_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,nat_size,Nat_1)),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_2940_coprime__bezout__strong,axiom,
! [A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
=> ( B != one_one(nat)
=> ? [X_2,Y_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),X_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),Y_2)),one_one(nat)) ) ) ).
fof(fact_2941_nat_Osize_I1_J,axiom,
hAPP(nat,nat,nat_size,zero_zero(nat)) = zero_zero(nat) ).
fof(fact_2942_nat_Osize_I4_J,axiom,
! [Nat_1] : hAPP(nat,nat,size_size(nat),hAPP(nat,nat,suc,Nat_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,size_size(nat),Nat_1)),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_2943_complex__expi__Ex,axiom,
! [Z] :
? [A_3,R_2] : Z = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_2)),hAPP(complex,complex,expi,A_3)) ).
fof(fact_2944_ResSet__finite,axiom,
! [X_3,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,M),X_3))
=> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),X_3)) ) ) ).
fof(fact_2945_nat__size,axiom,
! [N_1] : hAPP(nat,nat,size_size(nat),N_1) = N_1 ).
fof(fact_2946_ResSet__def,axiom,
! [M,X_3] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,M),X_3))
<=> ! [Y1,Y2] :
( ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y1),X_3))
& hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y2),X_3))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Y1),Y2),M)) )
=> Y1 = Y2 ) ) ).
fof(fact_2947_nat_Osize_I3_J,axiom,
hAPP(nat,nat,size_size(nat),zero_zero(nat)) = zero_zero(nat) ).
fof(fact_2948_ResSet__SRStar__prop,axiom,
! [Pa] : hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,Pa),hAPP(int,fun(int,bool),sRStar,Pa))) ).
fof(fact_2949_code__numeral_Osize_I1_J,axiom,
hAPP(code_code_numeral,nat,code_c271388182l_size,zero_zero(code_code_numeral)) = zero_zero(nat) ).
fof(fact_2950_complex__mod__mult__cnj,axiom,
! [Z] : hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z),hAPP(complex,complex,cnj,Z))) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,norm_norm(complex),Z)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) ).
fof(fact_2951_in__lex__prod,axiom,
! [X_a,X_b,Aa,Ba,A_7,B_6,R_1,S_3] :
( hBOOL(hAPP(fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool),bool,hAPP(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),fun(fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool),bool),member(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b))),hAPP(product_prod(X_a,X_b),product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),hAPP(product_prod(X_a,X_b),fun(product_prod(X_a,X_b),product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b))),product_Pair(product_prod(X_a,X_b),product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Aa),Ba)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_7),B_6))),hAPP(fun(product_prod(X_b,X_b),bool),fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool),hAPP(fun(product_prod(X_a,X_a),bool),fun(fun(product_prod(X_b,X_b),bool),fun(product_prod(product_prod(X_a,X_b),product_prod(X_a,X_b)),bool)),lex_prod(X_a,X_b),R_1),S_3)))
<=> ( hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),Aa),A_7)),R_1))
| ( ti(X_a,Aa) = ti(X_a,A_7)
& hBOOL(hAPP(fun(product_prod(X_b,X_b),bool),bool,hAPP(product_prod(X_b,X_b),fun(fun(product_prod(X_b,X_b),bool),bool),member(product_prod(X_b,X_b)),hAPP(X_b,product_prod(X_b,X_b),hAPP(X_b,fun(X_b,product_prod(X_b,X_b)),product_Pair(X_b,X_b),Ba),B_6)),S_3)) ) ) ) ).
fof(fact_2952_complex__cnj__complex__of__real,axiom,
! [X] : hAPP(complex,complex,cnj,hAPP(real,complex,of_real(complex),X)) = hAPP(real,complex,of_real(complex),X) ).
fof(fact_2953_complex__cnj__divide,axiom,
! [X,Y] : hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),X),Y)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) ).
fof(fact_2954_complex__cnj__power,axiom,
! [X,N_1] : hAPP(complex,complex,cnj,hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),X),N_1)) = hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(complex,complex,cnj,X)),N_1) ).
fof(fact_2955_complex__cnj__of__nat,axiom,
! [N_1] : hAPP(complex,complex,cnj,hAPP(nat,complex,semiring_1_of_nat(complex),N_1)) = hAPP(nat,complex,semiring_1_of_nat(complex),N_1) ).
fof(fact_2956_complex__cnj__one,axiom,
hAPP(complex,complex,cnj,one_one(complex)) = one_one(complex) ).
fof(fact_2957_complex__cnj__mult,axiom,
! [X,Y] : hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) ).
fof(fact_2958_cnj_Oadd,axiom,
! [X,Y] : hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) ).
fof(fact_2959_complex__cnj__number__of,axiom,
! [W] : hAPP(complex,complex,cnj,hAPP(int,complex,number_number_of(complex),W)) = hAPP(int,complex,number_number_of(complex),W) ).
fof(fact_2960_cnj_Ozero,axiom,
hAPP(complex,complex,cnj,zero_zero(complex)) = zero_zero(complex) ).
fof(fact_2961_complex__cnj__zero__iff,axiom,
! [Z_1] :
( hAPP(complex,complex,cnj,Z_1) = zero_zero(complex)
<=> Z_1 = zero_zero(complex) ) ).
fof(fact_2962_complex__cnj__cnj,axiom,
! [Z] : hAPP(complex,complex,cnj,hAPP(complex,complex,cnj,Z)) = Z ).
fof(fact_2963_complex__cnj__cancel__iff,axiom,
! [X_1,Y_1] :
( hAPP(complex,complex,cnj,X_1) = hAPP(complex,complex,cnj,Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_2964_complex__cnj__inverse,axiom,
! [X] : hAPP(complex,complex,cnj,hAPP(complex,complex,inverse_inverse(complex),X)) = hAPP(complex,complex,inverse_inverse(complex),hAPP(complex,complex,cnj,X)) ).
fof(fact_2965_complex__mod__cnj,axiom,
! [Z] : hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,Z)) = hAPP(complex,real,norm_norm(complex),Z) ).
fof(fact_2966_cnj_Odiff,axiom,
! [X,Y] : hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),hAPP(complex,complex,cnj,X)),hAPP(complex,complex,cnj,Y)) ).
fof(fact_2967_cnj_Ominus,axiom,
! [X] : hAPP(complex,complex,cnj,hAPP(complex,complex,uminus_uminus(complex),X)) = hAPP(complex,complex,uminus_uminus(complex),hAPP(complex,complex,cnj,X)) ).
fof(fact_2968_complex__cnj,axiom,
! [A_1,B] : hAPP(complex,complex,cnj,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),hAPP(real,real,uminus_uminus(real),B)) ).
fof(fact_2969_complex__cnj__i,axiom,
hAPP(complex,complex,cnj,ii) = hAPP(complex,complex,uminus_uminus(complex),ii) ).
fof(fact_2970_cnj_Opos__bounded,axiom,
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_2)),K_2))) ) ).
fof(fact_2971_cnj_Ononneg__bounded,axiom,
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_2)),K_2))) ) ).
fof(fact_2972_cnj_Obounded,axiom,
? [K_2] :
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,cnj,X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_2)),K_2))) ).
fof(fact_2973_complex__diff__cnj,axiom,
! [Z] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),Z),hAPP(complex,complex,cnj,Z)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(complex,real,im,Z)))),ii) ).
fof(fact_2974_complex__add__cnj,axiom,
! [Z] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),Z),hAPP(complex,complex,cnj,Z)) = hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(complex,real,re,Z))) ).
fof(fact_2975_code__numeral_Osize_I2_J,axiom,
! [Code_numeral_2] : hAPP(code_code_numeral,nat,code_c271388182l_size,hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(code_code_numeral,nat,code_c271388182l_size,Code_numeral_2)),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_2976_Re__complex__of__real,axiom,
! [Z] : hAPP(complex,real,re,hAPP(real,complex,of_real(complex),Z)) = Z ).
fof(fact_2977_complex__add__def,axiom,
! [X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y))) ).
fof(fact_2978_complex__Im__mult,axiom,
! [X,Y] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,im,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,re,Y))) ).
fof(fact_2979_code__numeral_Oinject,axiom,
! [Code_numeral,Code_numeral_5] :
( hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral) = hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_5)
<=> Code_numeral = Code_numeral_5 ) ).
fof(fact_2980_complex__eq__iff,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
<=> ( hAPP(complex,real,re,X_1) = hAPP(complex,real,re,Y_1)
& hAPP(complex,real,im,X_1) = hAPP(complex,real,im,Y_1) ) ) ).
fof(fact_2981_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_2982_complex__surj,axiom,
! [Z] : hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(complex,real,re,Z)),hAPP(complex,real,im,Z)) = Z ).
fof(fact_2983_Re,axiom,
! [X,Y] : hAPP(complex,real,re,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = X ).
fof(fact_2984_Im,axiom,
! [X,Y] : hAPP(complex,real,im,hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,X),Y)) = Y ).
fof(fact_2985_complex__Re__mult,axiom,
! [X,Y] : hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y))) ).
fof(fact_2986_complex__minus__def,axiom,
! [X] : hAPP(complex,complex,uminus_uminus(complex),X) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,uminus_uminus(real),hAPP(complex,real,re,X))),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X))) ).
fof(fact_2987_cnj__def,axiom,
! [Z] : hAPP(complex,complex,cnj,Z) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(complex,real,re,Z)),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,Z))) ).
fof(fact_2988_complex__Re__cnj,axiom,
! [X] : hAPP(complex,real,re,hAPP(complex,complex,cnj,X)) = hAPP(complex,real,re,X) ).
fof(fact_2989_complex__Re__le__cmod,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(complex,real,re,X)),hAPP(complex,real,norm_norm(complex),X))) ).
fof(fact_2990_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_2991_complex__Re__i,axiom,
hAPP(complex,real,re,ii) = zero_zero(real) ).
fof(fact_2992_Im__complex__of__real,axiom,
! [Z] : hAPP(complex,real,im,hAPP(real,complex,of_real(complex),Z)) = zero_zero(real) ).
fof(fact_2993_complex__Im__i,axiom,
hAPP(complex,real,im,ii) = one_one(real) ).
fof(fact_2994_code__numeral_Osimps_I3_J,axiom,
! [Code_numeral_4] : hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_4) != zero_zero(code_code_numeral) ).
fof(fact_2995_code__numeral_Osimps_I2_J,axiom,
! [Code_numeral_3] : zero_zero(code_code_numeral) != hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_3) ).
fof(fact_2996_Re_Ozero,axiom,
hAPP(complex,real,re,zero_zero(complex)) = zero_zero(real) ).
fof(fact_2997_Im_Ozero,axiom,
hAPP(complex,real,im,zero_zero(complex)) = zero_zero(real) ).
fof(fact_2998_complex__Im__one,axiom,
hAPP(complex,real,im,one_one(complex)) = zero_zero(real) ).
fof(fact_2999_Re_Oadd,axiom,
! [X,Y] : hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y)) ).
fof(fact_3000_Im_Oadd,axiom,
! [X,Y] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),plus_plus(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y)) ).
fof(fact_3001_complex__Re__one,axiom,
hAPP(complex,real,re,one_one(complex)) = one_one(real) ).
fof(fact_3002_complex__Im__number__of,axiom,
! [V] : hAPP(complex,real,im,hAPP(int,complex,number_number_of(complex),V)) = zero_zero(real) ).
fof(fact_3003_Re_Ominus,axiom,
! [X] : hAPP(complex,real,re,hAPP(complex,complex,uminus_uminus(complex),X)) = hAPP(real,real,uminus_uminus(real),hAPP(complex,real,re,X)) ).
fof(fact_3004_complex__Re__number__of,axiom,
! [V] : hAPP(complex,real,re,hAPP(int,complex,number_number_of(complex),V)) = hAPP(int,real,number_number_of(real),V) ).
fof(fact_3005_Im_Ominus,axiom,
! [X] : hAPP(complex,real,im,hAPP(complex,complex,uminus_uminus(complex),X)) = hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X)) ).
fof(fact_3006_Re_Odiff,axiom,
! [X,Y] : hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y)) ).
fof(fact_3007_Im_Odiff,axiom,
! [X,Y] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y)) ).
fof(fact_3008_complex__mult__def,axiom,
! [X,Y] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),X),Y) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,re,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,im,Y)))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,re,X)),hAPP(complex,real,im,Y))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,im,X)),hAPP(complex,real,re,Y)))) ).
fof(fact_3009_complex__Im__of__nat,axiom,
! [N_1] : hAPP(complex,real,im,hAPP(nat,complex,semiring_1_of_nat(complex),N_1)) = zero_zero(real) ).
fof(fact_3010_complex__Re__of__nat,axiom,
! [N_1] : hAPP(complex,real,re,hAPP(nat,complex,semiring_1_of_nat(complex),N_1)) = hAPP(nat,real,semiring_1_of_nat(real),N_1) ).
fof(fact_3011_abs__Re__le__cmod,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,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_3012_abs__Im__le__cmod,axiom,
! [X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,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_3013_complex__In__mult__cnj__zero,axiom,
! [Z] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z),hAPP(complex,complex,cnj,Z))) = zero_zero(real) ).
fof(fact_3014_Re__sgn,axiom,
! [Z] : hAPP(complex,real,re,hAPP(complex,complex,sgn_sgn(complex),Z)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,re,Z)),hAPP(complex,real,norm_norm(complex),Z)) ).
fof(fact_3015_Im__sgn,axiom,
! [Z] : hAPP(complex,real,im,hAPP(complex,complex,sgn_sgn(complex),Z)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,im,Z)),hAPP(complex,real,norm_norm(complex),Z)) ).
fof(fact_3016_cmod__def,axiom,
! [Z] : hAPP(complex,real,norm_norm(complex),Z) = hAPP(real,real,sqrt,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,Z)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,Z)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_3017_complex__Re__inverse,axiom,
! [X] : hAPP(complex,real,re,hAPP(complex,complex,inverse_inverse(complex),X)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,re,X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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_3018_Suc__code__numeral__minus__one,axiom,
! [N_1] : hAPP(code_code_numeral,code_code_numeral,hAPP(code_code_numeral,fun(code_code_numeral,code_code_numeral),minus_minus(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,N_1)),one_one(code_code_numeral)) = N_1 ).
fof(fact_3019_complex__Im__inverse,axiom,
! [X] : hAPP(complex,real,im,hAPP(complex,complex,inverse_inverse(complex),X)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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_3020_complex__mod__sqrt__Re__mult__cnj,axiom,
! [Z] : hAPP(complex,real,norm_norm(complex),Z) = hAPP(real,real,sqrt,hAPP(complex,real,re,hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z),hAPP(complex,complex,cnj,Z)))) ).
fof(fact_3021_complex__mult__cnj,axiom,
! [Z] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),Z),hAPP(complex,complex,cnj,Z)) = hAPP(real,complex,of_real(complex),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,re,Z)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(complex,real,im,Z)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_3022_complex__inverse__def,axiom,
! [X] : hAPP(complex,complex,inverse_inverse(complex),X) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(complex,real,re,X)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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(real,fun(real,real),inverse_divide(real),hAPP(real,real,uminus_uminus(real),hAPP(complex,real,im,X))),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,hAPP(real,fun(nat,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(real,fun(nat,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_3023_Im_Opos__bounded,axiom,
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,im,X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_2)),K_2))) ) ).
fof(fact_3024_Re_Opos__bounded,axiom,
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,re,X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_2)),K_2))) ) ).
fof(fact_3025_Im_Ononneg__bounded,axiom,
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,im,X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_2)),K_2))) ) ).
fof(fact_3026_Re_Ononneg__bounded,axiom,
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,re,X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_2)),K_2))) ) ).
fof(fact_3027_Im_Obounded,axiom,
? [K_2] :
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,im,X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_2)),K_2))) ).
fof(fact_3028_Re_Obounded,axiom,
? [K_2] :
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,norm_norm(real),hAPP(complex,real,re,X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(complex,real,norm_norm(complex),X_2)),K_2))) ).
fof(fact_3029_code__numeral_Osize_I4_J,axiom,
! [Code_numeral_2] : hAPP(code_code_numeral,nat,size_size(code_code_numeral),hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(code_code_numeral,nat,size_size(code_code_numeral),Code_numeral_2)),hAPP(nat,nat,suc,zero_zero(nat))) ).
fof(fact_3030_code__numeral_Oexhaust,axiom,
! [Y] :
( Y != zero_zero(code_code_numeral)
=> ~ ! [Code_numeral_1] : Y != hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral_1) ) ).
fof(fact_3031_expi__def,axiom,
! [Z] : hAPP(complex,complex,expi,Z) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),hAPP(real,real,exp(real),hAPP(complex,real,re,Z)))),hAPP(real,complex,cis,hAPP(complex,real,im,Z))) ).
fof(fact_3032_Re__cis,axiom,
! [A_1] : hAPP(complex,real,re,hAPP(real,complex,cis,A_1)) = hAPP(real,real,cos,A_1) ).
fof(fact_3033_Im__cis,axiom,
! [A_1] : hAPP(complex,real,im,hAPP(real,complex,cis,A_1)) = hAPP(real,real,sin,A_1) ).
fof(fact_3034_cis__mult,axiom,
! [A_1,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,cis,A_1)),hAPP(real,complex,cis,B)) = hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)) ).
fof(fact_3035_cis__zero,axiom,
hAPP(real,complex,cis,zero_zero(real)) = one_one(complex) ).
fof(fact_3036_cis__inverse,axiom,
! [A_1] : hAPP(complex,complex,inverse_inverse(complex),hAPP(real,complex,cis,A_1)) = hAPP(real,complex,cis,hAPP(real,real,uminus_uminus(real),A_1)) ).
fof(fact_3037_cis__divide,axiom,
! [A_1,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(real,complex,cis,A_1)),hAPP(real,complex,cis,B)) = hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)) ).
fof(fact_3038_cis__def,axiom,
! [A_1] : hAPP(real,complex,cis,A_1) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,cos,A_1)),hAPP(real,real,sin,A_1)) ).
fof(fact_3039_DeMoivre,axiom,
! [A_1,N_1] : hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,cis,A_1)),N_1) = hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),A_1)) ).
fof(fact_3040_cis__real__of__nat__Suc__mult,axiom,
! [N_1,A_1] : hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_1))),A_1)) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,cis,A_1)),hAPP(real,complex,cis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),A_1))) ).
fof(fact_3041_code__numeral_Osize_I3_J,axiom,
hAPP(code_code_numeral,nat,size_size(code_code_numeral),zero_zero(code_code_numeral)) = zero_zero(nat) ).
fof(fact_3042_cos__n__Re__cis__pow__n,axiom,
! [N_1,A_1] : hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),A_1)) = hAPP(complex,real,re,hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,cis,A_1)),N_1)) ).
fof(fact_3043_sin__n__Im__cis__pow__n,axiom,
! [N_1,A_1] : hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),A_1)) = hAPP(complex,real,im,hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,cis,A_1)),N_1)) ).
fof(fact_3044_Diff__infinite__finite,axiom,
! [X_a,S,T_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_4))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),S),T_4))) ) ) ).
fof(fact_3045_divides__cases,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),M_1))
=> ( M_1 = zero_zero(nat)
| M_1 = N_1
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_1)),M_1)) ) ) ).
fof(fact_3046_infinite__super,axiom,
! [X_a,S,T_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_4))
=> ( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_4)) ) ) ).
fof(fact_3047_dvd_Oorder__refl,axiom,
! [X] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X)) ).
fof(fact_3048_dvd__0__right,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),zero_zero(X_a))) ) ).
fof(fact_3049_dvd__1__left,axiom,
! [K_4] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,suc,zero_zero(nat))),K_4)) ).
fof(fact_3050_coprime__divisors,axiom,
! [E_1,B,D,A_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E_1),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),E_1)) ) ) ) ).
fof(fact_3051_divides__exp,axiom,
! [N_1,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y),N_1))) ) ).
fof(fact_3052_dvd__power__same,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [N_1,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N_1))) ) ) ).
fof(fact_3053_gcd__lcm__complete__lattice__nat_Onot__top__less,axiom,
! [A_1] :
~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat))) ) ).
fof(fact_3054_gcd__lcm__complete__lattice__nat_Otop__greatest,axiom,
! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),zero_zero(nat))) ).
fof(fact_3055_gcd__lcm__complete__lattice__nat_Oless__top,axiom,
! [Aa] :
( Aa != zero_zero(nat)
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Aa),zero_zero(nat)))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),Aa)) ) ) ).
fof(fact_3056_gcd__lcm__complete__lattice__nat_Otop__unique,axiom,
! [Aa] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),Aa))
<=> Aa = zero_zero(nat) ) ).
fof(fact_3057_gcd__lcm__complete__lattice__nat_Otop__le,axiom,
! [A_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A_1))
=> A_1 = zero_zero(nat) ) ).
fof(fact_3058_gcd__lcm__complete__lattice__nat_Ole__bot,axiom,
! [A_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),one_one(nat)))
=> A_1 = one_one(nat) ) ).
fof(fact_3059_gcd__lcm__complete__lattice__nat_Obot__unique,axiom,
! [Aa] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Aa),one_one(nat)))
<=> Aa = one_one(nat) ) ).
fof(fact_3060_gcd__lcm__complete__lattice__nat_Obot__less,axiom,
! [Aa] :
( Aa != one_one(nat)
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),Aa))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Aa),one_one(nat))) ) ) ).
fof(fact_3061_gcd__lcm__complete__lattice__nat_Obot__least,axiom,
! [A_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1)) ).
fof(fact_3062_gcd__lcm__complete__lattice__nat_Onot__less__bot,axiom,
! [A_1] :
~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),one_one(nat)))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),one_one(nat)),A_1)) ) ).
fof(fact_3063_one__dvd,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),one_one(X_a)),A_1)) ) ).
fof(fact_3064_nat__dvd__1__iff__1,axiom,
! [M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),one_one(nat)))
<=> M = one_one(nat) ) ).
fof(fact_3065_dvd__add,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),C))) ) ) ) ).
fof(fact_3066_dvd__triv__left,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B))) ) ).
fof(fact_3067_dvd__triv__right,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),A_1))) ) ).
fof(fact_3068_dvd__mult2,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))) ) ) ).
fof(fact_3069_dvd__mult,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [B,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C))) ) ) ).
fof(fact_3070_mult__dvd__mono,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,D,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),C)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),D))) ) ) ) ).
fof(fact_3071_dvdI,axiom,
! [X_a] :
( dvd(X_a)
=> ! [A_1,B,K_4] :
( ti(X_a,A_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),K_4)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),A_1)) ) ) ).
fof(fact_3072_dvd__mult__left,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)) ) ) ).
fof(fact_3073_dvd__mult__right,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,B,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),C)) ) ) ).
fof(fact_3074_divides__mul__r,axiom,
! [C,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(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),C))) ) ).
fof(fact_3075_divides__mul__l,axiom,
! [C,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),A_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),C),B))) ) ).
fof(fact_3076_divides__add__revr,axiom,
! [B,D,A_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),A_1),B)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D),B)) ) ) ).
fof(fact_3077_dvd__reduce,axiom,
! [K,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),N)) ) ).
fof(fact_3078_dvd__0__left,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),zero_zero(X_a)),A_1))
=> ti(X_a,A_1) = zero_zero(X_a) ) ) ).
fof(fact_3079_dvd__mod__iff,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [M,K,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),N))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),M),N)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K),M)) ) ) ) ).
fof(fact_3080_mod__mod__cancel,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,C,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C),B))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),A_1),C) ) ) ).
fof(fact_3081_dvd__mod,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [N_1,K_4,M_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),M_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),M_1),N_1))) ) ) ) ).
fof(fact_3082_dvd__mod__imp__dvd,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [K_4,M_1,N_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),M_1),N_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),K_4),M_1)) ) ) ) ).
fof(fact_3083_div__dvd__div,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [C_1,Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Aa),C_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),Ba),Aa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),C_1),Aa)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Ba),C_1)) ) ) ) ) ).
fof(fact_3084_dvd_Oless__irrefl,axiom,
! [X] :
~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),X)) ) ).
fof(fact_3085_dvd_Oeq__iff,axiom,
! [X_1,Y_1] :
( X_1 = Y_1
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) ) ) ).
fof(fact_3086_dvd_Ole__less,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
<=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) )
| X_1 = Y_1 ) ) ).
fof(fact_3087_dvd_Oless__le,axiom,
! [X_1,Y_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) )
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
& X_1 != Y_1 ) ) ).
fof(fact_3088_dvd_Oless__le__not__le,axiom,
! [X_1,Y_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) )
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) ) ) ).
fof(fact_3089_dvd_Oneq__le__trans,axiom,
! [A_1,B] :
( A_1 != B
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) ) ) ) ).
fof(fact_3090_dvd_Oeq__refl,axiom,
! [X,Y] :
( X = Y
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ).
fof(fact_3091_dvd_Oantisym__conv,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
<=> X_1 = Y_1 ) ) ).
fof(fact_3092_dvd_Ole__imp__less__or__eq,axiom,
! [X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
| X = Y ) ) ).
fof(fact_3093_dvd_Ole__neq__trans,axiom,
! [A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
=> ( A_1 != B
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) ) ) ) ).
fof(fact_3094_dvd_Oord__eq__le__trans,axiom,
! [C,A_1,B] :
( A_1 = B
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C)) ) ) ).
fof(fact_3095_dvd_Oord__le__eq__trans,axiom,
! [C,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
=> ( B = C
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C)) ) ) ).
fof(fact_3096_dvd__antisym,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M_1),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),M_1))
=> M_1 = N_1 ) ) ).
fof(fact_3097_dvd_Oantisym,axiom,
! [X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
=> X = Y ) ) ).
fof(fact_3098_dvd_Oorder__trans,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z)) ) ) ).
fof(fact_3099_dvd_Oord__eq__less__trans,axiom,
! [C,A_1,B] :
( A_1 = B
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),C))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),B)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),A_1)) ) ) ) ).
fof(fact_3100_dvd_Ole__less__trans,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),Y)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X)) ) ) ) ).
fof(fact_3101_dvd_Oless__imp__neq,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> X != Y ) ).
fof(fact_3102_dvd_Oless__not__sym,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).
fof(fact_3103_dvd_Oless__imp__le,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ).
fof(fact_3104_dvd_Oless__imp__not__less,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).
fof(fact_3105_dvd_Oless__imp__not__eq,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> X != Y ) ).
fof(fact_3106_dvd_Oless__imp__not__eq2,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> Y != X ) ).
fof(fact_3107_dvd_Oless__imp__triv,axiom,
! [P_1,X_1,Y_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) )
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) )
=> hBOOL(P_1) ) ) ).
fof(fact_3108_dvd_Oord__less__eq__trans,axiom,
! [C,A_1,B] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) )
=> ( B = C
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),C),A_1)) ) ) ) ).
fof(fact_3109_dvd_Oless__le__trans,axiom,
! [Z,X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X)) ) ) ) ).
fof(fact_3110_dvd_Oless__asym_H,axiom,
! [A_1,B] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1)) )
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),A_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) ) ) ).
fof(fact_3111_dvd_Oless__trans,axiom,
! [Z,X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),Z))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),Y)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Z),X)) ) ) ) ).
fof(fact_3112_dvd_Oless__asym,axiom,
! [X,Y] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X)) )
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).
fof(fact_3113_dvd__refl,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),A_1)) ) ).
fof(fact_3114_dvd__trans,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),C))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),C)) ) ) ) ).
fof(fact_3115_divides__antisym,axiom,
! [X_1,Y_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) )
<=> X_1 = Y_1 ) ).
fof(fact_3116_dvd__if__abs__eq,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [L_4,K_4] :
( hAPP(X_a,X_a,abs_abs(X_a),L_4) = hAPP(X_a,X_a,abs_abs(X_a),K_4)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L_4),K_4)) ) ) ).
fof(fact_3117_abs__dvd__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [M,K] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,abs_abs(X_a),M)),K))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),M),K)) ) ) ).
fof(fact_3118_dvd__abs__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [M,K] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),M),hAPP(X_a,X_a,abs_abs(X_a),K)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),M),K)) ) ) ).
fof(fact_3119_dvd__diff,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),Z))) ) ) ) ).
fof(fact_3120_dvd__diff__nat,axiom,
! [N_1,K_4,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1))) ) ) ).
fof(fact_3121_minus__dvd__iff,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X_1)),Y_1))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y_1)) ) ) ).
fof(fact_3122_dvd__minus__iff,axiom,
! [X_a] :
( comm_ring_1(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),hAPP(X_a,X_a,uminus_uminus(X_a),Y_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X_1),Y_1)) ) ) ).
fof(fact_3123_dvd__mult__cancel__left,axiom,
! [X_a] :
( idom(X_a)
=> ! [C_1,Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Aa)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Ba)))
<=> ( ti(X_a,C_1) = zero_zero(X_a)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Aa),Ba)) ) ) ) ).
fof(fact_3124_dvd__mult__cancel__right,axiom,
! [X_a] :
( idom(X_a)
=> ! [Aa,C_1,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),C_1)))
<=> ( ti(X_a,C_1) = zero_zero(X_a)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Aa),Ba)) ) ) ) ).
fof(fact_3125_dvd__imp__mod__0,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),B),A_1) = zero_zero(X_a) ) ) ).
fof(fact_3126_dvd__eq__mod__eq__0,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Aa),Ba))
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),Ba),Aa) = zero_zero(X_a) ) ) ).
fof(fact_3127_dvd__mult__div__cancel,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),A_1)) = ti(X_a,B) ) ) ).
fof(fact_3128_div__mult__swap,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,C,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C),B))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)),C) ) ) ).
fof(fact_3129_dvd__div__mult__self,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),A_1)),A_1) = ti(X_a,B) ) ) ).
fof(fact_3130_dvd__div__mult,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [C,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),A_1),B))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),B),A_1)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),C)),A_1) ) ) ).
fof(fact_3131_div__mult__div__if__dvd,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [Z,W,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Z),W))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),W),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),W)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y),Z)) ) ) ) ).
fof(fact_3132_div__add,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [Y,Z,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Z),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Z),Y))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),Y),Z)) ) ) ) ).
fof(fact_3133_le__imp__power__dvd,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [A_1,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1))) ) ) ).
fof(fact_3134_dvd__power__le,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [N_1,M_1,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),Y))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),M_1))) ) ) ) ).
fof(fact_3135_power__le__dvd,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [M_1,A_1,N_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),N_1)),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),A_1),M_1)),B)) ) ) ) ).
fof(fact_3136_div__power,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [N_1,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)),N_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y),N_1)) ) ) ).
fof(fact_3137_dvd__neg__div,axiom,
! [X_a] :
( ring_div(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),Y) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)) ) ) ).
fof(fact_3138_dvd__div__neg,axiom,
! [X_a] :
( ring_div(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Y),X))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),X),Y)) ) ) ).
fof(fact_3139_dvd__1__iff__1,axiom,
! [M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),hAPP(nat,nat,suc,zero_zero(nat))))
<=> M = hAPP(nat,nat,suc,zero_zero(nat)) ) ).
fof(fact_3140_nat__dvd__not__less,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),M_1)) ) ) ).
fof(fact_3141_dvd__pos__nat,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M_1)) ) ) ).
fof(fact_3142_divides__ge,axiom,
! [A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B))
=> ( B = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),A_1),B)) ) ) ).
fof(fact_3143_divides__le,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M_1),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
| N_1 = zero_zero(nat) ) ) ).
fof(fact_3144_nat__mult__dvd__cancel__disj_H,axiom,
! [M,K,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K)))
<=> ( K = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).
fof(fact_3145_nat__mult__dvd__cancel__disj,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)))
<=> ( K = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).
fof(fact_3146_dvd__diffD1,axiom,
! [K_4,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N_1)) ) ) ) ).
fof(fact_3147_dvd__diffD,axiom,
! [K_4,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M_1)) ) ) ) ).
fof(fact_3148_divides__rev,axiom,
! [A_1,N_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),N_1)))
=> ( N_1 != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),B)) ) ) ).
fof(fact_3149_divides__exp2,axiom,
! [X,Y,N_1] :
( N_1 != zero_zero(nat)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N_1)),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y)) ) ) ).
fof(fact_3150_pow__divides__eq__nat,axiom,
! [Aa,Ba,N] :
( N != zero_zero(nat)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Aa),N)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Ba),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Aa),Ba)) ) ) ).
fof(fact_3151_divides__rexp,axiom,
! [N_1,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Y),hAPP(nat,nat,suc,N_1)))) ) ).
fof(fact_3152_fact__dvd,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,fact_fact(nat),N_1)),hAPP(nat,nat,fact_fact(nat),M_1))) ) ).
fof(fact_3153_divides__mul,axiom,
! [N_1,M_1,R_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M_1),R_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),R_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)),R_4)) ) ) ) ).
fof(fact_3154_coprime__divprod,axiom,
! [D,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,D),A_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D),B)) ) ) ).
fof(fact_3155_coprime,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,Aa),Ba))
<=> ! [D_3] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),Aa))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),Ba)) )
<=> D_3 = one_one(nat) ) ) ).
fof(fact_3156_unity__coeff__ex,axiom,
! [X_a] :
( ( dvd(X_a)
& semiring_0(X_a) )
=> ! [P_1,L] :
( ? [X_2] : hBOOL(hAPP(X_a,bool,P_1,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),L),X_2)))
<=> ? [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),L),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),zero_zero(X_a))))
& hBOOL(hAPP(X_a,bool,P_1,X_2)) ) ) ) ).
fof(fact_3157_inf__period_I4_J,axiom,
! [X_a] :
( ( comm_ring(X_a)
& dvd(X_a) )
=> ! [T_3,D_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),D_2))
=> ! [X_2,K_1] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),T_3)))
<=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),K_1),D_2))),T_3))) ) ) ) ).
fof(fact_3158_inf__period_I3_J,axiom,
! [X_a] :
( ( comm_ring(X_a)
& dvd(X_a) )
=> ! [T_3,D_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),D_2))
=> ! [X_2,K_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),T_3)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_2),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),K_1),D_2))),T_3))) ) ) ) ).
fof(fact_3159_dvd__div__div__eq__mult,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [D_1,Ba,C_1,Aa] :
( ti(X_a,Aa) != zero_zero(X_a)
=> ( ti(X_a,C_1) != zero_zero(X_a)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),C_1),D_1))
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),Ba),Aa) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),D_1),C_1)
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Ba),C_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),D_1) ) ) ) ) ) ) ).
fof(fact_3160_dvd__div__eq__mult,axiom,
! [X_a] :
( semiring_div(X_a)
=> ! [C_1,Ba,Aa] :
( ti(X_a,Aa) != zero_zero(X_a)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),Aa),Ba))
=> ( hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_div(X_a),Ba),Aa) = ti(X_a,C_1)
<=> ti(X_a,Ba) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Aa) ) ) ) ) ).
fof(fact_3161_dvd__eq__mod__eq__0__number__of,axiom,
! [X_a] :
( ( semiring_div(X_a)
& number(X_a) )
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),hAPP(int,X_a,number_number_of(X_a),X_1)),hAPP(int,X_a,number_number_of(X_a),Y_1)))
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),div_mod(X_a),hAPP(int,X_a,number_number_of(X_a),Y_1)),hAPP(int,X_a,number_number_of(X_a),X_1)) = zero_zero(X_a) ) ) ).
fof(fact_3162_dvd__imp__le,axiom,
! [K_4,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_4),N_1)) ) ) ).
fof(fact_3163_dvd__mult__cancel,axiom,
! [K_4,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N_1)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M_1),N_1)) ) ) ).
fof(fact_3164_nat__mult__dvd__cancel1,axiom,
! [M,N,K] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),M)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).
fof(fact_3165_dvd__fact__nat,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M_1),hAPP(nat,nat,fact_fact(nat),N_1))) ) ) ).
fof(fact_3166_dvd__power,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [X,N_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
| ti(X_a,X) = one_one(X_a) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),X),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X),N_1))) ) ) ).
fof(fact_3167_divides__div__not,axiom,
! [X,Q_2,N_1,R_4] :
( X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Q_2),N_1)),R_4)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),R_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_4),N_1))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),X)) ) ) ) ).
fof(fact_3168_dvd__mult__cancel2,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),M)),M))
<=> N = one_one(nat) ) ) ).
fof(fact_3169_dvd__mult__cancel1,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)),M))
<=> N = one_one(nat) ) ) ).
fof(fact_3170_power__dvd__imp__le,axiom,
! [I_1,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),I_1),N_1)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1)) ) ) ).
fof(fact_3171_even__dvd,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),N))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)) ) ).
fof(fact_3172_nat__even__iff__2__dvd,axiom,
! [X_1] :
( hBOOL(hAPP(nat,bool,even_odd_even(nat),X_1))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X_1)) ) ).
fof(fact_3173_real__of__nat__div,axiom,
! [N_1,D] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),D))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D),N_1))
=> hAPP(nat,real,real_1(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),N_1),D)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),N_1)),hAPP(nat,real,real_1(nat),D)) ) ) ).
fof(fact_3174_dvd_Olift__Suc__mono__le,axiom,
! [N,N_5,F] :
( ! [N_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_2)),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_2))))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),N_5))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N)),hAPP(nat,nat,F,N_5))) ) ) ).
fof(fact_3175_dvd_Olift__Suc__mono__less__iff,axiom,
! [N,M,F] :
( ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_2)),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_2))))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_2))),hAPP(nat,nat,F,N_2))) )
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N)),hAPP(nat,nat,F,M)))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,M)),hAPP(nat,nat,F,N))) )
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ).
fof(fact_3176_dvd_Olift__Suc__mono__less,axiom,
! [N,N_5,F] :
( ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_2)),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_2))))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,hAPP(nat,nat,suc,N_2))),hAPP(nat,nat,F,N_2))) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),N_5))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N)),hAPP(nat,nat,F,N_5)))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,F,N_5)),hAPP(nat,nat,F,N))) ) ) ) ).
fof(fact_3177_zdvd__self__abs2,axiom,
! [D] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,abs_abs(int),D)),D)) ).
fof(fact_3178_zdvd__self__abs1,axiom,
! [D] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D),hAPP(int,int,abs_abs(int),D))) ).
fof(fact_3179_zdvd__zmod,axiom,
! [N_1,F_4,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F_4),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F_4),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),F_4),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M_1),N_1))) ) ) ).
fof(fact_3180_zdvd__zmod__imp__zdvd,axiom,
! [K_4,M_1,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M_1),N_1)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M_1)) ) ) ).
fof(fact_3181_zdvd__antisym__abs,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_1),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),B),A_1))
=> hAPP(int,int,abs_abs(int),A_1) = hAPP(int,int,abs_abs(int),B) ) ) ).
fof(fact_3182_zdvd__zdiffD,axiom,
! [K_4,M_1,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M_1),N_1)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M_1)) ) ) ).
fof(fact_3183_uminus__dvd__conv_I1_J,axiom,
! [D_1,T_3] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),T_3))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,uminus_uminus(int),D_1)),T_3)) ) ).
fof(fact_3184_uminus__dvd__conv_I2_J,axiom,
! [D_1,T_3] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),T_3))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,uminus_uminus(int),T_3))) ) ).
fof(fact_3185_zdvd__not__zless,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M_1),N_1))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N_1),M_1)) ) ) ).
fof(fact_3186_zdvd__antisym__nonneg,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_1),N_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N_1),M_1))
=> M_1 = N_1 ) ) ) ) ).
fof(fact_3187_zdvd__bounds,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N_1),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M_1),zero_zero(int)))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),N_1),M_1)) ) ) ).
fof(fact_3188_zdvd__mono,axiom,
! [M,T_3,K] :
( K != zero_zero(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),T_3))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),M)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),T_3))) ) ) ).
fof(fact_3189_zdvd__mult__cancel,axiom,
! [K_4,M_1,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N_1)))
=> ( K_4 != zero_zero(int)
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_1),N_1)) ) ) ).
fof(fact_3190_zdvd__period,axiom,
! [C_1,X_1,T_3,Aa,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Aa),D_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),T_3)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),D_1))),T_3))) ) ) ).
fof(fact_3191_zdvd__reduce,axiom,
! [K,N,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),M))))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),N)) ) ).
fof(fact_3192_zcong__zero__equiv__div,axiom,
! [Aa,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),zero_zero(int)),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),Aa)) ) ).
fof(fact_3193_zcong__eq__zdvd__prop,axiom,
! [X_1,Pa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X_1),zero_zero(int)),Pa))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Pa),X_1)) ) ).
fof(fact_3194_Nat__Transfer_Otransfer__int__nat__relations_I4_J,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)),hAPP(nat,int,semiring_1_of_nat(int),Y_1)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1)) ) ).
fof(fact_3195_zdvd__int,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)),hAPP(nat,int,semiring_1_of_nat(int),Y_1))) ) ).
fof(fact_3196_pow__divides__pow__int,axiom,
! [A_1,N_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N_1)))
=> ( N_1 != zero_zero(nat)
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),A_1),B)) ) ) ).
fof(fact_3197_pow__divides__eq__int,axiom,
! [Aa,Ba,N] :
( N != zero_zero(nat)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Aa),N)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Ba),N)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Aa),Ba)) ) ) ).
fof(fact_3198_zdvd__mult__div__cancel,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N_1),M_1))
=> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N_1),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),M_1),N_1)) = M_1 ) ).
fof(fact_3199_zdvd1__eq,axiom,
! [X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),X_1),one_one(int)))
<=> hAPP(int,int,abs_abs(int),X_1) = one_one(int) ) ).
fof(fact_3200_zcong__def,axiom,
! [Aa,Ba,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Aa),Ba))) ) ).
fof(fact_3201_zmod__eq__dvd__iff,axiom,
! [X_1,N,Y_1] :
( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),X_1),N) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),Y_1),N)
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_1),Y_1))) ) ).
fof(fact_3202_zprime__zdvd__zmult__better,axiom,
! [M_1,N_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),N_1)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),M_1))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),N_1)) ) ) ) ).
fof(fact_3203_abs__div,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_3204_zprime__zdvd__power,axiom,
! [A_1,N_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N_1)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),A_1)) ) ) ).
fof(fact_3205_zdvd__imp__le,axiom,
! [Z,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z),N_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z),N_1)) ) ) ).
fof(fact_3206_zpower__zdvd__prop1,axiom,
! [P_4,Y,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y),N_1))) ) ) ).
fof(fact_3207_zmod__zdvd__zmod,axiom,
! [A_1,B,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_1),B))
=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),B)),M_1) = hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),A_1),M_1) ) ) ).
fof(fact_3208_zdvd__iff__zmod__eq__0__number__of,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,number_number_of(int),X_1)),hAPP(int,int,number_number_of(int),Y_1)))
<=> hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,number_number_of(int),Y_1)),hAPP(int,int,number_number_of(int),X_1)) = zero_zero(int) ) ).
fof(fact_3209_dvd__imp__le__int,axiom,
! [D,I_1] :
( I_1 != zero_zero(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D),I_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,abs_abs(int),D)),hAPP(int,int,abs_abs(int),I_1))) ) ) ).
fof(fact_3210_zprime__power__zdvd__cancel__right,axiom,
! [N_1,A_1,B,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),B))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_4),N_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_4),N_1)),A_1)) ) ) ) ).
fof(fact_3211_zprime__power__zdvd__cancel__left,axiom,
! [N_1,B,A_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_4),N_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),P_4),N_1)),B)) ) ) ) ).
fof(fact_3212_transfer__nat__int__relations_I4_J,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,nat_1,X_1)),hAPP(int,nat,nat_1,Y_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),X_1),Y_1)) ) ) ) ).
fof(fact_3213_zdvd__mult__cancel1,axiom,
! [N,M] :
( M != zero_zero(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)),M))
<=> hAPP(int,int,abs_abs(int),N) = one_one(int) ) ) ).
fof(fact_3214_zprime__zdvd__zmult,axiom,
! [N_1,P_4,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_1))
=> ( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),N_1)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),M_1))
| hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),N_1)) ) ) ) ) ).
fof(fact_3215_real__of__int__div,axiom,
! [N_1,D] :
( D != zero_zero(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D),N_1))
=> hAPP(int,real,real_1(int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),N_1),D)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,real_1(int),N_1)),hAPP(int,real,real_1(int),D)) ) ) ).
fof(fact_3216_dvd__int__iff,axiom,
! [Z_1,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),hAPP(nat,int,semiring_1_of_nat(int),M)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Z_1))),M)) ) ).
fof(fact_3217_int__dvd__iff,axiom,
! [M,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(nat,int,semiring_1_of_nat(int),M)),Z_1))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Z_1)))) ) ).
fof(fact_3218_zpower__zdvd__prop2,axiom,
! [Y,N_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Y),N_1)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),Y)) ) ) ) ).
fof(fact_3219_zprime__def,axiom,
! [Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),Pa))
& ! [M_3] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_3))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_3),Pa)) )
=> ( M_3 = one_one(int)
| M_3 = Pa ) ) ) ) ).
fof(fact_3220_int__even__iff__2__dvd,axiom,
! [X_1] :
( hBOOL(hAPP(int,bool,even_odd_even(int),X_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),X_1)) ) ).
fof(fact_3221_nat__dvd__iff,axiom,
! [Z_1,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(int,nat,nat_1,Z_1)),M))
<=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),Z_1),hAPP(nat,int,semiring_1_of_nat(int),M))) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1))
=> M = zero_zero(nat) ) ) ) ).
fof(fact_3222_Little__Fermat,axiom,
! [X,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),P_4),one_one(int))))),one_one(int)),P_4)) ) ) ).
fof(fact_3223_Euler_Oaux____1,axiom,
! [Y,X,P_4] :
( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X),zero_zero(int)),P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,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_4))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),Y)) ) ) ).
fof(fact_3224_zprime__factor__exists,axiom,
! [A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),A_1))
=> ? [P_5] :
( hBOOL(hAPP(int,bool,zprime,P_5))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),A_1)) ) ) ).
fof(fact_3225_minf_I10_J,axiom,
! [X_a] :
( ( plus(X_a)
& linorder(X_a)
& dvd(X_a) )
=> ! [D_1,S_3] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Z_2))
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),S_3)))
<=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),S_3))) ) ) ) ).
fof(fact_3226_pinf_I10_J,axiom,
! [X_a] :
( ( plus(X_a)
& linorder(X_a)
& dvd(X_a) )
=> ! [D_1,S_3] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_2))
=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),S_3)))
<=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),S_3))) ) ) ) ).
fof(fact_3227_pinf_I9_J,axiom,
! [X_a] :
( ( plus(X_a)
& linorder(X_a)
& dvd(X_a) )
=> ! [D_1,S_3] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_2),X_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),S_3)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),S_3))) ) ) ) ).
fof(fact_3228_minf_I9_J,axiom,
! [X_a] :
( ( plus(X_a)
& linorder(X_a)
& dvd(X_a) )
=> ! [D_1,S_3] :
? [Z_2] :
! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),Z_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),S_3)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),D_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_2),S_3))) ) ) ) ).
fof(fact_3229_dvdE,axiom,
! [X_a] :
( dvd(X_a)
=> ! [B,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),dvd_dvd(X_a),B),A_1))
=> ~ ! [K_1] : ti(X_a,A_1) != hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),B),K_1) ) ) ).
fof(fact_3230_prime__dvd__power__two,axiom,
! [M_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),M_1)) ) ) ).
fof(fact_3231_one__code__int__code,axiom,
one_one(quickcheck_code_int) = hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit1,pls)) ).
fof(fact_3232_Quickcheck__Narrowing_Oint__of__code,axiom,
! [K_4] :
( ( K_4 = zero_zero(quickcheck_code_int)
=> hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4) = zero_zero(int) )
& ( K_4 != zero_zero(quickcheck_code_int)
=> ( ( hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = zero_zero(quickcheck_code_int)
=> hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4) = 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(quickcheck_code_int,int,quickcheck_int_of,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) )
& ( hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) != zero_zero(quickcheck_code_int)
=> hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4) = 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(quickcheck_code_int,int,quickcheck_int_of,hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),K_4),hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),one_one(int)) ) ) ) ) ).
fof(fact_3233_distinct__prime__coprime,axiom,
! [Q_2,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,prime,Q_2))
=> ( P_4 != Q_2
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,P_4),Q_2)) ) ) ) ).
fof(fact_3234_less__eq__code__int__def,axiom,
! [N,M] :
( hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less_eq(quickcheck_code_int),N),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M))) ) ).
fof(fact_3235_less__code__int__def,axiom,
! [N,M] :
( hBOOL(hAPP(quickcheck_code_int,bool,hAPP(quickcheck_code_int,fun(quickcheck_code_int,bool),ord_less(quickcheck_code_int),N),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M))) ) ).
fof(fact_3236_prime__1,axiom,
~ hBOOL(hAPP(nat,bool,prime,one_one(nat))) ).
fof(fact_3237_prime__0,axiom,
~ hBOOL(hAPP(nat,bool,prime,zero_zero(nat))) ).
fof(fact_3238_int__of__number,axiom,
! [K_4] : hAPP(quickcheck_code_int,int,quickcheck_int_of,hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),K_4)) = hAPP(int,int,number_number_of(int),K_4) ).
fof(fact_3239_code__int_Oint__of__inject,axiom,
! [X_1,Y_1] :
( hAPP(quickcheck_code_int,int,quickcheck_int_of,X_1) = hAPP(quickcheck_code_int,int,quickcheck_int_of,Y_1)
<=> X_1 = Y_1 ) ).
fof(fact_3240_Quickcheck__Narrowing_Oint__of__inject,axiom,
! [K,L] :
( hAPP(quickcheck_code_int,int,quickcheck_int_of,K) = hAPP(quickcheck_code_int,int,quickcheck_int_of,L)
<=> K = L ) ).
fof(fact_3241_primes__eq,axiom,
! [Q_2,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,prime,Q_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),Q_2))
=> P_4 = Q_2 ) ) ) ).
fof(fact_3242_prime__Suc0,axiom,
~ hBOOL(hAPP(nat,bool,prime,hAPP(nat,nat,suc,zero_zero(nat)))) ).
fof(fact_3243_prime__g__zero,axiom,
! [P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),P_4)) ) ).
fof(fact_3244_prime__divprod__eq,axiom,
! [Aa,Ba,Pa] :
( hBOOL(hAPP(nat,bool,prime,Pa))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Pa),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Aa),Ba)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Pa),Aa))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Pa),Ba)) ) ) ) ).
fof(fact_3245_prime__divprod,axiom,
! [A_1,B,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),A_1))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),B)) ) ) ) ).
fof(fact_3246_prime__dvd__mult,axiom,
! [M_1,N_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),M_1))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),N_1)) ) ) ) ).
fof(fact_3247_prime__product,axiom,
! [P_4,Q_2] :
( hBOOL(hAPP(nat,bool,prime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_4),Q_2)))
=> ( P_4 = one_one(nat)
| Q_2 = one_one(nat) ) ) ).
fof(fact_3248_prime__dvd__power,axiom,
! [A_1,N_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),A_1)) ) ) ).
fof(fact_3249_prime__divexp,axiom,
! [X,N_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),X)) ) ) ).
fof(fact_3250_prime__divexp__n,axiom,
! [X,N_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N_1))) ) ) ).
fof(fact_3251_prime__exp,axiom,
! [Pa,N] :
( hBOOL(hAPP(nat,bool,prime,hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Pa),N)))
<=> ( hBOOL(hAPP(nat,bool,prime,Pa))
& N = one_one(nat) ) ) ).
fof(fact_3252_coprime__prime,axiom,
! [P_4,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
=> ~ ( hBOOL(hAPP(nat,bool,prime,P_4))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),A_1))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),B)) ) ) ).
fof(fact_3253_prime__coprime__strong,axiom,
! [N_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),N_1))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,P_4),N_1)) ) ) ).
fof(fact_3254_coprime__prime__eq,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,Aa),Ba))
<=> ! [P_5] :
~ ( hBOOL(hAPP(nat,bool,prime,P_5))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),Aa))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),Ba)) ) ) ).
fof(fact_3255_prime__impl__zprime__int,axiom,
! [A_1] :
( hBOOL(hAPP(nat,bool,prime,A_1))
=> hBOOL(hAPP(int,bool,zprime,hAPP(nat,int,semiring_1_of_nat(int),A_1))) ) ).
fof(fact_3256_zero__code__int__code,axiom,
zero_zero(quickcheck_code_int) = hAPP(int,quickcheck_code_int,number_number_of(quickcheck_code_int),pls) ).
fof(fact_3257_prime__g__one,axiom,
! [P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),P_4)) ) ).
fof(fact_3258_prime__nd__one,axiom,
! [P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,suc,zero_zero(nat)))) ) ).
fof(fact_3259_prime__factor__lt,axiom,
! [M_1,N_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( N_1 != zero_zero(nat)
=> ( N_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_4),M_1)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_1),N_1)) ) ) ) ).
fof(fact_3260_prime__def,axiom,
! [Pa] :
( hBOOL(hAPP(nat,bool,prime,Pa))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),Pa))
& ! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M_3),Pa))
=> ( M_3 = one_one(nat)
| M_3 = Pa ) ) ) ) ).
fof(fact_3261_divides__primepow,axiom,
! [D_1,K,Pa] :
( hBOOL(hAPP(nat,bool,prime,Pa))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Pa),K)))
<=> ? [I_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),I_2),K))
& D_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),Pa),I_2) ) ) ) ).
fof(fact_3262_prime__power__dvd__cancel__right,axiom,
! [N_1,A_1,B,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),N_1)),A_1)) ) ) ) ).
fof(fact_3263_prime__coprime__lt,axiom,
! [X,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X),P_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,X),P_4)) ) ) ) ).
fof(fact_3264_prime__coprime,axiom,
! [N_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( N_1 = one_one(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),N_1))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,P_4),N_1)) ) ) ).
fof(fact_3265_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_3266_prime__dvd__square,axiom,
! [M_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),M_1),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),M_1)) ) ) ).
fof(fact_3267_prime__divprod__pow,axiom,
! [N_1,A_1,B,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),N_1)),A_1))
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),N_1)),B)) ) ) ) ) ).
fof(fact_3268_prime__ge__2,axiom,
! [P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),P_4)) ) ).
fof(fact_3269_prime__odd,axiom,
! [P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( P_4 = 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_4)) ) ) ).
fof(fact_3270_div__mod__code__int__def,axiom,
! [N_1,M_1] : hAPP(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int),hAPP(quickcheck_code_int,fun(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int)),quickc495462417de_int,N_1),M_1) = hAPP(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int),hAPP(quickcheck_code_int,fun(quickcheck_code_int,product_prod(quickcheck_code_int,quickcheck_code_int)),product_Pair(quickcheck_code_int,quickcheck_code_int),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),N_1),M_1)),hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),N_1),M_1)) ).
fof(fact_3271_nat__of__def,axiom,
! [I_1] : hAPP(quickcheck_code_int,nat,quickcheck_nat_of,I_1) = hAPP(int,nat,nat_1,hAPP(quickcheck_code_int,int,quickcheck_int_of,I_1)) ).
fof(fact_3272_coprime__prime__dvd__ex,axiom,
! [X,Y] :
( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,X),Y))
=> ? [P_5] :
( hBOOL(hAPP(nat,bool,prime,P_5))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),X))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),Y)) ) ) ).
fof(fact_3273_prime__power__exp,axiom,
! [X,K_4,N_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( N_1 != zero_zero(nat)
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),X),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),K_4)
=> ? [I_2] : X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),I_2) ) ) ) ).
fof(fact_3274_prime__factor,axiom,
! [N_1] :
( N_1 != one_one(nat)
=> ? [P_5] :
( hBOOL(hAPP(nat,bool,prime,P_5))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_5),N_1)) ) ) ).
fof(fact_3275_bezout__prime,axiom,
! [A_1,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),A_1))
=> ? [X_2,Y_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),X_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),P_4),Y_2)),one_one(nat)) ) ) ).
fof(fact_3276_not__prime__ex__mk,axiom,
! [N_1] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),N_1))
& ~ hBOOL(hAPP(nat,bool,prime,N_1)) )
=> ? [M_3,K_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),M_3))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,zero_zero(nat))),K_1))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N_1))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),K_1),N_1))
& N_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_3),K_1) ) ) ).
fof(fact_3277_euclid,axiom,
! [N_1] :
? [P_5] :
( hBOOL(hAPP(nat,bool,prime,P_5))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),P_5)) ) ).
fof(fact_3278_around__zero_Opinduct,axiom,
! [P_1,A0] :
( hBOOL(hAPP(int,bool,hAPP(fun(int,fun(int,bool)),fun(int,bool),accp(int),quickc1265749348ro_rel),A0))
=> ( ! [I_2] :
( hBOOL(hAPP(int,bool,hAPP(fun(int,fun(int,bool)),fun(int,bool),accp(int),quickc1265749348ro_rel),I_2))
=> ( ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_3279_DeMoivre2,axiom,
! [R_4,A_1,N_1] : hAPP(nat,complex,hAPP(complex,fun(nat,complex),power_power(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_4),A_1)),N_1) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),R_4),N_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N_1)),A_1)) ).
fof(fact_3280_fact__int__def,axiom,
! [X] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hAPP(int,int,fact_fact(int),X) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,fact_fact(nat),hAPP(int,nat,nat_1,X))) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hAPP(int,int,fact_fact(int),X) = zero_zero(int) ) ) ).
fof(fact_3281_in__finite__psubset,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),bool,hAPP(product_prod(fun(X_a,bool),fun(X_a,bool)),fun(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),bool),member(product_prod(fun(X_a,bool),fun(X_a,bool))),hAPP(fun(X_a,bool),product_prod(fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),product_prod(fun(X_a,bool),fun(X_a,bool))),product_Pair(fun(X_a,bool),fun(X_a,bool)),A_2),B_1)),finite_psubset(X_a)))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),A_2),B_1))
& hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) ) ) ).
fof(fact_3282_fact__mono__int,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M_1),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,fact_fact(int),M_1)),hAPP(int,int,fact_fact(int),N_1))) ) ).
fof(fact_3283_fact__1__int,axiom,
hAPP(int,int,fact_fact(int),one_one(int)) = one_one(int) ).
fof(fact_3284_fact__neg__int,axiom,
! [M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M_1),zero_zero(int)))
=> hAPP(int,int,fact_fact(int),M_1) = zero_zero(int) ) ).
fof(fact_3285_fact__less__mono__int,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),M_1),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,fact_fact(int),M_1)),hAPP(int,int,fact_fact(int),N_1))) ) ) ).
fof(fact_3286_fact__ge__zero__int,axiom,
! [M_1] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,fact_fact(int),M_1))) ).
fof(fact_3287_fact__nonzero__int,axiom,
! [N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N_1))
=> hAPP(int,int,fact_fact(int),N_1) != zero_zero(int) ) ).
fof(fact_3288_transfer__nat__int__factorial__closure,axiom,
! [X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,fact_fact(int),X))) ) ).
fof(fact_3289_fact__0__int,axiom,
hAPP(int,int,fact_fact(int),zero_zero(int)) = one_one(int) ).
fof(fact_3290_transfer__int__nat__factorial,axiom,
! [X] : hAPP(int,int,fact_fact(int),hAPP(nat,int,semiring_1_of_nat(int),X)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,fact_fact(nat),X)) ).
fof(fact_3291_rcis__zero__arg,axiom,
! [R_4] : hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_4),zero_zero(real)) = hAPP(real,complex,of_real(complex),R_4) ).
fof(fact_3292_complex__mod__rcis,axiom,
! [R_4,A_1] : hAPP(complex,real,norm_norm(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_4),A_1)) = hAPP(real,real,abs_abs(real),R_4) ).
fof(fact_3293_cis__rcis__eq,axiom,
! [A_1] : hAPP(real,complex,cis,A_1) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,one_one(real)),A_1) ).
fof(fact_3294_rcis__zero__mod,axiom,
! [A_1] : hAPP(real,complex,hAPP(real,fun(real,complex),rcis,zero_zero(real)),A_1) = zero_zero(complex) ).
fof(fact_3295_fact__gt__zero__int,axiom,
! [N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,fact_fact(int),N_1))) ) ).
fof(fact_3296_fact__ge__one__int,axiom,
! [N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),hAPP(int,int,fact_fact(int),N_1))) ) ).
fof(fact_3297_fact__mono__int__aux,axiom,
! [M_1,K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,fact_fact(int),M_1)),hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M_1),K_4)))) ) ).
fof(fact_3298_dvd__fact__int,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),one_one(int)),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),M_1),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_1),hAPP(int,int,fact_fact(int),N_1))) ) ) ).
fof(fact_3299_Re__rcis,axiom,
! [R_4,A_1] : hAPP(complex,real,re,hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_4),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),hAPP(real,real,cos,A_1)) ).
fof(fact_3300_Im__rcis,axiom,
! [R_4,A_1] : hAPP(complex,real,im,hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_4),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),hAPP(real,real,sin,A_1)) ).
fof(fact_3301_rcis__mult,axiom,
! [R1,A_1,R2,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R1),A_1)),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R2),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R1),R2)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)) ).
fof(fact_3302_transfer__nat__int__factorial,axiom,
! [X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> hAPP(nat,nat,fact_fact(nat),hAPP(int,nat,nat_1,X)) = hAPP(int,nat,nat_1,hAPP(int,int,fact_fact(int),X)) ) ).
fof(fact_3303_rcis__divide,axiom,
! [R1,A_1,R2,B] : hAPP(complex,complex,hAPP(complex,fun(complex,complex),inverse_divide(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R1),A_1)),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R2),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),R1),R2)),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)) ).
fof(fact_3304_rcis__def,axiom,
! [R_4,A_1] : hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_4),A_1) = hAPP(complex,complex,hAPP(complex,fun(complex,complex),times_times(complex),hAPP(real,complex,of_real(complex),R_4)),hAPP(real,complex,cis,A_1)) ).
fof(fact_3305_fact__less__mono__int__aux,axiom,
! [M_1,K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,fact_fact(int),M_1)),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_1),one_one(int))),K_4)))) ) ) ).
fof(fact_3306_fact__plus__one__int,axiom,
! [N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N_1))
=> hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int))),hAPP(int,int,fact_fact(int),N_1)) ) ).
fof(fact_3307_fact__reduce__int,axiom,
! [N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N_1))
=> hAPP(int,int,fact_fact(int),N_1) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N_1),hAPP(int,int,fact_fact(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),N_1),one_one(int)))) ) ).
fof(fact_3308_rcis__inverse,axiom,
! [R_4,A_1] : hAPP(complex,complex,inverse_inverse(complex),hAPP(real,complex,hAPP(real,fun(real,complex),rcis,R_4),A_1)) = hAPP(real,complex,hAPP(real,fun(real,complex),rcis,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),R_4)),hAPP(real,real,uminus_uminus(real),A_1)) ).
fof(fact_3309_card__Plus__conv__if,axiom,
! [X_b,X_a,B_1,A_2] :
( ( ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) )
=> hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_1)) )
& ( ~ ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) )
=> hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)) = zero_zero(nat) ) ) ).
fof(fact_3310_length__around__zero,axiom,
! [I_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I_1))
=> hAPP(list(int),nat,size_size(list(int)),hAPP(int,list(int),quickc666637781d_zero,I_1)) = 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_1,I_1))),one_one(nat)) ) ).
fof(fact_3311_Bnor__prime,axiom,
! [Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),Pa))
=> hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),Pa)) = hAPP(int,nat,nat_1,Aa) ) ) ).
fof(fact_3312_finite__Plus__iff,axiom,
! [X_a,X_b,A_2,B_1] :
( hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) ) ) ).
fof(fact_3313_finite__Plus,axiom,
! [X_b,X_a,B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1))
=> hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1))) ) ) ).
fof(fact_3314_finite__PlusD_I1_J,axiom,
! [X_a,X_b,A_2,B_1] :
( hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).
fof(fact_3315_finite__PlusD_I2_J,axiom,
! [X_a,X_b,A_2,B_1] :
( hBOOL(hAPP(fun(sum_sum(X_a,X_b),bool),bool,finite_finite_1(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) ) ).
fof(fact_3316_Bnor__mem__zle,axiom,
! [Ba,Aa,M] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),M)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),Aa)) ) ).
fof(fact_3317_Bnor__mem__zle__swap,axiom,
! [M,Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),Ba))
=> ~ hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),M))) ) ).
fof(fact_3318_Bnor__fin,axiom,
! [Aa,M] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),M))) ).
fof(fact_3319_Bnor__mem__zg,axiom,
! [Ba,Aa,M] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),M)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba)) ) ).
fof(fact_3320_card__Plus,axiom,
! [X_b,X_a,B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1))
=> hAPP(fun(sum_sum(X_a,X_b),bool),nat,finite_card(sum_sum(X_a,X_b)),hAPP(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(sum_sum(X_a,X_b),bool)),sum_Plus(X_a,X_b),A_2),B_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)),hAPP(fun(X_b,bool),nat,finite_card(X_b),B_1)) ) ) ).
fof(fact_3321_lemma__BOLZANO,axiom,
! [Aa,Ba,P_1] :
( ! [A_3,B_3,C_4] :
( ( hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_3)))
& hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_3),C_4)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_3))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_3),C_4)) )
=> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),C_4))) )
=> ( ! [X_2] :
? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [A_3,B_3] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),B_3))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_3),A_3)),D_3)) )
=> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_3))) ) )
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),Aa),Ba))) ) ) ) ).
fof(fact_3322_lemma__BOLZANO2,axiom,
! [P_1] :
( ( ! [A_3,B_3,C_4] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_3))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),B_3),C_4))
& hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_3)))
& hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),B_3),C_4))) )
=> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),C_4))) )
& ! [X_2] :
? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [A_3,B_3] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),B_3))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),B_3),A_3)),D_3)) )
=> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_3))) ) ) )
=> ! [A_3,B_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_3),B_3))
=> hBOOL(hAPP(product_prod(real,real),bool,P_1,hAPP(real,product_prod(real,real),hAPP(real,fun(real,product_prod(real,real)),product_Pair(real,real),A_3),B_3))) ) ) ).
fof(fact_3323_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(product_prod(int,int),int,product_fst(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y))),hAPP(nat,int,semiring_1_of_nat(int),X))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_snd(int,int),hAPP(nat,product_prod(int,int),hAPP(nat,fun(nat,product_prod(int,int)),bezw,X),Y))),hAPP(nat,int,semiring_1_of_nat(int),Y))) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) ).
fof(fact_3324_gcd__dvd2__nat,axiom,
! [M_1,N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1)),N_1)) ).
fof(fact_3325_gcd__dvd1__nat,axiom,
! [M_1,N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1)),M_1)) ).
fof(fact_3326_gcd__dvd__prod__nat,axiom,
! [M_1,N_1,K_4] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N_1))) ).
fof(fact_3327_coprime__exp2__nat,axiom,
! [N_1,M_1,A_1,B] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) = 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_1),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),M_1)) = one_one(nat) ) ).
fof(fact_3328_gcd__le2__nat,axiom,
! [A_1,B] :
( B != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),B)) ) ).
fof(fact_3329_gcd__le1__nat,axiom,
! [B,A_1] :
( A_1 != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),A_1)) ) ).
fof(fact_3330_gcd__Suc__0,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),hAPP(nat,nat,suc,zero_zero(nat))) = hAPP(nat,nat,suc,zero_zero(nat)) ).
fof(fact_3331_gcd__pos__nat,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)))
<=> ( M != zero_zero(nat)
| N != zero_zero(nat) ) ) ).
fof(fact_3332_gcd__exp__nat,axiom,
! [A_1,N_1,B] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),B),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),N_1) ).
fof(fact_3333_gcd__semilattice__nat_Oinf_Oidem,axiom,
! [A_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),A_1) = A_1 ).
fof(fact_3334_gcd__idem__nat,axiom,
! [X] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),X) = X ).
fof(fact_3335_gcd__commute__nat,axiom,
! [A_1,B] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),A_1) ).
fof(fact_3336_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_3337_gcd__semilattice__nat_Oinf_Oleft__idem,axiom,
! [A_1,B] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) ).
fof(fact_3338_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_3339_gcd__left__commute__nat,axiom,
! [B,A_1,C] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),C)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),C)) ).
fof(fact_3340_gcd__semilattice__nat_Oinf__left__commute,axiom,
! [X,Y,Z] : 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)) = 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)) ).
fof(fact_3341_gcd__assoc__nat,axiom,
! [A_1,B,C] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),C) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),B),C)) ).
fof(fact_3342_gcd__semilattice__nat_Oinf__assoc,axiom,
! [X,Y,Z] : 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) = 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)) ).
fof(fact_3343_gcd__mult__distrib__nat,axiom,
! [K_4,M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),N_1)) ).
fof(fact_3344_gcd__add1__nat,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1) ).
fof(fact_3345_gcd__add2__nat,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M_1),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1) ).
fof(fact_3346_gcd__semilattice__nat_Oless__infI2,axiom,
! [A_1,B,X] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))) ) ) ).
fof(fact_3347_gcd__semilattice__nat_Oless__infI1,axiom,
! [B,A_1,X] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1)) )
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))) ) ) ).
fof(fact_3348_gcd__semilattice__nat_Ole__infE,axiom,
! [X,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)))
=> ~ ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1))
=> ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B)) ) ) ).
fof(fact_3349_dvd__gcd__D2__nat,axiom,
! [K_4,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N_1)) ) ).
fof(fact_3350_dvd__gcd__D1__nat,axiom,
! [K_4,M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M_1)) ) ).
fof(fact_3351_gcd__semilattice__nat_Oinf__mono,axiom,
! [B,D,A_1,C] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),C))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),D))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),C),D))) ) ) ).
fof(fact_3352_gcd__semilattice__nat_Oinf__greatest,axiom,
! [Z,X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Z))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y),Z))) ) ) ).
fof(fact_3353_gcd__semilattice__nat_Ole__infI,axiom,
! [B,X,A_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))) ) ) ).
fof(fact_3354_gcd__greatest__nat,axiom,
! [N_1,K_4,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1))) ) ) ).
fof(fact_3355_gcd__proj2__if__dvd__nat,axiom,
! [Y,X] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y),X))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = Y ) ).
fof(fact_3356_gcd__proj1__if__dvd__nat,axiom,
! [X,Y] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),Y))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = X ) ).
fof(fact_3357_gcd__semilattice__nat_Ole__infI2,axiom,
! [A_1,B,X] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),B),X))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X)) ) ).
fof(fact_3358_gcd__semilattice__nat_Ole__infI1,axiom,
! [B,A_1,X] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),A_1),X))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)),X)) ) ).
fof(fact_3359_gcd__unique__nat,axiom,
! [Ba,D_1,Aa] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),Aa))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_1),Ba))
& ! [E] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E),Aa))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E),Ba)) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),E),D_1)) ) )
<=> D_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Aa),Ba) ) ).
fof(fact_3360_gcd__semilattice__nat_Ole__inf__iff,axiom,
! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Y_1),Z_1)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Z_1)) ) ) ).
fof(fact_3361_gcd__greatest__iff__nat,axiom,
! [K,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N)))
<=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),M))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),N)) ) ) ).
fof(fact_3362_gcd__semilattice__nat_Ole__iff__inf,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
<=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X_1),Y_1) = X_1 ) ).
fof(fact_3363_gcd__semilattice__nat_Oinf__le2,axiom,
! [X,Y] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)),Y)) ).
fof(fact_3364_gcd__semilattice__nat_Oinf__le1,axiom,
! [X,Y] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)),X)) ).
fof(fact_3365_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_3366_gcd__1__nat,axiom,
! [M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),one_one(nat)) = one_one(nat) ).
fof(fact_3367_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_3368_gcd__lcm__complete__lattice__nat_Oinf__eq__top__iff,axiom,
! [X_1,Y_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X_1),Y_1) = zero_zero(nat)
<=> ( X_1 = zero_zero(nat)
& Y_1 = zero_zero(nat) ) ) ).
fof(fact_3369_gcd__zero__nat,axiom,
! [M,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M),N) = zero_zero(nat)
<=> ( M = zero_zero(nat)
& N = zero_zero(nat) ) ) ).
fof(fact_3370_gcd__0__nat,axiom,
! [X] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),zero_zero(nat)) = X ).
fof(fact_3371_gcd__0__left__nat,axiom,
! [X] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),X) = X ).
fof(fact_3372_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,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) ).
fof(fact_3373_coprime__Suc__nat,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,suc,N_1)),N_1) = one_one(nat) ).
fof(fact_3374_gcd__add__mult__nat,axiom,
! [M_1,K_4,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M_1)),N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1) ).
fof(fact_3375_coprime__common__divisor__nat,axiom,
! [X,A_1,B] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) = one_one(nat)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),A_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X),B))
=> X = one_one(nat) ) ) ) ).
fof(fact_3376_coprime__nat,axiom,
! [Aa,Ba] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Aa),Ba) = one_one(nat)
<=> ! [D_3] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),Aa))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),Ba)) )
<=> D_3 = one_one(nat) ) ) ).
fof(fact_3377_coprime__plus__one__nat,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_1),one_one(nat))),N_1) = one_one(nat) ).
fof(fact_3378_coprime__rmult__nat,axiom,
! [D,A_1,B] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)) = one_one(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D),B) = one_one(nat) ) ).
fof(fact_3379_coprime__lmult__nat,axiom,
! [D,A_1,B] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)) = one_one(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D),A_1) = one_one(nat) ) ).
fof(fact_3380_coprime__crossproduct__nat,axiom,
! [Ba,C_1,Aa,D_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Aa),D_1) = one_one(nat)
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Ba),C_1) = one_one(nat)
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Aa),C_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Ba),D_1)
<=> ( Aa = Ba
& C_1 = D_1 ) ) ) ) ).
fof(fact_3381_coprime__mult__nat,axiom,
! [B,D,A_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D),A_1) = one_one(nat)
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D),B) = one_one(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B)) = one_one(nat) ) ) ).
fof(fact_3382_gcd__mult__cancel__nat,axiom,
! [M_1,K_4,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),K_4),N_1) = one_one(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K_4),M_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1) ) ).
fof(fact_3383_coprime__mul__eq__nat,axiom,
! [D_1,Aa,Ba] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),Aa),Ba)) = one_one(nat)
<=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_1),Aa) = one_one(nat)
& hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D_1),Ba) = one_one(nat) ) ) ).
fof(fact_3384_gcd__diff1__nat,axiom,
! [N_1,M_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_1),M_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),N_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1) ) ).
fof(fact_3385_gcd__diff2__nat,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),M_1)),N_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1) ) ).
fof(fact_3386_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,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) ) ) ).
fof(fact_3387_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,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) ) ).
fof(fact_3388_coprime__exp__nat,axiom,
! [N_1,D,A_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D),A_1) = one_one(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),D),hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),A_1),N_1)) = one_one(nat) ) ).
fof(fact_3389_gcd__coprime__nat,axiom,
! [B_4,A_5,A_1,B] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) != zero_zero(nat)
=> ( A_1 = 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_1),B))
=> ( B = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_5),B_4) = one_one(nat) ) ) ) ).
fof(fact_3390_coprime__minus__one__nat,axiom,
! [N_1] :
( N_1 != zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),one_one(nat))),N_1) = one_one(nat) ) ).
fof(fact_3391_divides__mult__nat,axiom,
! [N_1,M_1,R_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),M_1),R_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),N_1),R_4))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_1),N_1) = one_one(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)),R_4)) ) ) ) ).
fof(fact_3392_coprime__dvd__mult__iff__nat,axiom,
! [M,K,N] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),K),N) = one_one(nat)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M),N)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K),M)) ) ) ).
fof(fact_3393_coprime__dvd__mult__nat,axiom,
! [M_1,K_4,N_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),K_4),N_1) = one_one(nat)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),M_1),N_1)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),K_4),M_1)) ) ) ).
fof(fact_3394_div__gcd__coprime__nat,axiom,
! [B,A_1] :
( ( A_1 != zero_zero(nat)
| B != 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_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),B),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))) = one_one(nat) ) ).
fof(fact_3395_invertible__coprime__nat,axiom,
! [X,Y,M_1] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),X),Y)),M_1) = one_one(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),M_1) = one_one(nat) ) ).
fof(fact_3396_coprime__Suc__0__nat,axiom,
! [Aa,Ba] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),Aa),Ba) = one_one(nat)
<=> ! [D_3] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),Aa))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),D_3),Ba)) )
<=> D_3 = hAPP(nat,nat,suc,zero_zero(nat)) ) ) ).
fof(fact_3397_gcd__coprime__exists__nat,axiom,
! [A_1,B] :
( hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B) != zero_zero(nat)
=> ? [A_4,B_2] :
( A_1 = 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_1),B))
& B = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B))
& hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_4),B_2) = one_one(nat) ) ) ).
fof(fact_3398_bezout__nat,axiom,
! [B,A_1] :
( A_1 != zero_zero(nat)
=> ? [X_2,Y_2] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),X_2) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),B),Y_2)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),A_1),B)) ) ).
fof(fact_3399_Cauchy__iff2,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,real),bool,cauchy(real),X_3))
<=> ! [J_1] :
? [M_4] :
! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),M_3))
=> ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,X_3,M_3)),hAPP(nat,real,X_3,N_2)))),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,J_1))))) ) ) ) ).
fof(fact_3400_CauchyD,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [E_2,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E_2))
=> ? [M_4] :
! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),M_3))
=> ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_3,M_3)),hAPP(nat,X_a,X_3,N_2)))),E_2)) ) ) ) ) ) ).
fof(fact_3401_Cauchy__iff,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3))
<=> ! [E] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
=> ? [M_4] :
! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),M_3))
=> ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_3,M_3)),hAPP(nat,X_a,X_3,N_2)))),E)) ) ) ) ) ) ).
fof(fact_3402_CauchyI,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( ! [E] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
=> ? [M_4] :
! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),M_3))
=> ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_3,M_3)),hAPP(nat,X_a,X_3,N_2)))),E)) ) ) )
=> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3)) ) ) ).
fof(fact_3403_in__measure,axiom,
! [X_a,X_1,Y_1,F] :
( hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),X_1),Y_1)),hAPP(fun(X_a,nat),fun(product_prod(X_a,X_a),bool),measure(X_a),F)))
<=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(X_a,nat,F,X_1)),hAPP(X_a,nat,F,Y_1))) ) ).
fof(fact_3404_ln__powr__bound,axiom,
! [A_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,ln,X)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),A_1))) ) ) ).
fof(fact_3405_ln__powr__bound2,axiom,
! [A_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,ln,X)),A_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),powr,A_1),A_1)),X))) ) ) ).
fof(fact_3406_powr__one__eq__one,axiom,
! [A_1] : hAPP(real,real,hAPP(real,fun(real,real),powr,one_one(real)),A_1) = one_one(real) ).
fof(fact_3407_powr__not__zero,axiom,
! [X,A_1] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1) != zero_zero(real) ).
fof(fact_3408_powr__powr__swap,axiom,
! [X,A_1,B] : hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),B) = hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)),A_1) ).
fof(fact_3409_powr__powr,axiom,
! [X,A_1,B] : hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),B) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B)) ).
fof(fact_3410_powr__ge__pzero,axiom,
! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),Y))) ).
fof(fact_3411_powr__less__mono2__neg,axiom,
! [Y,X,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),zero_zero(real)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1))) ) ) ) ).
fof(fact_3412_powr__less__mono2,axiom,
! [Y,X,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1))) ) ) ) ).
fof(fact_3413_powr__gt__zero,axiom,
! [X,A_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1))) ).
fof(fact_3414_powr__mono,axiom,
! [X,A_1,B] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),A_1),B))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B))) ) ) ).
fof(fact_3415_powr__zero__eq__one,axiom,
! [X] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),zero_zero(real)) = one_one(real) ).
fof(fact_3416_powr__less__cancel,axiom,
! [X,A_1,B] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),B)) ) ) ).
fof(fact_3417_powr__less__mono,axiom,
! [X,A_1,B] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),A_1),B))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B))) ) ) ).
fof(fact_3418_powr__less__cancel__iff,axiom,
! [Aa,Ba,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X_1),Aa)),hAPP(real,real,hAPP(real,fun(real,real),powr,X_1),Ba)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),Ba)) ) ) ).
fof(fact_3419_powr__add,axiom,
! [X,A_1,B] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)) ).
fof(fact_3420_powr__divide2,axiom,
! [X,A_1,B] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),B)) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)) ).
fof(fact_3421_powr__minus,axiom,
! [X,A_1] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,uminus_uminus(real),A_1)) = hAPP(real,real,inverse_inverse(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)) ).
fof(fact_3422_powr__mono2,axiom,
! [Y,X,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X),Y))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1))) ) ) ) ).
fof(fact_3423_powr__mult,axiom,
! [A_1,Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),X),Y)),A_1) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1)) ) ) ).
fof(fact_3424_ge__one__powr__ge__zero,axiom,
! [A_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1))) ) ) ).
fof(fact_3425_powr__le__cancel__iff,axiom,
! [Aa,Ba,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X_1),Aa)),hAPP(real,real,hAPP(real,fun(real,real),powr,X_1),Ba)))
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba)) ) ) ).
fof(fact_3426_powr__one__gt__zero__iff,axiom,
! [X_1] :
( hAPP(real,real,hAPP(real,fun(real,real),powr,X_1),one_one(real)) = X_1
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1)) ) ).
fof(fact_3427_powr__divide,axiom,
! [A_1,Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> hAPP(real,real,hAPP(real,fun(real,real),powr,hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),X),Y)),A_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)),hAPP(real,real,hAPP(real,fun(real,real),powr,Y),A_1)) ) ) ).
fof(fact_3428_powr__minus__divide,axiom,
! [X,A_1] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(real,real,uminus_uminus(real),A_1)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1)) ).
fof(fact_3429_powr__def,axiom,
! [X,A_1] : hAPP(real,real,hAPP(real,fun(real,real),powr,X),A_1) = hAPP(real,real,exp(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),hAPP(real,real,ln,X))) ).
fof(fact_3430_powr__realpow,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(nat,real,real_1(nat),N_1)) = hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N_1) ) ).
fof(fact_3431_ln__powr,axiom,
! [Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
=> hAPP(real,real,ln,hAPP(real,real,hAPP(real,fun(real,real),powr,X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),hAPP(real,real,ln,X)) ) ) ).
fof(fact_3432_log__powr__cancel,axiom,
! [Y,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ( A_1 != one_one(real)
=> hAPP(real,real,hAPP(real,fun(real,real),log,A_1),hAPP(real,real,hAPP(real,fun(real,real),powr,A_1),Y)) = Y ) ) ).
fof(fact_3433_powr__log__cancel,axiom,
! [X,A_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),A_1))
=> ( A_1 != one_one(real)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> hAPP(real,real,hAPP(real,fun(real,real),powr,A_1),hAPP(real,real,hAPP(real,fun(real,real),log,A_1),X)) = X ) ) ) ).
fof(fact_3434_log__powr,axiom,
! [B,Y,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
=> hAPP(real,real,hAPP(real,fun(real,real),log,B),hAPP(real,real,hAPP(real,fun(real,real),powr,X),Y)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),Y),hAPP(real,real,hAPP(real,fun(real,real),log,B),X)) ) ) ).
fof(fact_3435_powr__realpow2,axiom,
! [N_1,X] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_1))
=> ( ( X = zero_zero(real)
=> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N_1) = zero_zero(real) )
& ( X != zero_zero(real)
=> hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X),N_1) = hAPP(real,real,hAPP(real,fun(real,real),powr,X),hAPP(nat,real,real_1(nat),N_1)) ) ) ) ) ).
fof(fact_3436_coprime__pow,axiom,
! [C,N_1,A_1,B] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),coprime,A_1),B))
=> ( hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),A_1),B) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),C),N_1)
=> ? [R_2,S_1] :
( A_1 = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),R_2),N_1)
& B = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),S_1),N_1) ) ) ) ).
fof(fact_3437_prime__power__mult,axiom,
! [X,Y,K_4,P_4] :
( hBOOL(hAPP(nat,bool,prime,P_4))
=> ( 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_4),K_4)
=> ? [I_2,J_1] :
( X = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),I_2)
& Y = hAPP(nat,nat,hAPP(nat,fun(nat,nat),power_power(nat),P_4),J_1) ) ) ) ).
fof(fact_3438_Euler__Fermat,axiom,
! [X,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M_1))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,X),M_1) = one_one(int)
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X),hAPP(int,nat,phi,M_1))),one_one(int)),M_1)) ) ) ).
fof(fact_3439_zgcd__zdvd2,axiom,
! [I_1,J_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2)),J_2)) ).
fof(fact_3440_zgcd__zdvd1,axiom,
! [I_1,J_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2)),I_1)) ).
fof(fact_3441_zgcd__zmult__distrib2__abs,axiom,
! [K_4,M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1)) ).
fof(fact_3442_zgcd__pos,axiom,
! [I_1,J_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2))) ).
fof(fact_3443_zgcd__self,axiom,
! [M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_1))
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),M_1) = M_1 ) ).
fof(fact_3444_zgcd__geq__zero,axiom,
! [X,Y] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,X),Y))) ).
fof(fact_3445_zgcd__0,axiom,
! [M_1] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),zero_zero(int)) = hAPP(int,int,abs_abs(int),M_1) ).
fof(fact_3446_zgcd__0__left,axiom,
! [M_1] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,zero_zero(int)),M_1) = hAPP(int,int,abs_abs(int),M_1) ).
fof(fact_3447_zgcd__zdvd__zgcd__zmult,axiom,
! [M_1,N_1,K_4] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M_1)),N_1))) ).
fof(fact_3448_zgcd__1__power__left__distrib,axiom,
! [N_1,A_1,B] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N_1)),B) = one_one(int) ) ).
fof(fact_3449_zgcd__1__power__distrib,axiom,
! [N_1,A_1,B] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N_1)) = one_one(int) ) ).
fof(fact_3450_zgcd__power__distrib,axiom,
! [A_1,B,N_1] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B)),N_1) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N_1)) ).
fof(fact_3451_zgcd__commute,axiom,
! [I_1,J_2] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,J_2),I_1) ).
fof(fact_3452_zgcd__left__commute,axiom,
! [K_4,M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),N_1)) ).
fof(fact_3453_zgcd__assoc,axiom,
! [K_4,M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),M_1)),N_1) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1)) ).
fof(fact_3454_zgcd__zminus2,axiom,
! [I_1,J_2] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),hAPP(int,int,uminus_uminus(int),J_2)) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2) ).
fof(fact_3455_zgcd__zminus,axiom,
! [I_1,J_2] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,uminus_uminus(int),I_1)),J_2) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2) ).
fof(fact_3456_zgcd__zadd__zmult,axiom,
! [M_1,N_1,K_4] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),N_1),K_4))),N_1) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1) ).
fof(fact_3457_zgcd__greatest,axiom,
! [N_1,K_4,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1))) ) ) ).
fof(fact_3458_zgcd__greatest__iff,axiom,
! [K,M,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N)))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),M))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),N)) ) ) ).
fof(fact_3459_zgcd__zgcd__zmult,axiom,
! [N_1,K_4,M_1] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),M_1) = one_one(int)
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N_1),M_1) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N_1)),M_1) = one_one(int) ) ) ).
fof(fact_3460_zgcd__zmult__cancel,axiom,
! [M_1,K_4,N_1] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),N_1) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M_1)),N_1) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1) ) ).
fof(fact_3461_zgcd__1,axiom,
! [M_1] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),one_one(int)) = one_one(int) ).
fof(fact_3462_zgcd__1__left,axiom,
! [M_1] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,one_one(int)),M_1) = one_one(int) ).
fof(fact_3463_zgcd0,axiom,
! [I,J] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I),J) = zero_zero(int)
<=> ( I = zero_zero(int)
& J = zero_zero(int) ) ) ).
fof(fact_3464_zgcd__eq,axiom,
! [M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M_1),N_1)) ).
fof(fact_3465_zgcd__zmult__eq__self2,axiom,
! [N_1,K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N_1)),K_4) = K_4 ) ).
fof(fact_3466_zgcd__zmult__distrib2,axiom,
! [M_1,N_1,K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
=> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N_1)) ) ).
fof(fact_3467_zgcd__zmult__eq__self,axiom,
! [N_1,K_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K_4))
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N_1)) = K_4 ) ).
fof(fact_3468_zdvd__iff__zgcd,axiom,
! [N,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M),N))
<=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N),M) = M ) ) ).
fof(fact_3469_zgcd__non__0,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N_1))
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1) = hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N_1),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),M_1),N_1)) ) ).
fof(fact_3470_div__zgcd__relprime,axiom,
! [B,A_1] :
( ( A_1 != zero_zero(int)
| B != zero_zero(int) )
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B))) = one_one(int) ) ).
fof(fact_3471_zcong__zgcd__zmult__zmod,axiom,
! [N_1,A_1,B,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),N_1))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1) = one_one(int)
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),N_1))) ) ) ) ).
fof(fact_3472_zgcd__0__1__iff,axiom,
! [M] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,zero_zero(int)),M) = one_one(int)
<=> hAPP(int,int,abs_abs(int),M) = one_one(int) ) ).
fof(fact_3473_zgcd__zmult__zdvd__zgcd,axiom,
! [M_1,K_4,N_1] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K_4),N_1) = one_one(int)
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M_1)),N_1)),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1))) ) ).
fof(fact_3474_zrelprime__zdvd__zmult,axiom,
! [M_1,N_1,K_4] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N_1),K_4) = one_one(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),N_1)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M_1)) ) ) ).
fof(fact_3475_zrelprime__dvd__mult,axiom,
! [K_4,I_1,J_2] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,I_1),J_2) = one_one(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),J_2)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),K_4)) ) ) ).
fof(fact_3476_zgcd__code,axiom,
! [K,L] : hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K),L) = hAPP(int,int,abs_abs(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),L),zero_zero(int))),K),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,L),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,abs_abs(int),K)),hAPP(int,int,abs_abs(int),L))))) ).
fof(fact_3477_zprime__imp__zrelprime,axiom,
! [N_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_4),N_1))
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N_1),P_4) = one_one(int) ) ) ).
fof(fact_3478_zgcd1__iff__no__common__primedivisor,axiom,
! [Aa,Ba] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Aa),Ba) = one_one(int)
<=> ~ ? [P_5] :
( hBOOL(hAPP(int,bool,zprime,P_5))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),Aa))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),P_5),Ba)) ) ) ).
fof(fact_3479_zgcd__zcong__zgcd,axiom,
! [B,A_1,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M_1))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),M_1) = one_one(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,A_1),B),M_1))
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,B),M_1) = one_one(int) ) ) ) ).
fof(fact_3480_zless__zprime__imp__zrelprime,axiom,
! [N_1,P_4] :
( hBOOL(hAPP(int,bool,zprime,P_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),N_1),P_4))
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N_1),P_4) = one_one(int) ) ) ) ).
fof(fact_3481_zcong__cancel2,axiom,
! [Aa,Ba,K,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K),M) = one_one(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),Aa)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K),Ba)),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M)) ) ) ) ).
fof(fact_3482_zcong__cancel,axiom,
! [Aa,Ba,K,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,K),M) = one_one(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Aa),K)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ba),K)),M))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M)) ) ) ) ).
fof(fact_3483_zrelprime__zdvd__zmult__aux,axiom,
! [M_1,N_1,K_4] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,N_1),K_4) = one_one(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),N_1)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M_1)) ) ) ) ).
fof(fact_3484_Bnor__mem__if,axiom,
! [Aa,Ba,M] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Ba),M) = one_one(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),Aa))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),M))) ) ) ) ).
fof(fact_3485_int__relprime__odd__power__divisors,axiom,
! [Aa,Ba,C_1,X_1] :
( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Aa),Ba) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C_1),hAPP(int,nat,nat_1,X_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(int,nat,nat_1,X_1)))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Aa),Ba) = one_one(int)
=> ? [K_1] : Aa = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),hAPP(int,nat,nat_1,X_1)) ) ) ) ) ).
fof(fact_3486_xzgcd__correct,axiom,
! [M,K,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M),N) = K
<=> ? [S_1,T_5] : hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),xzgcd,M),N) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),K),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),S_1),T_5)) ) ) ).
fof(fact_3487_int__relprime__power__divisors,axiom,
! [A_1,B,C,N_1] :
( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),C),N_1)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N_1))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) = one_one(int)
=> ? [K_1] : hAPP(int,int,abs_abs(int),A_1) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),N_1) ) ) ) ).
fof(fact_3488_xzgcd__correct__aux2,axiom,
! [M_1,N_1,R_5,R_4,S_5,S_4,T_7,T_6,K_4] :
( ? [Sn,Tn] : hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M_1),N_1),R_5),R_4),S_5),S_4),T_7),T_6) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),K_4),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Sn),Tn))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),R_4))
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,R_5),R_4) = K_4 ) ) ).
fof(fact_3489_zcong__lineq__unique,axiom,
! [B,A_1,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N_1))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),N_1) = one_one(int)
=> ? [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),N_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),X_2)),B),N_1))
& ! [Y_2] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_2))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Y_2),N_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),Y_2)),B),N_1)) )
=> Y_2 = X_2 ) ) ) ) ).
fof(fact_3490_zcong__lineq__ex,axiom,
! [A_1,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N_1))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),N_1) = one_one(int)
=> ? [X_2] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),X_2)),one_one(int)),N_1)) ) ) ).
fof(fact_3491_xzgcd__correct__aux1,axiom,
! [M_1,N_1,S_5,S_4,T_7,T_6,R_5,R_4,K_4] :
( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,R_5),R_4) = K_4
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),R_4))
=> ? [Sn,Tn] : hAPP(int,product_prod(int,product_prod(int,int)),hAPP(int,fun(int,product_prod(int,product_prod(int,int))),hAPP(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))),hAPP(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int)))))))),hAPP(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,fun(int,product_prod(int,product_prod(int,int))))))))),xzgcda,M_1),N_1),R_5),R_4),S_5),S_4),T_7),T_6) = hAPP(product_prod(int,int),product_prod(int,product_prod(int,int)),hAPP(int,fun(product_prod(int,int),product_prod(int,product_prod(int,int))),product_Pair(int,product_prod(int,int)),K_4),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Sn),Tn)) ) ) ).
fof(fact_3492_int__triple__relprime__odd__power__divisors,axiom,
! [Aa,Ba,C_1,D_1,X_1] :
( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Aa),Ba)),C_1) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),D_1),hAPP(int,nat,nat_1,X_1))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),hAPP(int,nat,nat_1,X_1)))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),zOdd))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Aa),Ba) = one_one(int)
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Ba),C_1) = one_one(int)
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,C_1),Aa) = one_one(int)
=> ? [K_1,L_3,M_3] :
( Aa = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),hAPP(int,nat,nat_1,X_1))
& Ba = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),L_3),hAPP(int,nat,nat_1,X_1))
& C_1 = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),M_3),hAPP(int,nat,nat_1,X_1)) ) ) ) ) ) ) ) ).
fof(fact_3493_zgcd__ex__linear,axiom,
! [M_1,K_4,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N_1))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,M_1),N_1) = K_4
=> ? [S_1,T_5] : K_4 = 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_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),T_5),N_1)) ) ) ).
fof(fact_3494_int__triple__relprime__power__divisors,axiom,
! [A_1,B,C,D,N_1] :
( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)),C) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),D),N_1)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N_1))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B) = one_one(int)
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,B),C) = one_one(int)
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,C),A_1) = one_one(int)
=> ? [K_1,L_3,M_3] :
( hAPP(int,int,abs_abs(int),A_1) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),K_1),N_1)
& hAPP(int,int,abs_abs(int),B) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),L_3),N_1)
& hAPP(int,int,abs_abs(int),C) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),M_3),N_1) ) ) ) ) ) ) ).
fof(fact_3495_make__zrelprime,axiom,
! [B,A_1] :
( ( A_1 != zero_zero(int)
| B != zero_zero(int) )
=> ? [C_4,D_3] :
( A_1 = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B)),C_4)
& B = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,A_1),B)),D_3)
& hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,C_4),D_3) = one_one(int) ) ) ).
fof(fact_3496_Bseq__iff2,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3))
<=> ? [K_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_1))
& ? [X_2] :
! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,X_3,N_2)),hAPP(X_a,X_a,uminus_uminus(X_a),X_2)))),K_1)) ) ) ) ).
fof(fact_3497_Cauchy__Bseq,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3)) ) ) ).
fof(fact_3498_Bseq__def,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3))
<=> ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_3,N_2))),K_2)) ) ) ) ).
fof(fact_3499_Bseq__iff,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3))
<=> ? [N_3] :
! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_3,N_2))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_3)))) ) ) ).
fof(fact_3500_Bseq__iff1a,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3))
<=> ? [N_3] :
! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_3,N_2))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,suc,N_3)))) ) ) ).
fof(fact_3501_Bseq__iff3,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3))
<=> ? [K_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_1))
& ? [N_3] :
! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,X_3,N_2)),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,X_3,N_3))))),K_1)) ) ) ) ).
fof(fact_3502_BseqD,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3))
=> ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_3,N_2))),K_2)) ) ) ) ).
fof(fact_3503_BseqI,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3,K_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_3))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_3,N_2))),K_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3)) ) ) ) ).
fof(fact_3504_BseqI2_H,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3,K_3,N_4] :
( ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_4),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_3,N_2))),K_3)) )
=> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3)) ) ) ).
fof(fact_3505_BseqI_H,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3,K_3] :
( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_3,N_2))),K_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3)) ) ) ).
fof(fact_3506_BseqE,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3))
=> ~ ! [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
=> ~ ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,X_3,N_2))),K_2)) ) ) ) ).
fof(fact_3507_polar__Ex,axiom,
! [Y,X] :
? [R_2,A_3] :
( X = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_2),hAPP(real,real,cos,A_3))
& Y = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_2),hAPP(real,real,sin,A_3)) ) ).
fof(fact_3508_dist__triangle__half__l,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X2_1,X1_1,Y,E_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X1_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X2_1),Y)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X1_1),X2_1)),E_1)) ) ) ) ).
fof(fact_3509_dist__triangle__half__r,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X2_1,Y,X1_1,E_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),X1_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),X2_1)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),E_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X1_1),X2_1)),E_1)) ) ) ) ).
fof(fact_3510_dist__not__less__zero,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X,Y] : ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),zero_zero(real))) ) ).
fof(fact_3511_dist__nz,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_1,Y_1] :
( ti(X_a,X_1) != ti(X_a,Y_1)
<=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),Y_1))) ) ) ).
fof(fact_3512_zero__less__dist__iff,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),Y_1)))
<=> ti(X_a,X_1) != ti(X_a,Y_1) ) ) ).
fof(fact_3513_dist__pos__lt,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X,Y] :
( ti(X_a,X) != ti(X_a,Y)
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y))) ) ) ).
fof(fact_3514_dist__norm,axiom,
! [X_a] :
( dist_norm(X_a)
=> ! [X,Y] : hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y) = hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)) ) ).
fof(fact_3515_dist__eq__0__iff,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_1,Y_1] :
( hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),Y_1) = zero_zero(real)
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ).
fof(fact_3516_dist__self,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X] : hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),X) = zero_zero(real) ) ).
fof(fact_3517_dist__commute,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X,Y] : hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y) = hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),X) ) ).
fof(fact_3518_dist__le__zero__iff,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_1),Y_1)),zero_zero(real)))
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ).
fof(fact_3519_zero__le__dist,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y))) ) ).
fof(fact_3520_dist__triangle__lt,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X,Z,Y,E_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z))),E_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),E_1)) ) ) ).
fof(fact_3521_dist__triangle__le,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X,Z,Y,E_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z))),E_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),E_1)) ) ) ).
fof(fact_3522_dist__triangle__alt,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [Y,Z,X] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z)))) ) ).
fof(fact_3523_dist__triangle,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X,Z,Y] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z)))) ) ).
fof(fact_3524_dist__triangle2,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X,Y,Z] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Z)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),Y),Z)))) ) ).
fof(fact_3525_dist__triangle3,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X,Y,A_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),Y)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),A_1),X)),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),A_1),Y)))) ) ).
fof(fact_3526_dist__real__def,axiom,
! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),dist_dist(real),X),Y) = hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)) ).
fof(fact_3527_dist__complex__def,axiom,
! [X,Y] : hAPP(complex,real,hAPP(complex,fun(complex,real),dist_dist(complex),X),Y) = hAPP(complex,real,norm_norm(complex),hAPP(complex,complex,hAPP(complex,fun(complex,complex),minus_minus(complex),X),Y)) ).
fof(fact_3528_Cauchy__def,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3))
<=> ! [E] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
=> ? [M_4] :
! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),M_3))
=> ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_3,M_3)),hAPP(nat,X_a,X_3,N_2))),E)) ) ) ) ) ) ).
fof(fact_3529_metric__CauchyD,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [E_2,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E_2))
=> ? [M_4] :
! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),M_3))
=> ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_3,M_3)),hAPP(nat,X_a,X_3,N_2))),E_2)) ) ) ) ) ) ).
fof(fact_3530_norm__conv__dist,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X] : hAPP(X_a,real,norm_norm(X_a),X) = hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X),zero_zero(X_a)) ) ).
fof(fact_3531_metric__CauchyI,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_3] :
( ! [E] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
=> ? [M_4] :
! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),M_3))
=> ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_4),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_3,M_3)),hAPP(nat,X_a,X_3,N_2))),E)) ) ) )
=> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3)) ) ) ).
fof(fact_3532_int__of__nat__induct,axiom,
! [Z_1,P_1] :
( ! [N_2] : hBOOL(hAPP(int,bool,P_1,hAPP(nat,int,semiring_1_of_nat(int),N_2)))
=> ( ! [N_2] : hBOOL(hAPP(int,bool,P_1,hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_2)))))
=> hBOOL(hAPP(int,bool,P_1,Z_1)) ) ) ).
fof(fact_3533_int__cases,axiom,
! [Z] :
( ! [N_2] : Z != hAPP(nat,int,semiring_1_of_nat(int),N_2)
=> ~ ! [N_2] : Z != hAPP(int,int,uminus_uminus(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,suc,N_2))) ) ).
fof(fact_3534_DERIV__nonneg__imp__nonincreasing,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Ba)) )
=> ? [Y_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y_2)) ) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,Aa)),hAPP(real,real,F,Ba))) ) ) ).
fof(fact_3535_DERIV__nonpos__imp__nonincreasing,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Ba)) )
=> ? [Y_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_2),zero_zero(real))) ) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,F,Ba)),hAPP(real,real,F,Aa))) ) ) ).
fof(fact_3536_gcd__nat__induct,axiom,
! [M,N,P_1] :
( ! [M_3] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,M_3),zero_zero(nat)))
=> ( ! [M_3,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N_2))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,N_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),M_3),N_2)))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,M_3),N_2)) ) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,M),N)) ) ) ).
fof(fact_3537_code__numeral_Osimps_I5_J,axiom,
! [X_a,F1,F2,Code_numeral] : hAPP(code_code_numeral,X_a,hAPP(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a),hAPP(X_a,fun(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a)),code_c600343653l_case(X_a),F1),F2),hAPP(code_code_numeral,code_code_numeral,code_S1047413653umeral,Code_numeral)) = hAPP(code_code_numeral,X_a,F2,Code_numeral) ).
fof(fact_3538_divides__fact,axiom,
! [N_1,P_4] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P_4))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),P_4),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),P_4),hAPP(nat,nat,fact,N_1))) ) ) ).
fof(fact_3539_mod__code__int__def,axiom,
! [N_1,M_1] : hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_mod(quickcheck_code_int),N_1),M_1) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N_1)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M_1))) ).
fof(fact_3540_Quickcheck__Narrowing_Oof__int__inject,axiom,
! [N,M] :
( hAPP(int,quickcheck_code_int,quickcheck_of_int,N) = hAPP(int,quickcheck_code_int,quickcheck_of_int,M)
<=> N = M ) ).
fof(fact_3541_int__of__inverse,axiom,
! [X] : hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(quickcheck_code_int,int,quickcheck_int_of,X)) = X ).
fof(fact_3542_of__int__int__of,axiom,
! [K_4] : hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(quickcheck_code_int,int,quickcheck_int_of,K_4)) = K_4 ).
fof(fact_3543_int__of__of__int,axiom,
! [N_1] : hAPP(quickcheck_code_int,int,quickcheck_int_of,hAPP(int,quickcheck_code_int,quickcheck_of_int,N_1)) = N_1 ).
fof(fact_3544_number__of__code__int__def,axiom,
number_number_of(quickcheck_code_int) = quickcheck_of_int ).
fof(fact_3545_fact__mono,axiom,
! [M_1,N_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_1),N_1))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,fact,M_1)),hAPP(nat,nat,fact,N_1))) ) ).
fof(fact_3546_times__code__int__def,axiom,
! [N_1,M_1] : hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),times_times(quickcheck_code_int),N_1),M_1) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N_1)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M_1))) ).
fof(fact_3547_plus__code__int__def,axiom,
! [N_1,M_1] : hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),plus_plus(quickcheck_code_int),N_1),M_1) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N_1)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M_1))) ).
fof(fact_3548_minus__code__int__def,axiom,
! [N_1,M_1] : hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),minus_minus(quickcheck_code_int),N_1),M_1) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N_1)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M_1))) ).
fof(fact_3549_zero__code__int__def,axiom,
zero_zero(quickcheck_code_int) = hAPP(int,quickcheck_code_int,quickcheck_of_int,zero_zero(int)) ).
fof(fact_3550_fact__lt,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(nat,nat,fact,N_1))) ).
fof(fact_3551_fact_Osimps_I1_J,axiom,
hAPP(nat,nat,fact,zero_zero(nat)) = one_one(nat) ).
fof(fact_3552_fact_Osimps_I2_J,axiom,
! [N_1] : hAPP(nat,nat,fact,hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,N_1)),hAPP(nat,nat,fact,N_1)) ).
fof(fact_3553_fact__le,axiom,
! [N_1] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),hAPP(nat,nat,fact,N_1))) ).
fof(fact_3554_one__code__int__def,axiom,
one_one(quickcheck_code_int) = hAPP(int,quickcheck_code_int,quickcheck_of_int,one_one(int)) ).
fof(fact_3555_code__numeral_Osimps_I4_J,axiom,
! [X_a,F1,F2] : hAPP(code_code_numeral,X_a,hAPP(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a),hAPP(X_a,fun(fun(code_code_numeral,X_a),fun(code_code_numeral,X_a)),code_c600343653l_case(X_a),F1),F2),zero_zero(code_code_numeral)) = ti(X_a,F1) ).
fof(fact_3556_div__code__int__def,axiom,
! [N_1,M_1] : hAPP(quickcheck_code_int,quickcheck_code_int,hAPP(quickcheck_code_int,fun(quickcheck_code_int,quickcheck_code_int),div_div(quickcheck_code_int),N_1),M_1) = hAPP(int,quickcheck_code_int,quickcheck_of_int,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(quickcheck_code_int,int,quickcheck_int_of,N_1)),hAPP(quickcheck_code_int,int,quickcheck_int_of,M_1))) ).
fof(fact_3557_euclid__bound,axiom,
! [N_1] :
? [P_5] :
( hBOOL(hAPP(nat,bool,prime,P_5))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N_1),P_5))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),P_5),hAPP(nat,nat,suc,hAPP(nat,nat,fact,N_1)))) ) ).
fof(fact_3558_weak__decr__stable,axiom,
! [F] :
( ! [I_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,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_3559_norm__frac_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),norm_frac_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
=> ( ! [A_3,B_3] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),norm_frac_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_3),B_3)))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B_3),zero_zero(int)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,hAPP(int,int,uminus_uminus(int),A_3)),hAPP(int,int,uminus_uminus(int),B_3))) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A_3),B_3)) ) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1)) ) ) ).
fof(fact_3560_Nitpick_Orefl_H__def,axiom,
! [X_a,R_1] :
( hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,refl(X_a),R_1))
<=> ! [X_2] : hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),X_2),X_2)),R_1)) ) ).
fof(fact_3561_upto_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),upto_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
=> ( ! [I_2,J_1] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),upto_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),I_2),J_1)))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),I_2),J_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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(int,fun(int,bool),P_1,I_2),J_1)) ) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1)) ) ) ).
fof(fact_3562_small__lazy_H_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),lazy_small_lazy_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
=> ( ! [D_3,I_2] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),lazy_small_lazy_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),D_3),I_2)))
=> ( ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),D_3),I_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,D_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),I_2),one_one(int)))) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,D_3),I_2)) ) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),P_1,A0),A1)) ) ) ).
fof(fact_3563_neq__if__length__neq,axiom,
! [X_a,Xs,Ys] :
( hAPP(list(X_a),nat,size_size(list(X_a)),Xs) != hAPP(list(X_a),nat,size_size(list(X_a)),Ys)
=> Xs != Ys ) ).
fof(fact_3564_finite__maxlen,axiom,
! [X_a,M_2] :
( hBOOL(hAPP(fun(list(X_a),bool),bool,finite_finite_1(list(X_a)),M_2))
=> ? [N_2] :
! [X_2] :
( hBOOL(hAPP(fun(list(X_a),bool),bool,hAPP(list(X_a),fun(fun(list(X_a),bool),bool),member(list(X_a)),X_2),M_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(list(X_a),nat,size_size(list(X_a)),X_2)),N_2)) ) ) ).
fof(fact_3565_of__int__of__nat,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [K_4] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_4),zero_zero(int)))
=> hAPP(int,X_a,ring_1_of_int(X_a),K_4) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,hAPP(int,int,uminus_uminus(int),K_4)))) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K_4),zero_zero(int)))
=> hAPP(int,X_a,ring_1_of_int(X_a),K_4) = hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,K_4)) ) ) ) ).
fof(fact_3566_norRRset__def,axiom,
! [M] : hAPP(int,fun(int,bool),norRRset,M) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),M),one_one(int))),M) ).
fof(fact_3567_le__of__int__ceiling,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archimedean_ceiling(X_a),X)))) ) ).
fof(fact_3568_of__int__floor__le,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archim2085082626_floor(X_a),X))),X)) ) ).
fof(fact_3569_of__int__minus,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [Z] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,uminus_uminus(int),Z)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)) ) ).
fof(fact_3570_of__int__power,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [Z,N_1] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z),N_1)) = hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),N_1) ) ).
fof(fact_3571_of__int__less__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [W_1,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W_1)),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),W_1),Z_1)) ) ) ).
fof(fact_3572_of__int__le__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [W_1,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W_1)),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),W_1),Z_1)) ) ) ).
fof(fact_3573_of__int__int__eq,axiom,
! [N_1] : hAPP(int,int,ring_1_of_int(int),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,int,semiring_1_of_nat(int),N_1) ).
fof(fact_3574_of__real__of__int__eq,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [Z] : hAPP(real,X_a,of_real(X_a),hAPP(int,real,ring_1_of_int(real),Z)) = hAPP(int,X_a,ring_1_of_int(X_a),Z) ) ).
fof(fact_3575_real__eq__of__int,axiom,
real_1(int) = ring_1_of_int(real) ).
fof(fact_3576_real__of__int__def,axiom,
real_1(int) = ring_1_of_int(real) ).
fof(fact_3577_floor__of__int,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [Z] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)) = Z ) ).
fof(fact_3578_of__int__eq__iff,axiom,
! [X_a] :
( ring_char_0(X_a)
=> ! [W_1,Z_1] :
( hAPP(int,X_a,ring_1_of_int(X_a),W_1) = hAPP(int,X_a,ring_1_of_int(X_a),Z_1)
<=> W_1 = Z_1 ) ) ).
fof(fact_3579_ceiling__of__int,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [Z] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)) = Z ) ).
fof(fact_3580_Ints__of__int,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [Z_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),ring_1_Ints(X_a))) ) ).
fof(fact_3581_norm__of__int,axiom,
! [X_a] :
( real_n2089651433ebra_1(X_a)
=> ! [Z] : hAPP(X_a,real,norm_norm(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)) = hAPP(real,real,abs_abs(real),hAPP(int,real,ring_1_of_int(real),Z)) ) ).
fof(fact_3582_complex__cnj__of__int,axiom,
! [Z] : hAPP(complex,complex,cnj,hAPP(int,complex,ring_1_of_int(complex),Z)) = hAPP(int,complex,ring_1_of_int(complex),Z) ).
fof(fact_3583_complex__Re__of__int,axiom,
! [Z] : hAPP(complex,real,re,hAPP(int,complex,ring_1_of_int(complex),Z)) = hAPP(int,real,ring_1_of_int(real),Z) ).
fof(fact_3584_of__int__1,axiom,
! [X_a] :
( ring_1(X_a)
=> hAPP(int,X_a,ring_1_of_int(X_a),one_one(int)) = one_one(X_a) ) ).
fof(fact_3585_of__int__number__of__eq,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [V] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,number_number_of(int),V)) = hAPP(int,X_a,number_number_of(X_a),V) ) ).
fof(fact_3586_complex__number__of__def,axiom,
! [W] : hAPP(int,complex,number_number_of(complex),W) = hAPP(int,complex,ring_1_of_int(complex),W) ).
fof(fact_3587_number__of__eq,axiom,
! [X_a] :
( number_ring(X_a)
=> ! [K_4] : hAPP(int,X_a,number_number_of(X_a),K_4) = hAPP(int,X_a,ring_1_of_int(X_a),K_4) ) ).
fof(fact_3588_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_3589_number__of__real__def,axiom,
! [X] : hAPP(int,real,number_number_of(real),X) = hAPP(int,real,ring_1_of_int(real),X) ).
fof(fact_3590_of__int__0,axiom,
! [X_a] :
( ring_1(X_a)
=> hAPP(int,X_a,ring_1_of_int(X_a),zero_zero(int)) = zero_zero(X_a) ) ).
fof(fact_3591_of__int__0__eq__iff,axiom,
! [X_a] :
( ring_char_0(X_a)
=> ! [Z_1] :
( zero_zero(X_a) = hAPP(int,X_a,ring_1_of_int(X_a),Z_1)
<=> Z_1 = zero_zero(int) ) ) ).
fof(fact_3592_of__int__eq__0__iff,axiom,
! [X_a] :
( ring_char_0(X_a)
=> ! [Z_1] :
( hAPP(int,X_a,ring_1_of_int(X_a),Z_1) = zero_zero(X_a)
<=> Z_1 = zero_zero(int) ) ) ).
fof(fact_3593_of__int__of__nat__eq,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [N_1] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(nat,int,semiring_1_of_nat(int),N_1)) = hAPP(nat,X_a,semiring_1_of_nat(X_a),N_1) ) ).
fof(fact_3594_complex__Im__of__int,axiom,
! [Z] : hAPP(complex,real,im,hAPP(int,complex,ring_1_of_int(complex),Z)) = zero_zero(real) ).
fof(fact_3595_of__int__mult,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [W,Z] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),W),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W)),hAPP(int,X_a,ring_1_of_int(X_a),Z)) ) ).
fof(fact_3596_of__int__add,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [W,Z] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),W),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W)),hAPP(int,X_a,ring_1_of_int(X_a),Z)) ) ).
fof(fact_3597_of__int__diff,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [W,Z] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),W),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),W)),hAPP(int,X_a,ring_1_of_int(X_a),Z)) ) ).
fof(fact_3598_of__int__m1,axiom,
! [X_a] :
( number_ring(X_a)
=> hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,number_number_of(int),min)) = hAPP(int,X_a,number_number_of(X_a),min) ) ).
fof(fact_3599_floor__less__iff,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)),Z_1))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))) ) ) ).
fof(fact_3600_le__floor__iff,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [Z_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),X_1)) ) ) ).
fof(fact_3601_less__ceiling__iff,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [Z_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),X_1)) ) ) ).
fof(fact_3602_ceiling__le__iff,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)),Z_1))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(int,X_a,ring_1_of_int(X_a),Z_1))) ) ) ).
fof(fact_3603_floor__add__of__int,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,Z] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),Z) ) ).
fof(fact_3604_ceiling__add__of__int,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,Z] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z))) = hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),Z) ) ).
fof(fact_3605_floor__diff__of__int,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,Z] : hAPP(X_a,int,archim2085082626_floor(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),Z) ) ).
fof(fact_3606_ceiling__diff__of__int,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X,Z] : hAPP(X_a,int,archimedean_ceiling(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(X_a,int,archimedean_ceiling(X_a),X)),Z) ) ).
fof(fact_3607_of__int__less__0__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),zero_zero(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),zero_zero(int))) ) ) ).
fof(fact_3608_of__int__0__less__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),zero_zero(X_a)),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Z_1)) ) ) ).
fof(fact_3609_of__int__le__0__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),zero_zero(X_a)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),zero_zero(int))) ) ) ).
fof(fact_3610_of__int__0__le__iff,axiom,
! [X_a] :
( linordered_idom(X_a)
=> ! [Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z_1)) ) ) ).
fof(fact_3611_of__nat__nat,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [Z] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
=> hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(int,nat,nat_1,Z)) = hAPP(int,X_a,ring_1_of_int(X_a),Z) ) ) ).
fof(fact_3612_floor__unique,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [Z,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),one_one(X_a))))
=> hAPP(X_a,int,archim2085082626_floor(X_a),X) = Z ) ) ) ).
fof(fact_3613_ceiling__unique,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [Z,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z)),one_one(X_a))),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z)))
=> hAPP(X_a,int,archimedean_ceiling(X_a),X) = Z ) ) ) ).
fof(fact_3614_ceiling__correct,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archimedean_ceiling(X_a),X))),one_one(X_a))),X))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archimedean_ceiling(X_a),X)))) ) ) ).
fof(fact_3615_less__floor__iff,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [Z_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_1),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),one_one(X_a))),X_1)) ) ) ).
fof(fact_3616_floor__le__iff,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(X_a,int,archim2085082626_floor(X_a),X_1)),Z_1))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),one_one(X_a)))) ) ) ).
fof(fact_3617_ceiling__less__iff,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X_1,Z_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)),Z_1))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),one_one(X_a)))) ) ) ).
fof(fact_3618_le__ceiling__iff,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [Z_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Z_1),hAPP(X_a,int,archimedean_ceiling(X_a),X_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),one_one(X_a))),X_1)) ) ) ).
fof(fact_3619_floor__correct,axiom,
! [X_a] :
( archim1727834104eiling(X_a)
=> ! [X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(X_a,int,archim2085082626_floor(X_a),X))),X))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(X_a,int,archim2085082626_floor(X_a),X)),one_one(int))))) ) ) ).
fof(fact_3620_phi__def,axiom,
! [M] : hAPP(int,nat,phi,M) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),norRRset,M)) ).
fof(fact_3621_norR__mem__unique,axiom,
! [Aa,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Aa),M) = one_one(int)
=> ? [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),X_2),M))
& hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_2),hAPP(int,fun(int,bool),norRRset,M)))
& ! [Y_2] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Y_2),M))
& hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Y_2),hAPP(int,fun(int,bool),norRRset,M))) )
=> Y_2 = X_2 ) ) ) ) ).
fof(fact_3622_card__nor__eq__noX,axiom,
! [X_1,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,X_1),M) = one_one(int)
=> hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),noXRRset,M),X_1)) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),norRRset,M)) ) ) ).
fof(fact_3623_floor__exists,axiom,
! [X_a] :
( archim1804426504_field(X_a)
=> ! [X] :
? [Z_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_2)),X))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Z_2),one_one(int))))) ) ) ).
fof(fact_3624_floor__exists1,axiom,
! [X_a] :
( archim1804426504_field(X_a)
=> ! [X] :
? [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),X_2)),X))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),one_one(int)))))
& ! [Y_2] :
( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Y_2)),X))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Y_2),one_one(int))))) )
=> Y_2 = X_2 ) ) ) ).
fof(fact_3625_noX__is__RRset,axiom,
! [X_1,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,X_1),M) = one_one(int)
=> hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),noXRRset,M),X_1)),M)) ) ) ).
fof(fact_3626_RRset__gcd,axiom,
! [Aa,A_2,M] :
( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),M))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),A_2))
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,Aa),M) = one_one(int) ) ) ).
fof(fact_3627_RRset__zcong__eq,axiom,
! [Aa,Ba,A_2,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M))
=> ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),M))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),Ba),M))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),A_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Ba),A_2))
=> Aa = Ba ) ) ) ) ) ).
fof(fact_3628_RRset2norRR__correct,axiom,
! [Aa,A_2,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M))
=> ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),M))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),A_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),M),Aa)),M))
& hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),M),Aa)),hAPP(int,fun(int,bool),norRRset,M))) ) ) ) ) ).
fof(fact_3629_RRset2norRR__correct2,axiom,
! [Aa,A_2,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M))
=> ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),M))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),A_2))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),M),Aa)),hAPP(int,fun(int,bool),norRRset,M))) ) ) ) ).
fof(fact_3630_RRset2norRR__correct1,axiom,
! [Aa,A_2,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M))
=> ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),M))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),A_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),M),Aa)),M)) ) ) ) ).
fof(fact_3631_Ints__induct,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [P_1,Qa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Qa),ring_1_Ints(X_a)))
=> ( ! [Z_2] : hBOOL(hAPP(X_a,bool,P_1,hAPP(int,X_a,ring_1_of_int(X_a),Z_2)))
=> hBOOL(hAPP(X_a,bool,P_1,Qa)) ) ) ) ).
fof(fact_3632_Ints__cases,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [Qa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Qa),ring_1_Ints(X_a)))
=> ~ ! [Z_2] : ti(X_a,Qa) != hAPP(int,X_a,ring_1_of_int(X_a),Z_2) ) ) ).
fof(fact_3633_ex__less__of__int,axiom,
! [X_a] :
( archim1804426504_field(X_a)
=> ! [X] :
? [Z_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_2))) ) ).
fof(fact_3634_ex__of__int__less,axiom,
! [X_a] :
( archim1804426504_field(X_a)
=> ! [X] :
? [Z_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_2)),X)) ) ).
fof(fact_3635_ex__le__of__int,axiom,
! [X_a] :
( archim1804426504_field(X_a)
=> ! [X] :
? [Z_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(int,X_a,ring_1_of_int(X_a),Z_2))) ) ).
fof(fact_3636_is__RRset__def,axiom,
! [A_2,M] :
( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),M))
<=> ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),A_2),hAPP(int,fun(fun(int,bool),bool),rsetR,M)))
& hAPP(fun(int,bool),nat,finite_card(int),A_2) = hAPP(int,nat,phi,M) ) ) ).
fof(fact_3637_RsetR__fin,axiom,
! [A_2,M] :
( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),A_2),hAPP(int,fun(fun(int,bool),bool),rsetR,M)))
=> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),A_2)) ) ).
fof(fact_3638_Bnor__in__RsetR,axiom,
! [Aa,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),M))
=> hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),M)),hAPP(int,fun(fun(int,bool),bool),rsetR,M))) ) ).
fof(fact_3639_RRset2norRR__eq__norR,axiom,
! [A_2,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M))
=> ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),M))
=> hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),M)),A_2) = hAPP(int,fun(int,bool),norRRset,M) ) ) ).
fof(fact_3640_Nitpick_Oint__lcm__def,axiom,
! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),int_lcm,X),Y) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_lcm,hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Y)))) ).
fof(fact_3641_image__eqI,axiom,
! [X_a,X_b,A_2,Ba,F,X_1] :
( ti(X_a,Ba) = hAPP(X_b,X_a,F,X_1)
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_1),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ba),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2))) ) ) ).
fof(fact_3642_finite__imageI,axiom,
! [X_b,X_a,H,F_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),H),F_1))) ) ).
fof(fact_3643_finite__surj,axiom,
! [X_b,X_a,B_1,F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),B_1),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) ) ) ).
fof(fact_3644_image__diff__subset,axiom,
! [X_a,X_b,F,A_2,B_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),B_1))),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,bool),fun(fun(X_b,bool),fun(X_b,bool)),minus_minus(fun(X_b,bool)),A_2),B_1)))) ).
fof(fact_3645_image__Pow__surj,axiom,
! [X_b,X_a,F,A_2,B_1] :
( hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2) = ti(fun(X_a,bool),B_1)
=> hAPP(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_b,bool),fun(X_a,bool)),fun(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool)),image(fun(X_b,bool),fun(X_a,bool)),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),pow(X_b),A_2)) = hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1) ) ).
fof(fact_3646_rev__image__eqI,axiom,
! [X_b,X_a,Ba,F,X_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> ( ti(X_b,Ba) = hAPP(X_a,X_b,F,X_1)
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),Ba),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2))) ) ) ).
fof(fact_3647_imageI,axiom,
! [X_b,X_a,F,X_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_1),A_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_a,X_b,F,X_1)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2))) ) ).
fof(fact_3648_image__iff,axiom,
! [X_a,X_b,Z_1,F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z_1),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))
<=> ? [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_2))
& ti(X_a,Z_1) = hAPP(X_b,X_a,F,X_2) ) ) ).
fof(fact_3649_image__Pow__mono,axiom,
! [X_a,X_b,F,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)),B_1))
=> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_b,bool),fun(X_a,bool)),fun(fun(fun(X_b,bool),bool),fun(fun(X_a,bool),bool)),image(fun(X_b,bool),fun(X_a,bool)),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F)),hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),pow(X_b),A_2))),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),B_1))) ) ).
fof(fact_3650_image__mono,axiom,
! [X_b,X_a,F,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),B_1))) ) ).
fof(fact_3651_subset__image__iff,axiom,
! [X_a,X_b,B_1,F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))
<=> ? [AA] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),AA),A_2))
& ti(fun(X_a,bool),B_1) = hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),AA) ) ) ).
fof(fact_3652_card__image__le,axiom,
! [X_b,X_a,F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(fun(X_b,bool),nat,finite_card(X_b),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2))),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))) ) ).
fof(fact_3653_ResSet__image,axiom,
! [F,A_2,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,M),A_2))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_2),A_2))
=> ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),A_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(int,int,F,X_2)),hAPP(int,int,F,Xa)),M))
=> X_2 = Xa ) ) )
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),resSet,M),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),F),A_2))) ) ) ) ).
fof(fact_3654_finite__subset__image,axiom,
! [X_b,X_a,F,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))
=> ? [C_3] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),C_3),A_2))
& hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),C_3))
& ti(fun(X_a,bool),B_1) = hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),C_3) ) ) ) ).
fof(fact_3655_transfer__nat__int__set__relations_I1_J,axiom,
! [A_2] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),A_2))
<=> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2))) ) ).
fof(fact_3656_Nat__Transfer_Otransfer__nat__int__set__functions_I1_J,axiom,
! [A_2] : hAPP(fun(nat,bool),nat,finite_card(nat),A_2) = hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) ).
fof(fact_3657_transfer__nat__int__set__relations_I2_J,axiom,
! [X_1,A_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_1),A_2))
<=> hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(nat,int,semiring_1_of_nat(int),X_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2))) ) ).
fof(fact_3658_transfer__nat__int__set__relations_I3_J,axiom,
! [A_2,B_1] :
( A_2 = B_1
<=> hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_1) ) ).
fof(fact_3659_transfer__int__nat__set__return__embed,axiom,
! [A_2] : hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) = A_2 ).
fof(fact_3660_transfer__nat__int__set__relations_I4_J,axiom,
! [A_2,B_1] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less(fun(nat,bool)),A_2),B_1))
<=> hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less(fun(int,bool)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_1))) ) ).
fof(fact_3661_transfer__nat__int__set__relations_I5_J,axiom,
! [A_2,B_1] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),A_2),B_1))
<=> hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),B_1))) ) ).
fof(fact_3662_SetS__def,axiom,
! [Aa,Pa] : hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,Aa),Pa) = hAPP(fun(int,bool),fun(fun(int,bool),bool),hAPP(fun(int,fun(int,bool)),fun(fun(int,bool),fun(fun(int,bool),bool)),image(int,fun(int,bool)),hAPP(int,fun(int,fun(int,bool)),hAPP(int,fun(int,fun(int,fun(int,bool))),multInvPair,Aa),Pa)),hAPP(int,fun(int,bool),sRStar,Pa)) ).
fof(fact_3663_finite__surjI,axiom,
! [X_a,X_b,B_1,F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),A_2))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1)) ) ) ).
fof(fact_3664_image__subsetI,axiom,
! [X_b,X_a,F,B_1,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),hAPP(X_a,X_b,F,X_2)),B_1)) )
=> hBOOL(hAPP(fun(X_b,bool),bool,hAPP(fun(X_b,bool),fun(fun(X_b,bool),bool),ord_less_eq(fun(X_b,bool)),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)),B_1)) ) ).
fof(fact_3665_infinite__nat__iff__unbounded__le,axiom,
! [S] :
( ~ hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S))
<=> ! [M_3] :
? [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_2))
& hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),N_2),S)) ) ) ).
fof(fact_3666_infinite__nat__iff__unbounded,axiom,
! [S] :
( ~ hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S))
<=> ! [M_3] :
? [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N_2))
& hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),N_2),S)) ) ) ).
fof(fact_3667_Nitpick_Onat__lcm__def,axiom,
! [X,Y] : hAPP(nat,nat,hAPP(nat,fun(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,hAPP(nat,fun(nat,nat),nat_gcd,X),Y)) ).
fof(fact_3668_transfer__int__nat__set__relations_I4_J,axiom,
! [B_1,A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less(fun(int,bool)),A_2),B_1))
<=> hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less(fun(nat,bool)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B_1))) ) ) ) ).
fof(fact_3669_gcd__eq__nitpick__gcd,axiom,
! [X,Y] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y) ).
fof(fact_3670_Nat__Transfer_Otransfer__nat__int__set__function__closures_I6_J,axiom,
! [X_1,A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),A_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) ) ) ).
fof(fact_3671_nat__set__def,axiom,
! [S] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,S))
<=> ! [X_2] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_2),S))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2)) ) ) ).
fof(fact_3672_Nat__Transfer_Otransfer__int__nat__set__function__closures_I5_J,axiom,
! [C_2] : hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),C_2))) ).
fof(fact_3673_transfer__int__nat__set__relations_I3_J,axiom,
! [B_1,A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1))
=> ( A_2 = B_1
<=> hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B_1) ) ) ) ).
fof(fact_3674_nat__gcd_Osimps,axiom,
! [X,Y] :
( ( Y = zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y) = X )
& ( Y != zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X),Y) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,Y),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X),Y)) ) ) ).
fof(fact_3675_transfer__nat__int__set__return__embed,axiom,
! [A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)) = A_2 ) ).
fof(fact_3676_transfer__int__nat__set__relations_I1_J,axiom,
! [A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),A_2))
<=> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2))) ) ) ).
fof(fact_3677_Nat__Transfer_Otransfer__int__nat__set__functions_I1_J,axiom,
! [A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> hAPP(fun(int,bool),nat,finite_card(int),A_2) = hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)) ) ).
fof(fact_3678_transfer__int__nat__set__relations_I5_J,axiom,
! [B_1,A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),A_2),B_1))
<=> hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),B_1))) ) ) ) ).
fof(fact_3679_nat__gcd_Opsimps,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(product_prod(nat,nat),bool),accp(product_prod(nat,nat)),nat_gcd_rel),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X_1),Y_1)))
=> ( ( Y_1 = zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X_1),Y_1) = X_1 )
& ( Y_1 != zero_zero(nat)
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,X_1),Y_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,Y_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X_1),Y_1)) ) ) ) ).
fof(fact_3680_Nitpick_Oint__gcd__def,axiom,
! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),int_gcd,X),Y) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),nat_gcd,hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Y)))) ).
fof(fact_3681_nat__gcd_Opinduct,axiom,
! [P_1,A0,A1] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(product_prod(nat,nat),bool),accp(product_prod(nat,nat)),nat_gcd_rel),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A0),A1)))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(product_prod(nat,nat),bool,hAPP(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(product_prod(nat,nat),bool),accp(product_prod(nat,nat)),nat_gcd_rel),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X_2),Y_2)))
=> ( ( Y_2 != zero_zero(nat)
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,Y_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_mod(nat),X_2),Y_2))) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,X_2),Y_2)) ) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),P_1,A0),A1)) ) ) ).
fof(fact_3682_finite__nat__set__iff__bounded__le,axiom,
! [N_4] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_4))
<=> ? [M_3] :
! [X_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),N_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),X_2),M_3)) ) ) ).
fof(fact_3683_finite__nat__set__iff__bounded,axiom,
! [N_4] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_4))
<=> ? [M_3] :
! [X_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),N_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_2),M_3)) ) ) ).
fof(fact_3684_bounded__nat__set__is__finite,axiom,
! [N,N_4] :
( ! [X_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),N_4))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),X_2),N)) )
=> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),N_4)) ) ).
fof(fact_3685_card__greaterThanLessThan__int,axiom,
! [L,U_1] : hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),L),U_1)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).
fof(fact_3686_finite__greaterThanLessThan__int,axiom,
! [L,U_1] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),L),U_1))) ).
fof(fact_3687_image__uminus__greaterThanLessThan,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [X_1,Y_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,X_a),fun(fun(X_a,bool),fun(X_a,bool)),image(X_a,X_a),uminus_uminus(X_a)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),X_1),Y_1)) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y_1)),hAPP(X_a,X_a,uminus_uminus(X_a),X_1)) ) ).
fof(fact_3688_Rats__abs__nat__div__natE,axiom,
! [X_1] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),field_char_0_Rats(real)))
=> ~ ! [M_3,N_2] :
( N_2 != zero_zero(nat)
=> ( hAPP(real,real,abs_abs(real),X_1) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,real_1(nat),M_3)),hAPP(nat,real,real_1(nat),N_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),M_3),N_2) != one_one(nat) ) ) ) ).
fof(fact_3689_Nitpick_Ozero__frac__def,axiom,
! [X_a] : zero_frac(X_a) = hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) ).
fof(fact_3690_finite__greaterThanLessThan,axiom,
! [L,U_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr676655987ssThan(nat),L),U_1))) ).
fof(fact_3691_Rats__real__nat,axiom,
! [N] : hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(nat,real,real_1(nat),N)),field_char_0_Rats(real))) ).
fof(fact_3692_card__greaterThanLessThan,axiom,
! [L,U_1] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr676655987ssThan(nat),L),U_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),U_1),hAPP(nat,nat,suc,L)) ).
fof(fact_3693_Nitpick_Oone__frac__def,axiom,
! [X_a] : one_frac(X_a) = hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).
fof(fact_3694_Nitpick_Onumber__of__frac__def,axiom,
! [X_a,N_1] : hAPP(int,X_a,number_of_frac(X_a),N_1) = hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),N_1),one_one(int))) ).
fof(fact_3695_Rats__dense__in__nn__real,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1))
=> ? [X_2] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_2),field_char_0_Rats(real)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Y_1)) ) ) ) ).
fof(fact_3696_DERIV__isconst3,axiom,
! [F,Y_1,X_1,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),Ba))
=> ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),Aa),Ba)))
=> ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Y_1),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),Aa),Ba)))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_2),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),Aa),Ba)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),zero_zero(real))) )
=> hAPP(real,real,F,X_1) = hAPP(real,real,F,Y_1) ) ) ) ) ).
fof(fact_3697_nonzero__Rats__inverse,axiom,
! [X_a] :
( field_char_0(X_a)
=> ! [Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),field_char_0_Rats(X_a)))
=> ( ti(X_a,Aa) != zero_zero(X_a)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Aa)),field_char_0_Rats(X_a))) ) ) ) ).
fof(fact_3698_nonzero__Rats__divide,axiom,
! [X_a] :
( field_char_0(X_a)
=> ! [Ba,Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),field_char_0_Rats(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ba),field_char_0_Rats(X_a)))
=> ( ti(X_a,Ba) != zero_zero(X_a)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Ba)),field_char_0_Rats(X_a))) ) ) ) ) ).
fof(fact_3699_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_3700_Rats__0,axiom,
! [X_a] :
( field_char_0(X_a)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),zero_zero(X_a)),field_char_0_Rats(X_a))) ) ).
fof(fact_3701_Rats__mult,axiom,
! [X_a] :
( field_char_0(X_a)
=> ! [Ba,Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),field_char_0_Rats(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ba),field_char_0_Rats(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Ba)),field_char_0_Rats(X_a))) ) ) ) ).
fof(fact_3702_Rats__add,axiom,
! [X_a] :
( field_char_0(X_a)
=> ! [Ba,Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),field_char_0_Rats(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ba),field_char_0_Rats(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Ba)),field_char_0_Rats(X_a))) ) ) ) ).
fof(fact_3703_Rats__number__of,axiom,
! [X_a] :
( ( number_ring(X_a)
& field_char_0(X_a) )
=> ! [W_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,number_number_of(X_a),W_1)),field_char_0_Rats(X_a))) ) ).
fof(fact_3704_Rats__1,axiom,
! [X_a] :
( field_char_0(X_a)
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),one_one(X_a)),field_char_0_Rats(X_a))) ) ).
fof(fact_3705_Rats__diff,axiom,
! [X_a] :
( field_char_0(X_a)
=> ! [Ba,Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),field_char_0_Rats(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ba),field_char_0_Rats(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba)),field_char_0_Rats(X_a))) ) ) ) ).
fof(fact_3706_Rats__power,axiom,
! [X_a] :
( field_char_0(X_a)
=> ! [N,Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),field_char_0_Rats(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),N)),field_char_0_Rats(X_a))) ) ) ).
fof(fact_3707_Rats__minus,axiom,
! [X_a] :
( field_char_0(X_a)
=> ! [Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),field_char_0_Rats(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),field_char_0_Rats(X_a))) ) ) ).
fof(fact_3708_Rats__divide,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& field_char_0(X_a) )
=> ! [Ba,Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),field_char_0_Rats(X_a)))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ba),field_char_0_Rats(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Ba)),field_char_0_Rats(X_a))) ) ) ) ).
fof(fact_3709_Rats__of__nat,axiom,
! [X_a] :
( field_char_0(X_a)
=> ! [N] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),field_char_0_Rats(X_a))) ) ).
fof(fact_3710_Rats__inverse,axiom,
! [X_a] :
( ( field_inverse_zero(X_a)
& field_char_0(X_a) )
=> ! [Aa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),field_char_0_Rats(X_a)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),Aa)),field_char_0_Rats(X_a))) ) ) ).
fof(fact_3711_Rats__of__int,axiom,
! [X_a] :
( field_char_0(X_a)
=> ! [Z_1] : hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),hAPP(int,X_a,ring_1_of_int(X_a),Z_1)),field_char_0_Rats(X_a))) ) ).
fof(fact_3712_Rats__dense__in__real,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Y_1))
=> ? [X_2] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_2),field_char_0_Rats(real)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Y_1)) ) ) ).
fof(fact_3713_Sup__greaterThanLessThan,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),X_1))
=> hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),Y_1),X_1)) = X_1 ) ).
fof(fact_3714_Sup__upper__EX,axiom,
! [X_1,X_3] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_3))
=> ( ? [Z_2] :
! [X_2] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_2),X_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Z_2)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_3))) ) ) ).
fof(fact_3715_SupInf_OSup__upper,axiom,
! [Z_1,X_1,X_3] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_3))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_2),X_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Z_1)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_3))) ) ) ).
fof(fact_3716_Frct__code__post_I4_J,axiom,
! [K_4] : hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),K_4)),one_one(int))) = hAPP(int,rat,number_number_of(rat),K_4) ).
fof(fact_3717_Frct__code__post_I2_J,axiom,
! [K_4] : hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),K_4),zero_zero(int))) = zero_zero(rat) ).
fof(fact_3718_Frct__code__post_I1_J,axiom,
! [K_4] : hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K_4)) = zero_zero(rat) ).
fof(fact_3719_Frct__code__post_I3_J,axiom,
hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).
fof(fact_3720_Frct__code__post_I6_J,axiom,
! [K_4,L_4] : hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),K_4)),hAPP(int,int,number_number_of(int),L_4))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,number_number_of(rat),K_4)),hAPP(int,rat,number_number_of(rat),L_4)) ).
fof(fact_3721_Frct__code__post_I5_J,axiom,
! [K_4] : hAPP(product_prod(int,int),rat,frct,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),hAPP(int,int,number_number_of(int),K_4))) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),K_4)) ).
fof(fact_3722_Sup__eq__maximum,axiom,
! [Z_1,X_3] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),Z_1),X_3))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_2),X_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Z_1)) )
=> hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_3) = Z_1 ) ) ).
fof(fact_3723_SupInf_OSup__upper2,axiom,
! [Z_1,Y_1,X_1,X_3] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_1),X_3))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),X_1))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_2),X_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Z_1)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),hAPP(fun(real,bool),real,complete_Sup_Sup(real),X_3))) ) ) ) ).
fof(fact_3724_abs__rat__def,axiom,
! [Q_2] :
( ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Q_2),zero_zero(rat)))
=> hAPP(rat,rat,abs_abs(rat),Q_2) = hAPP(rat,rat,uminus_uminus(rat),Q_2) )
& ( ~ hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Q_2),zero_zero(rat)))
=> hAPP(rat,rat,abs_abs(rat),Q_2) = Q_2 ) ) ).
fof(fact_3725_sgn__rat__def,axiom,
! [Q_2] :
( ( Q_2 = zero_zero(rat)
=> hAPP(rat,rat,sgn_sgn(rat),Q_2) = zero_zero(rat) )
& ( Q_2 != zero_zero(rat)
=> ( ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),Q_2))
=> hAPP(rat,rat,sgn_sgn(rat),Q_2) = one_one(rat) )
& ( ~ hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),Q_2))
=> hAPP(rat,rat,sgn_sgn(rat),Q_2) = hAPP(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ).
fof(fact_3726_divide__rat__def,axiom,
! [Q_2,R_4] : hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),Q_2),R_4) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),Q_2),hAPP(rat,rat,inverse_inverse(rat),R_4)) ).
fof(fact_3727_normalize__negative,axiom,
! [P_4,Q_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Q_2),zero_zero(int)))
=> hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),Q_2)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,uminus_uminus(int),P_4)),hAPP(int,int,uminus_uminus(int),Q_2))) ) ).
fof(fact_3728_rat__sgn__code,axiom,
! [P_4] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,sgn_sgn(rat),P_4)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,sgn_sgn(int),hAPP(product_prod(int,int),int,product_fst(int,int),hAPP(rat,product_prod(int,int),quotient_of,P_4)))),one_one(int)) ).
fof(fact_3729_diff__rat__def,axiom,
! [Q_2,R_4] : hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),Q_2),R_4) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),Q_2),hAPP(rat,rat,uminus_uminus(rat),R_4)) ).
fof(fact_3730_quotient__of__inject__eq,axiom,
! [Aa,Ba] :
( hAPP(rat,product_prod(int,int),quotient_of,Aa) = hAPP(rat,product_prod(int,int),quotient_of,Ba)
<=> Aa = Ba ) ).
fof(fact_3731_quotient__of__inject,axiom,
! [A_1,B] :
( hAPP(rat,product_prod(int,int),quotient_of,A_1) = hAPP(rat,product_prod(int,int),quotient_of,B)
=> A_1 = B ) ).
fof(fact_3732_less__rat__def,axiom,
! [Z_1,W_1] :
( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),Z_1),W_1))
<=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),Z_1),W_1))
& Z_1 != W_1 ) ) ).
fof(fact_3733_quotient__of__denom__pos,axiom,
! [R_4,P_4,Q_2] :
( hAPP(rat,product_prod(int,int),quotient_of,R_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),Q_2)
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Q_2)) ) ).
fof(fact_3734_rat__one__code,axiom,
hAPP(rat,product_prod(int,int),quotient_of,one_one(rat)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).
fof(fact_3735_normalize__denom__pos,axiom,
! [R_4,P_4,Q_2] :
( hAPP(product_prod(int,int),product_prod(int,int),normalize,R_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),Q_2)
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Q_2)) ) ).
fof(fact_3736_normalize__denom__zero,axiom,
! [P_4] : hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),zero_zero(int))) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).
fof(fact_3737_normalize__crossproduct,axiom,
! [P_4,R_4,S_4,Q_2] :
( Q_2 != zero_zero(int)
=> ( S_4 != zero_zero(int)
=> ( hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),Q_2)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),R_4),S_4))
=> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),P_4),S_4) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),R_4),Q_2) ) ) ) ).
fof(fact_3738_rat__zero__code,axiom,
hAPP(rat,product_prod(int,int),quotient_of,zero_zero(rat)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).
fof(fact_3739_quotient__of__number_I3_J,axiom,
! [K_4] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(int,rat,number_number_of(rat),K_4)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,number_number_of(int),K_4)),one_one(int)) ).
fof(fact_3740_obtain__pos__sum,axiom,
! [R_4] :
( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R_4))
=> ~ ! [S_1] :
( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),S_1))
=> ! [T_5] :
( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),T_5))
=> R_4 != hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),S_1),T_5) ) ) ) ).
fof(fact_3741_ratrel__iff,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),bool,hAPP(product_prod(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),bool),member(product_prod(product_prod(int,int),product_prod(int,int))),hAPP(product_prod(int,int),product_prod(product_prod(int,int),product_prod(int,int)),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(product_prod(int,int),product_prod(int,int))),product_Pair(product_prod(int,int),product_prod(int,int)),X_1),Y_1)),ratrel))
<=> ( hAPP(product_prod(int,int),int,product_snd(int,int),X_1) != zero_zero(int)
& hAPP(product_prod(int,int),int,product_snd(int,int),Y_1) != zero_zero(int)
& hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_fst(int,int),X_1)),hAPP(product_prod(int,int),int,product_snd(int,int),Y_1)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,product_fst(int,int),Y_1)),hAPP(product_prod(int,int),int,product_snd(int,int),X_1)) ) ) ).
fof(fact_3742_Fract__1__number__of,axiom,
! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,one_one(int)),hAPP(int,int,number_number_of(int),K_4)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),one_one(rat)),hAPP(int,rat,number_number_of(rat),K_4)) ).
fof(fact_3743_Ratreal__number__of__quotient2,axiom,
! [R_4,S_4] : hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,number_number_of(rat),R_4)),hAPP(int,rat,number_number_of(rat),S_4))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),R_4)),hAPP(int,real,number_number_of(real),S_4)) ).
fof(fact_3744_quotient__of__eq,axiom,
! [A_1,B,P_4,Q_2] :
( hAPP(rat,product_prod(int,int),quotient_of,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),Q_2)
=> hAPP(int,rat,hAPP(int,fun(int,rat),fract,P_4),Q_2) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) ) ).
fof(fact_3745_minus__rat,axiom,
! [A_1,B] : hAPP(rat,rat,uminus_uminus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,uminus_uminus(int),A_1)),B) ).
fof(fact_3746_divide__rat,axiom,
! [A_1,B,C,D] : hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C),D)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),C)) ).
fof(fact_3747_Fract__of__int__quotient,axiom,
! [K_4,L_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),L_4) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,ring_1_of_int(rat),K_4)),hAPP(int,rat,ring_1_of_int(rat),L_4)) ).
fof(fact_3748_Fract__of__int__eq,axiom,
! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),one_one(int)) = hAPP(int,rat,ring_1_of_int(rat),K_4) ).
fof(fact_3749_of__int__rat,axiom,
! [K_4] : hAPP(int,rat,ring_1_of_int(rat),K_4) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),one_one(int)) ).
fof(fact_3750_rat__number__collapse_I1_J,axiom,
! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),K_4) = zero_zero(rat) ).
fof(fact_3751_eq__rat_I3_J,axiom,
! [A_1,C] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),A_1) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),C) ).
fof(fact_3752_rat__number__collapse_I4_J,axiom,
! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,K_4),zero_zero(int)) = zero_zero(rat) ).
fof(fact_3753_one__rat,axiom,
one_one(rat) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,one_one(int)),one_one(int)) ).
fof(fact_3754_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_3755_eq__rat_I2_J,axiom,
! [A_1] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),zero_zero(int)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).
fof(fact_3756_Fract__of__nat__eq,axiom,
! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(nat,int,semiring_1_of_nat(int),K_4)),one_one(int)) = hAPP(nat,rat,semiring_1_of_nat(rat),K_4) ).
fof(fact_3757_of__nat__rat,axiom,
! [K_4] : hAPP(nat,rat,semiring_1_of_nat(rat),K_4) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(nat,int,semiring_1_of_nat(int),K_4)),one_one(int)) ).
fof(fact_3758_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_3759_real__floor__code,axiom,
! [X] : hAPP(real,int,archim2085082626_floor(real),hAPP(rat,real,ratreal,X)) = hAPP(rat,int,archim2085082626_floor(rat),X) ).
fof(fact_3760_floor__Fract,axiom,
! [A_1,B] : hAPP(rat,int,archim2085082626_floor(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),B) ).
fof(fact_3761_inverse__rat,axiom,
! [A_1,B] : hAPP(rat,rat,inverse_inverse(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,B),A_1) ).
fof(fact_3762_minus__rat__cancel,axiom,
! [A_1,B] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) ).
fof(fact_3763_mult__rat__cancel,axiom,
! [A_1,B,C] :
( C != zero_zero(int)
=> hAPP(int,rat,hAPP(int,fun(int,rat),fract,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)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) ) ).
fof(fact_3764_eq__rat_I1_J,axiom,
! [Aa,C_1,D_1,Ba] :
( Ba != zero_zero(int)
=> ( D_1 != zero_zero(int)
=> ( hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_1)
<=> hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Aa),D_1) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),Ba) ) ) ) ).
fof(fact_3765_mult__rat,axiom,
! [A_1,B,C,D] : hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C),D)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,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),D)) ).
fof(fact_3766_abs__rat,axiom,
! [A_1,B] : hAPP(rat,rat,abs_abs(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,abs_abs(int),A_1)),hAPP(int,int,abs_abs(int),B)) ).
fof(fact_3767_normalize__eq,axiom,
! [A_1,B,P_4,Q_2] :
( hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_1),B)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),Q_2)
=> hAPP(int,rat,hAPP(int,fun(int,rat),fract,P_4),Q_2) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) ) ).
fof(fact_3768_real__less__code,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(rat,real,ratreal,X_1)),hAPP(rat,real,ratreal,Y_1)))
<=> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),X_1),Y_1)) ) ).
fof(fact_3769_zero__real__code,axiom,
zero_zero(real) = hAPP(rat,real,ratreal,zero_zero(rat)) ).
fof(fact_3770_Ratreal__number__collapse_I1_J,axiom,
hAPP(rat,real,ratreal,zero_zero(rat)) = zero_zero(real) ).
fof(fact_3771_number__of__real__code,axiom,
! [K_4] : hAPP(int,real,number_number_of(real),K_4) = hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),K_4)) ).
fof(fact_3772_Ratreal__number__collapse_I3_J,axiom,
! [K_4] : hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),K_4)) = hAPP(int,real,number_number_of(real),K_4) ).
fof(fact_3773_one__real__code,axiom,
one_one(real) = hAPP(rat,real,ratreal,one_one(rat)) ).
fof(fact_3774_Ratreal__number__collapse_I2_J,axiom,
hAPP(rat,real,ratreal,one_one(rat)) = one_one(real) ).
fof(fact_3775_real__less__eq__code,axiom,
! [X_1,Y_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(rat,real,ratreal,X_1)),hAPP(rat,real,ratreal,Y_1)))
<=> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),X_1),Y_1)) ) ).
fof(fact_3776_real__plus__code,axiom,
! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y)) = hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),X),Y)) ).
fof(fact_3777_real__minus__code,axiom,
! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y)) = hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),X),Y)) ).
fof(fact_3778_real__uminus__code,axiom,
! [X] : hAPP(real,real,uminus_uminus(real),hAPP(rat,real,ratreal,X)) = hAPP(rat,real,ratreal,hAPP(rat,rat,uminus_uminus(rat),X)) ).
fof(fact_3779_real__divide__code,axiom,
! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y)) = hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),X),Y)) ).
fof(fact_3780_real__times__code,axiom,
! [X,Y] : hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(rat,real,ratreal,X)),hAPP(rat,real,ratreal,Y)) = hAPP(rat,real,ratreal,hAPP(rat,rat,hAPP(rat,fun(rat,rat),times_times(rat),X),Y)) ).
fof(fact_3781_real__inverse__code,axiom,
! [X] : hAPP(real,real,inverse_inverse(real),hAPP(rat,real,ratreal,X)) = hAPP(rat,real,ratreal,hAPP(rat,rat,inverse_inverse(rat),X)) ).
fof(fact_3782_zero__rat,axiom,
zero_zero(rat) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).
fof(fact_3783_rat__number__expand_I3_J,axiom,
! [K_4] : hAPP(int,rat,number_number_of(rat),K_4) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,number_number_of(int),K_4)),one_one(int)) ).
fof(fact_3784_rat__number__collapse_I3_J,axiom,
! [K_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,number_number_of(int),K_4)),one_one(int)) = hAPP(int,rat,number_number_of(rat),K_4) ).
fof(fact_3785_Fract__number__of__quotient,axiom,
! [K_4,L_4] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,number_number_of(int),K_4)),hAPP(int,int,number_number_of(int),L_4)) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),inverse_divide(rat),hAPP(int,rat,number_number_of(rat),K_4)),hAPP(int,rat,number_number_of(rat),L_4)) ).
fof(fact_3786_quotient__of__Fract,axiom,
! [A_1,B] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),A_1),B)) ).
fof(fact_3787_Frct__def,axiom,
! [P_4] : hAPP(product_prod(int,int),rat,frct,P_4) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(product_prod(int,int),int,product_fst(int,int),P_4)),hAPP(product_prod(int,int),int,product_snd(int,int),P_4)) ).
fof(fact_3788_less__rat,axiom,
! [Aa,C_1,D_1,Ba] :
( Ba != zero_zero(int)
=> ( D_1 != zero_zero(int)
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),Aa),D_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ba),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_1),Ba)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ba),D_1)))) ) ) ) ).
fof(fact_3789_Ratreal__number__of__quotient,axiom,
! [R_4,S_4] : hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),R_4))),hAPP(rat,real,ratreal,hAPP(int,rat,number_number_of(rat),S_4))) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(int,real,number_number_of(real),R_4)),hAPP(int,real,number_number_of(real),S_4)) ).
fof(fact_3790_sgn__rat,axiom,
! [A_1,B] : hAPP(rat,rat,sgn_sgn(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)) = 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_1)),hAPP(int,int,sgn_sgn(int),B))) ).
fof(fact_3791_Fract__less__zero__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba)),zero_zero(rat)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),zero_zero(int))) ) ) ).
fof(fact_3792_zero__less__Fract__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Aa)) ) ) ).
fof(fact_3793_le__rat,axiom,
! [Aa,C_1,D_1,Ba] :
( Ba != zero_zero(int)
=> ( D_1 != zero_zero(int)
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C_1),D_1)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),Aa),D_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ba),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_1),Ba)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Ba),D_1)))) ) ) ) ).
fof(fact_3794_add__rat,axiom,
! [A_1,C,D,B] :
( B != zero_zero(int)
=> ( D != zero_zero(int)
=> hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C),D)) = 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_1),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D)) ) ) ).
fof(fact_3795_diff__rat,axiom,
! [A_1,C,D,B] :
( B != zero_zero(int)
=> ( D != zero_zero(int)
=> hAPP(rat,rat,hAPP(rat,fun(rat,rat),minus_minus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,C),D)) = 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_1),D)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C),B))),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B),D)) ) ) ).
fof(fact_3796_one__less__Fract__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),one_one(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ba),Aa)) ) ) ).
fof(fact_3797_Fract__less__one__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba)),one_one(rat)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),Ba)) ) ) ).
fof(fact_3798_Fract__add__one,axiom,
! [M_1,N_1] :
( N_1 != zero_zero(int)
=> hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M_1),N_1)),N_1) = hAPP(rat,rat,hAPP(rat,fun(rat,rat),plus_plus(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,M_1),N_1)),one_one(rat)) ) ).
fof(fact_3799_Fract__le__zero__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba)),zero_zero(rat)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Aa),zero_zero(int))) ) ) ).
fof(fact_3800_zero__le__Fract__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Aa)) ) ) ).
fof(fact_3801_rat__floor__lemma,axiom,
! [A_1,B] :
( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,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_1),B))),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)))
& hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B)),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_1),B)),one_one(int))))) ) ).
fof(fact_3802_one__le__Fract__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Ba),Aa)) ) ) ).
fof(fact_3803_Fract__le__one__iff,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less_eq(rat),hAPP(int,rat,hAPP(int,fun(int,rat),fract,Aa),Ba)),one_one(rat)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Aa),Ba)) ) ) ).
fof(fact_3804_Rat__induct__pos,axiom,
! [Qa,P_1] :
( ! [A_3,B_3] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3))
=> hBOOL(hAPP(rat,bool,P_1,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_3))) )
=> hBOOL(hAPP(rat,bool,P_1,Qa)) ) ).
fof(fact_3805_mlex__leq,axiom,
! [X_a,R_3,F,X_1,Y_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(X_a,nat,F,X_1)),hAPP(X_a,nat,F,Y_1)))
=> ( hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),X_1),Y_1)),R_3))
=> hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),X_1),Y_1)),hAPP(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool),hAPP(fun(X_a,nat),fun(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool)),mlex_prod(X_a),F),R_3))) ) ) ).
fof(fact_3806_mlex__less,axiom,
! [X_a,R_3,F,X_1,Y_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(X_a,nat,F,X_1)),hAPP(X_a,nat,F,Y_1)))
=> hBOOL(hAPP(fun(product_prod(X_a,X_a),bool),bool,hAPP(product_prod(X_a,X_a),fun(fun(product_prod(X_a,X_a),bool),bool),member(product_prod(X_a,X_a)),hAPP(X_a,product_prod(X_a,X_a),hAPP(X_a,fun(X_a,product_prod(X_a,X_a)),product_Pair(X_a,X_a),X_1),Y_1)),hAPP(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool),hAPP(fun(X_a,nat),fun(fun(product_prod(X_a,X_a),bool),fun(product_prod(X_a,X_a),bool)),mlex_prod(X_a),F),R_3))) ) ).
fof(fact_3807_adjust__eq,axiom,
! [Ba,Qa,R_1] : hAPP(product_prod(int,int),product_prod(int,int),hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,Ba),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Qa),R_1)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),hAPP(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),if(product_prod(int,int)),hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),R_1),Ba))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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)))),Qa)),one_one(int))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),R_1),Ba))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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)))),Qa)),R_1)) ).
fof(fact_3808_lemma__DERIV__sin__cos__minus,axiom,
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,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(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,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_2),zero_zero(real))) ).
fof(fact_3809_finite__Collect__conjI,axiom,
! [X_a,Q_1,P_1] :
( ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)))
| hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Q_1))) )
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),P_1)),Q_1)))) ) ).
fof(fact_3810_finite__Collect__less__nat,axiom,
! [K] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),K)))) ).
fof(fact_3811_finite__Collect__le__nat,axiom,
! [K] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),K)))) ).
fof(fact_3812_finite__Collect__subsets,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool),collect(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),combc(fun(X_a,bool),fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool))),A_2)))) ) ).
fof(fact_3813_transfer__morphism__int__nat,axiom,
hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,int),fun(fun(nat,bool),bool),nat_tr1645093318rphism(nat,int),semiring_1_of_nat(int)),hAPP(bool,fun(nat,bool),combk(bool,nat),fTrue))) ).
fof(fact_3814_Nat__Transfer_Otransfer__int__nat__set__functions_I5_J,axiom,
! [P_1] : hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_1)) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,int),fun(nat,bool),hAPP(fun(int,bool),fun(fun(nat,int),fun(nat,bool)),combb(int,bool,nat),P_1),semiring_1_of_nat(int)))) ).
fof(fact_3815_int__card__bdd__int__set__l__le,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),N))))) = N ) ).
fof(fact_3816_DERIV__divide,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [G,E_3,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),X_1),E_3))
=> ( hAPP(X_a,X_a,G,X_1) != zero_zero(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),F)),G)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D_2),hAPP(X_a,X_a,G,X_1))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,F,X_1)),E_3))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,G,X_1)),hAPP(X_a,X_a,G,X_1))))) ) ) ) ) ).
fof(fact_3817_card__bdd__int__set__l,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),N)))) = hAPP(int,nat,nat_1,N) ) ).
fof(fact_3818_card__bdd__int__set__l__le,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),N)))) = hAPP(int,nat,nat_1,N) ) ).
fof(fact_3819_bdd__int__set__l__le__finite,axiom,
! [N] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),N))))) ).
fof(fact_3820_bdd__int__set__l__finite,axiom,
! [N] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),N))))) ).
fof(fact_3821_DERIV__cos__add,axiom,
! [K,Xa_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),K))),Xa_2),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa_2),K))))) ).
fof(fact_3822_less__eq__Suc__le__raw,axiom,
! [X_2] : hAPP(nat,fun(nat,bool),ord_less(nat),X_2) = hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,X_2)) ).
fof(fact_3823_DERIV__add__minus,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [G,E_3,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),X_1),E_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),plus_plus(X_a)),F)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),uminus_uminus(X_a)),G))),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),D_2),hAPP(X_a,X_a,uminus_uminus(X_a),E_3)))) ) ) ) ).
fof(fact_3824_DERIV__mult,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [G,Db,F,X_1,Da] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),Da))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),X_1),Db))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),times_times(X_a)),F)),G)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Da),hAPP(X_a,X_a,G,X_1))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Db),hAPP(X_a,X_a,F,X_1))))) ) ) ) ).
fof(fact_3825_DERIV__mult_H,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [G,E_3,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),X_1),E_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),times_times(X_a)),F)),G)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,F,X_1)),E_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D_2),hAPP(X_a,X_a,G,X_1))))) ) ) ) ).
fof(fact_3826_DERIV__sin__sin__mult,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),sin)),X_1),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X_1)),hAPP(real,real,sin,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X_1)),hAPP(real,real,sin,X_1))))) ).
fof(fact_3827_DERIV__sin__add,axiom,
! [K,Xa_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),K))),Xa_2),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),Xa_2),K)))) ).
fof(fact_3828_DERIV__add,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [G,E_3,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),X_1),E_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),plus_plus(X_a)),F)),G)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),D_2),E_3))) ) ) ) ).
fof(fact_3829_DERIV__diff,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [G,E_3,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),X_1),E_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),F)),G)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),D_2),E_3))) ) ) ) ).
fof(fact_3830_Bseq__ignore__initial__segment,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [K,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),X_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))) ) ) ).
fof(fact_3831_Bseq__offset,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3,K] :
( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),X_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))))
=> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3)) ) ) ).
fof(fact_3832_curry__def,axiom,
! [X_b,X_c,X_a,X_2,Xa,Xb] : hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),hAPP(fun(product_prod(X_a,X_b),X_c),fun(X_a,fun(X_b,X_c)),product_curry(X_a,X_b,X_c),X_2),Xa),Xb) = hAPP(product_prod(X_a,X_b),X_c,X_2,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Xa),Xb)) ).
fof(fact_3833_card__bdd__nat__set__le,axiom,
! [X_1] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),X_1))) = hAPP(nat,nat,suc,X_1) ).
fof(fact_3834_card__Collect__le__nat,axiom,
! [N] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),N))) = hAPP(nat,nat,suc,N) ).
fof(fact_3835_bdd__nat__set__le__finite,axiom,
! [X_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less_eq(nat)),X_1)))) ).
fof(fact_3836_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_3837_Nat__Transfer_Otransfer__int__nat__set__function__closures_I4_J,axiom,
! [P_1] : hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_1)))) ).
fof(fact_3838_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J,axiom,
! [P_1] : hAPP(fun(nat,bool),fun(nat,bool),collect(nat),P_1) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(fun(int,nat),fun(int,bool),hAPP(fun(nat,bool),fun(fun(int,nat),fun(int,bool)),combb(nat,bool,int),P_1),nat_1)))) ).
fof(fact_3839_bdd__int__set__le__finite,axiom,
! [N] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),N))))) ).
fof(fact_3840_Bseq__realpow,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),one_one(real)))
=> hBOOL(hAPP(fun(nat,real),bool,bseq(real),hAPP(real,fun(nat,real),power_power(real),X_1))) ) ) ).
fof(fact_3841_diff__nat__eq__if,axiom,
! [Z_1,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_1,Z_1)),hAPP(int,nat,nat_1,Z_3)) = hAPP(int,nat,nat_1,Z_1) )
& ( ~ hBOOL(hAPP(int,bool,nat_neg,Z_3))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,Z_1)),hAPP(int,nat,nat_1,Z_3)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_1),Z_3))),zero_zero(nat)),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Z_1),Z_3))) ) ) ).
fof(fact_3842_DERIV__ident,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),combi(X_a)),X_1),one_one(X_a))) ) ).
fof(fact_3843_DERIV__const,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [K,X_1] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(X_a,fun(X_a,X_a),combk(X_a,X_a),K)),X_1),zero_zero(X_a))) ) ).
fof(fact_3844_DERIV__cdivide,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [C_1,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),F)),C_1)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),D_2),C_1))) ) ) ).
fof(fact_3845_finite__divisors__nat,axiom,
! [M] :
( M != zero_zero(nat)
=> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat)),M)))) ) ).
fof(fact_3846_finite__divisors__int,axiom,
! [I] :
( I != zero_zero(int)
=> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),dvd_dvd(int)),I)))) ) ).
fof(fact_3847_image__ident,axiom,
! [X_a,Y_3] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,X_a),fun(fun(X_a,bool),fun(X_a,bool)),image(X_a,X_a),combi(X_a)),Y_3) = ti(fun(X_a,bool),Y_3) ).
fof(fact_3848_image__image,axiom,
! [X_b,X_a,X_c,F,G,A_2] : hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),hAPP(fun(X_c,bool),fun(X_b,bool),hAPP(fun(X_c,X_b),fun(fun(X_c,bool),fun(X_b,bool)),image(X_c,X_b),G),A_2)) = hAPP(fun(X_c,bool),fun(X_a,bool),hAPP(fun(X_c,X_a),fun(fun(X_c,bool),fun(X_a,bool)),image(X_c,X_a),hAPP(fun(X_c,X_b),fun(X_c,X_a),hAPP(fun(X_b,X_a),fun(fun(X_c,X_b),fun(X_c,X_a)),combb(X_b,X_a,X_c),F),G)),A_2) ).
fof(fact_3849_primes__infinite,axiom,
~ hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),prime))) ).
fof(fact_3850_finite__Collect__disjI,axiom,
! [X_a,P_1,Q_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fdisj),P_1)),Q_1))))
<=> ( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)))
& hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),Q_1))) ) ) ).
fof(fact_3851_cnj_OCauchy,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),X_3))
=> hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_3))) ) ).
fof(fact_3852_Re_OCauchy,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),X_3))
=> hBOOL(hAPP(fun(nat,real),bool,cauchy(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_3))) ) ).
fof(fact_3853_Im_OCauchy,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,complex),bool,cauchy(complex),X_3))
=> hBOOL(hAPP(fun(nat,real),bool,cauchy(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_3))) ) ).
fof(fact_3854_bounded__bilinear_Oflip,axiom,
! [X_c,X_b,X_a] :
( ( real_normed_vector(X_a)
& real_normed_vector(X_b)
& real_normed_vector(X_c) )
=> ! [Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hBOOL(hAPP(fun(X_b,fun(X_a,X_c)),bool,bounded_bilinear(X_b,X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),Prod))) ) ) ).
fof(fact_3855_DERIV__fun__cos,axiom,
! [G,X_1,M] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),X_1),M))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),G)),X_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,hAPP(real,real,G,X_1)))),M))) ) ).
fof(fact_3856_DERIV__fun__sin,axiom,
! [G,X_1,M] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),X_1),M))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),G)),X_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,hAPP(real,real,G,X_1))),M))) ) ).
fof(fact_3857_DERIV__cmult,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [C_1,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1)),F)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),D_2))) ) ) ).
fof(fact_3858_DERIV__chain_H,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [G,E_3,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),hAPP(X_a,X_a,F,X_1)),E_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),G),F)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),E_3),D_2))) ) ) ) ).
fof(fact_3859_DERIV__chain2,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Db,F,G,X_1,Da] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),hAPP(X_a,X_a,G,X_1)),Da))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),X_1),Db))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),G)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Da),Db))) ) ) ) ).
fof(fact_3860_DERIV__fun__exp,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [G,X_1,M] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),X_1),M))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),exp(X_a)),G)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,exp(X_a),hAPP(X_a,X_a,G,X_1))),M))) ) ) ).
fof(fact_3861_mult_Oflip,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> hBOOL(hAPP(fun(X_a,fun(X_a,X_a)),bool,bounded_bilinear(X_a,X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),times_times(X_a)))) ) ).
fof(fact_3862_Bseq__minus__iff,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_3)))
<=> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),X_3)) ) ) ).
fof(fact_3863_Collect__neg__eq,axiom,
! [X_a,P_1] : hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(bool,bool),fun(fun(X_a,bool),fun(X_a,bool)),combb(bool,bool,X_a),fNot),P_1)) = hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)) ).
fof(fact_3864_DERIV__minus,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),uminus_uminus(X_a)),F)),X_1),hAPP(X_a,X_a,uminus_uminus(X_a),D_2))) ) ) ).
fof(fact_3865_card__Collect__less__nat,axiom,
! [N] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),N))) = N ).
fof(fact_3866_card__bdd__nat__set__l,axiom,
! [X_1] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),X_1))) = X_1 ).
fof(fact_3867_bdd__nat__set__l__finite,axiom,
! [X_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),X_1)))) ).
fof(fact_3868_nat__seg__image__imp__finite,axiom,
! [X_a,A_2,F,N] :
( ti(fun(X_a,bool),A_2) = hAPP(fun(nat,bool),fun(X_a,bool),hAPP(fun(nat,X_a),fun(fun(nat,bool),fun(X_a,bool)),image(nat,X_a),F),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),N)))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).
fof(fact_3869_finite__M__bounded__by__nat,axiom,
! [P_1,I] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),P_1)),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I))))) ).
fof(fact_3870_zpower__number__of__even,axiom,
! [Z,W] : hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z),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),hAPP(int,nat,number_number_of(nat),W))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z),hAPP(int,nat,number_number_of(nat),W))) ).
fof(fact_3871_power__number__of__even,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [Z,W] : hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,W))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),hAPP(int,nat,number_number_of(nat),W))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),hAPP(int,nat,number_number_of(nat),W))) ) ).
fof(fact_3872_power__number__of__even__number__of,axiom,
! [X_b] :
( ( monoid_mult(X_b)
& number(X_b) )
=> ! [V,W] : hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,W))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W))),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W))) ) ).
fof(fact_3873_card__less__Suc2,axiom,
! [I,M_2] :
( ~ hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),M_2))
=> hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),hAPP(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool)),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool))),combb(nat,fun(fun(nat,bool),bool),nat),member(nat)),suc)),M_2))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I)))) = hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),member(nat)),M_2))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),hAPP(nat,nat,suc,I))))) ) ).
fof(fact_3874_card__less,axiom,
! [I,M_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),M_2))
=> hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),member(nat)),M_2))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),hAPP(nat,nat,suc,I))))) != zero_zero(nat) ) ).
fof(fact_3875_card__less__Suc,axiom,
! [I,M_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),M_2))
=> hAPP(nat,nat,suc,hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),hAPP(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool)),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,nat),fun(nat,fun(fun(nat,bool),bool))),combb(nat,fun(fun(nat,bool),bool),nat),member(nat)),suc)),M_2))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I))))) = hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),hAPP(fun(nat,bool),fun(nat,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,fun(fun(nat,bool),bool)),fun(fun(nat,bool),fun(nat,bool)),combc(nat,fun(nat,bool),bool),member(nat)),M_2))),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),hAPP(nat,nat,suc,I))))) ) ).
fof(fact_3876_bdd__int__set__l__l__finite,axiom,
! [N] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),N))))) ).
fof(fact_3877_DERIV__inverse_H,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> ( hAPP(X_a,X_a,F,X_1) != zero_zero(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),inverse_inverse(X_a)),F)),X_1),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,F,X_1))),D_2)),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,F,X_1)))))) ) ) ) ).
fof(fact_3878_noXRRset__def,axiom,
! [M,X_1] : hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),noXRRset,M),X_1) = hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),times_times(int)),X_1)),hAPP(int,fun(int,bool),norRRset,M)) ).
fof(fact_3879_DERIV__fun__pow,axiom,
! [N,G,X_1,M] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),X_1),M))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),G)),N)),X_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,G,X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))),M))) ) ).
fof(fact_3880_card__bdd__int__set__le,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),N)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,N)),one_one(nat)) ) ).
fof(fact_3881_int__card__bdd__int__set__l__l,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),N))))) = hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),N),one_one(int)) ) ).
fof(fact_3882_card__bdd__int__set__l__l,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
=> hAPP(fun(int,bool),nat,finite_card(int),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),N)))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,nat_1,N)),one_one(nat)) ) ).
fof(fact_3883_DERIV__cos__cos__mult,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),cos)),X_1),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X_1))),hAPP(real,real,cos,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X_1))),hAPP(real,real,cos,X_1))))) ).
fof(fact_3884_eq__number__of__Suc,axiom,
! [V_1,N] :
( hAPP(int,nat,number_number_of(nat),V_1) = hAPP(nat,nat,suc,N)
<=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
=> hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))) = N ) ) ) ).
fof(fact_3885_Suc__eq__number__of,axiom,
! [N,V_1] :
( hAPP(nat,nat,suc,N) = hAPP(int,nat,number_number_of(nat),V_1)
<=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
=> hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))) = N ) ) ) ).
fof(fact_3886_DERIV__power__Suc,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [N,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(nat,fun(X_a,X_a)),combc(X_a,nat,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combb(X_a,fun(nat,X_a),X_a),power_power(X_a)),F)),hAPP(nat,nat,suc,N))),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),hAPP(nat,X_a,semiring_1_of_nat(X_a),N))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D_2),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,F,X_1)),N))))) ) ) ).
fof(fact_3887_DERIV__inverse,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [X_1] :
( ti(X_a,X_1) != zero_zero(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),inverse_inverse(X_a)),X_1),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,inverse_inverse(X_a),X_1)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))) ) ) ).
fof(fact_3888_DERIV__power,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [N,F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(nat,fun(X_a,X_a)),combc(X_a,nat,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combb(X_a,fun(nat,X_a),X_a),power_power(X_a)),F)),N)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D_2),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,F,X_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))))) ) ) ).
fof(fact_3889_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_3890_DERIV__pow,axiom,
! [N,X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),N)),X_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat))))))) ).
fof(fact_3891_DERIV__cos__cos__mult2,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),cos)),X_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,min))),hAPP(real,real,cos,X_1))),hAPP(real,real,sin,X_1)))) ).
fof(fact_3892_less__number__of__Suc,axiom,
! [V_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,number_number_of(nat),V_1)),hAPP(nat,nat,suc,N)))
<=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),N)) ) ) ).
fof(fact_3893_less__Suc__number__of,axiom,
! [N,V_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,N)),hAPP(int,nat,number_number_of(nat),V_1)))
<=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))) ) ) ) ).
fof(fact_3894_le__Suc__number__of,axiom,
! [N,V_1] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,N)),hAPP(int,nat,number_number_of(nat),V_1)))
<=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
& ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))) ) ) ) ).
fof(fact_3895_le__number__of__Suc,axiom,
! [V_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,number_number_of(nat),V_1)),hAPP(nat,nat,suc,N)))
<=> ( ~ hBOOL(hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),N)) ) ) ).
fof(fact_3896_SR__pos,axiom,
! [X_3,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> hBOOL(hAPP(fun(int,bool),bool,hAPP(fun(int,bool),fun(fun(int,bool),bool),ord_less_eq(fun(int,bool)),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),standardRes,M)),X_3)),hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),M))))) ) ).
fof(fact_3897_DERIV__quotient,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [G,E_2,F,X_1,D_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),G),X_1),E_2))
=> ( hAPP(X_a,X_a,G,X_1) != zero_zero(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),F)),G)),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D_1),hAPP(X_a,X_a,G,X_1))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),E_2),hAPP(X_a,X_a,F,X_1)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,G,X_1)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))) ) ) ) ) ).
fof(fact_3898_power__number__of__odd,axiom,
! [X_a] :
( monoid_mult(X_a)
=> ! [Z,W] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Z),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),hAPP(int,nat,number_number_of(nat),W)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),hAPP(int,nat,number_number_of(nat),W))) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
=> hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = one_one(X_a) ) ) ) ).
fof(fact_3899_DERIV__inverse__fun,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [F,X_1,D_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_1))
=> ( hAPP(X_a,X_a,F,X_1) != zero_zero(X_a)
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),inverse_inverse(X_a)),F)),X_1),hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),D_1),hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,F,X_1)),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat))))))))) ) ) ) ).
fof(fact_3900_RsetR__zmult__mono,axiom,
! [X_1,A_2,M] :
( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),A_2),hAPP(int,fun(fun(int,bool),bool),rsetR,M)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,X_1),M) = one_one(int)
=> hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),times_times(int)),X_1)),A_2)),hAPP(int,fun(fun(int,bool),bool),rsetR,M))) ) ) ) ).
fof(fact_3901_zpower__number__of__odd,axiom,
! [Z,W] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),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),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z),hAPP(int,nat,number_number_of(nat),W)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Z),hAPP(int,nat,number_number_of(nat),W))) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,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),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = one_one(int) ) ) ).
fof(fact_3902_DERIV__sin__sin__mult2,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),sin)),X_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,X_1))),hAPP(real,real,sin,X_1)))) ).
fof(fact_3903_DERIV__log,axiom,
! [Ba,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(real,fun(real,real),log,Ba)),X_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,ln,Ba)),X_1)))) ) ).
fof(fact_3904_power__number__of__odd__number__of,axiom,
! [X_b] :
( ( monoid_mult(X_b)
& number(X_b) )
=> ! [V,W] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
=> hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W)))),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),W))) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,number_number_of(int),W)))
=> hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),hAPP(int,X_b,number_number_of(X_b),V)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit1,W))) = one_one(X_b) ) ) ) ).
fof(fact_3905_DERIV__cos__cos__mult3,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),cos)),X_1),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,X_1))),hAPP(real,real,sin,X_1))))) ).
fof(fact_3906_DERIV__sin__circle__all__zero,axiom,
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_2),zero_zero(real))) ).
fof(fact_3907_DERIV__sin__realpow2a,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X_1),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,X_1))),hAPP(real,real,sin,X_1)))) ).
fof(fact_3908_DERIV__sin__realpow2,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),X_1),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X_1)),hAPP(real,real,sin,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,cos,X_1)),hAPP(real,real,sin,X_1))))) ).
fof(fact_3909_diff__nat__number__of,axiom,
! [V_1,V_2] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),V_2),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(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(int,fun(int,bool),ord_less(int),V_2),pls))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(int,nat,number_number_of(nat),V_1)),hAPP(int,nat,number_number_of(nat),V_2)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),V_1),hAPP(int,int,uminus_uminus(int),V_2))))),zero_zero(nat)),hAPP(int,nat,nat_1,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),V_2))))) ) ) ).
fof(fact_3910_DERIV__cos__realpow2a,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),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(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,min))),hAPP(real,real,cos,X_1))),hAPP(real,real,sin,X_1)))) ).
fof(fact_3911_DERIV__cos__realpow2b,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),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,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,X_1))),hAPP(real,real,sin,X_1))))) ).
fof(fact_3912_DERIV__cos__realpow2,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),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(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X_1))),hAPP(real,real,cos,X_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X_1))),hAPP(real,real,cos,X_1))))) ).
fof(fact_3913_lemma__DERIV__tan,axiom,
! [X_1] :
( hAPP(real,real,cos,X_1) != zero_zero(real)
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),sin)),cos)),X_1),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,cos,X_1)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ) ).
fof(fact_3914_lemma__DERIV__sin__cos__add,axiom,
! [Y_1,X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),sin),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),Y_1)))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),hAPP(real,real,cos,Y_1)))),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),hAPP(real,real,sin,Y_1)))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),cos),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),plus_plus(real)),Y_1)))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),cos)),hAPP(real,real,cos,Y_1)))),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),times_times(real)),sin)),hAPP(real,real,sin,Y_1)))))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_2),zero_zero(real))) ).
fof(fact_3915_DERIV__sin__circle__all,axiom,
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),sin)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),hAPP(fun(real,real),fun(real,fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(fun(real,real),fun(real,fun(nat,real))),combb(real,fun(nat,real),real),power_power(real)),cos)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),X_2),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,X_2))),hAPP(real,real,sin,X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(real,real,cos,X_2))),hAPP(real,real,sin,X_2))))) ).
fof(fact_3916_norm__frac_Opsimps,axiom,
! [Aa,Ba] :
( hBOOL(hAPP(product_prod(int,int),bool,hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),bool),accp(product_prod(int,int)),norm_frac_rel),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),Aa),Ba)))
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ba),zero_zero(int)))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,Aa),Ba) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,hAPP(int,int,uminus_uminus(int),Aa)),hAPP(int,int,uminus_uminus(int),Ba)) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Ba),zero_zero(int)))
=> ( ( ( Aa = zero_zero(int)
| Ba = zero_zero(int) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,Aa),Ba) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) )
& ( ~ ( Aa = zero_zero(int)
| Ba = zero_zero(int) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,Aa),Ba) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,Aa),Ba))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),Ba),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,Aa),Ba))) ) ) ) ) ) ).
fof(fact_3917_monoseq__arctan__series,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X_1)),one_one(real)))
=> hBOOL(hAPP(fun(nat,real),bool,monoseq(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),X_1)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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_3918_monoseq__minus,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [Aa] :
( hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),Aa))
=> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),Aa))) ) ) ).
fof(fact_3919_monoseq__Suc,axiom,
! [X_a] :
( order(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_3))
<=> ( ! [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_3,N_2)),hAPP(nat,X_a,X_3,hAPP(nat,nat,suc,N_2))))
| ! [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_3,hAPP(nat,nat,suc,N_2))),hAPP(nat,X_a,X_3,N_2))) ) ) ) ).
fof(fact_3920_monoseq__def,axiom,
! [X_a] :
( order(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_3))
<=> ( ! [M_3,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_3,M_3)),hAPP(nat,X_a,X_3,N_2))) )
| ! [M_3,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_3,N_2)),hAPP(nat,X_a,X_3,M_3))) ) ) ) ) ).
fof(fact_3921_monoseq__realpow,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),one_one(real)))
=> hBOOL(hAPP(fun(nat,real),bool,monoseq(real),hAPP(real,fun(nat,real),power_power(real),X_1))) ) ) ).
fof(fact_3922_norm__frac_Osimps,axiom,
! [A_1,B] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,hAPP(int,int,uminus_uminus(int),A_1)),hAPP(int,int,uminus_uminus(int),B)) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),B),zero_zero(int)))
=> ( ( ( A_1 = zero_zero(int)
| B = zero_zero(int) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) )
& ( ~ ( A_1 = zero_zero(int)
| B = zero_zero(int) )
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),int_gcd,A_1),B))) ) ) ) ) ).
fof(fact_3923_arg__def,axiom,
! [Z_1] : hAPP(complex,real,arg,Z_1) = hAPP(fun(real,bool),real,hilbert_Eps(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(fun(real,real),fun(real,bool),hAPP(fun(real,bool),fun(fun(real,real),fun(real,bool)),combb(real,bool,real),hAPP(real,fun(real,bool),fequal(real),hAPP(complex,real,re,hAPP(complex,complex,sgn_sgn(complex),Z_1)))),cos))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(fun(real,real),fun(real,bool),hAPP(fun(real,bool),fun(fun(real,real),fun(real,bool)),combb(real,bool,real),hAPP(real,fun(real,bool),fequal(real),hAPP(complex,real,im,hAPP(complex,complex,sgn_sgn(complex),Z_1)))),sin))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),pi)))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),pi))))) ).
fof(fact_3924_Union__SetS__setprod__prop1,axiom,
! [Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),zero_zero(int)),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,Pa),Aa))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,Aa),Pa)))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),Aa),hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),Pa)) ) ) ) ) ).
fof(fact_3925_of__int__setprod,axiom,
! [X_b,X_a] :
( ( comm_monoid_mult(X_a)
& ring_1(X_a) )
=> ! [F,A_2] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(fun(X_b,bool),int,hAPP(fun(X_b,int),fun(fun(X_b,bool),int),big_co302546470etprod(X_b,int),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,int),fun(X_b,X_a),hAPP(fun(int,X_a),fun(fun(X_b,int),fun(X_b,X_a)),combb(int,X_a,X_b),ring_1_of_int(X_a)),F)),A_2) ) ).
fof(fact_3926_of__nat__setprod,axiom,
! [X_b,X_a] :
( ( comm_monoid_mult(X_a)
& semiring_1(X_a) )
=> ! [F,A_2] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co302546470etprod(X_b,nat),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,nat),fun(X_b,X_a),hAPP(fun(nat,X_a),fun(fun(X_b,nat),fun(X_b,X_a)),combb(nat,X_a,X_b),semiring_1_of_nat(X_a)),F)),A_2) ) ).
fof(fact_3927_d22set__prod__zfact,axiom,
! [Aa] : hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),d22set,Aa)) = hAPP(int,int,zfact,Aa) ).
fof(fact_3928_Bnor__prod__zgcd,axiom,
! [Aa,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),M))
=> hAPP(int,int,hAPP(int,fun(int,int),legacy_zgcd,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),M))),M) = one_one(int) ) ).
fof(fact_3929_Bnor__prod__power,axiom,
! [Aa,M,X_1] :
( X_1 != zero_zero(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),M))
=> hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),times_times(int)),X_1)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),M))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),M))),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),X_1),hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),bnorRset,Aa),M)))) ) ) ).
fof(fact_3930_SetS__setprod__prop,axiom,
! [X_1,Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),zero_zero(int)),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),quadRes,Pa),Aa))
=> ( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_1),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,Aa),Pa)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),X_1)),Aa),Pa)) ) ) ) ) ) ).
fof(fact_3931_wset__zcong__prod__1,axiom,
! [Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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))))),Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),wset,Aa),Pa))),one_one(int)),Pa)) ) ) ) ).
fof(fact_3932_Union__SetS__setprod__prop2,axiom,
! [Aa,Pa] :
( hBOOL(hAPP(int,bool,zprime,Pa))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),Pa))
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),zero_zero(int)),Pa))
=> hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(fun(fun(int,bool),bool),fun(int,bool),complete_Sup_Sup(fun(int,bool)),hAPP(int,fun(fun(int,bool),bool),hAPP(int,fun(int,fun(fun(int,bool),bool)),setS,Aa),Pa))) = hAPP(int,int,zfact,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Pa),one_one(int))) ) ) ) ).
fof(fact_3933_setprod__gen__delta,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [Ba,C_1,Aa,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),Aa))),Ba)),C_1)),S) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(X_a,X_b,Ba,Aa)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),C_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),S)),one_one(nat)))) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),Aa))),Ba)),C_1)),S) = hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),C_1),hAPP(fun(X_a,bool),nat,finite_card(X_a),S)) ) ) ) ) ).
fof(fact_3934_transfer__nat__int__sum__prod__closure_I2_J,axiom,
! [F,A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> ( ! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,F,X_2))) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),F),A_2))) ) ) ).
fof(fact_3935_setprod__eq__1__iff,axiom,
! [X_a,F,F_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
=> ( hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),F_1) = one_one(nat)
<=> ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),F_1))
=> hAPP(X_a,nat,F,X_2) = one_one(nat) ) ) ) ).
fof(fact_3936_setprod__pos__nat__iff,axiom,
! [X_a,F,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),S)))
<=> ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),S))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(X_a,nat,F,X_2))) ) ) ) ).
fof(fact_3937_real__of__nat__setprod,axiom,
! [X_a,F,A_2] : hAPP(nat,real,real_1(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co302546470etprod(X_a,real),hAPP(fun(X_a,nat),fun(X_a,real),hAPP(fun(nat,real),fun(fun(X_a,nat),fun(X_a,real)),combb(nat,real,X_a),real_1(nat)),F)),A_2) ).
fof(fact_3938_real__of__int__setprod,axiom,
! [X_a,F,A_2] : hAPP(int,real,real_1(int),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co302546470etprod(X_a,real),hAPP(fun(X_a,int),fun(X_a,real),hAPP(fun(int,real),fun(fun(X_a,int),fun(X_a,real)),combb(int,real,X_a),real_1(int)),F)),A_2) ).
fof(fact_3939_int__setprod,axiom,
! [X_a,F,A_2] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2)) = hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2) ).
fof(fact_3940_transfer__nat__int__sum__prod2_I2_J,axiom,
! [X_a,F,A_2] : hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2) = hAPP(int,nat,nat_1,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2)) ).
fof(fact_3941_transfer__nat__int__sum__prod_I2_J,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& comm_monoid_mult(X_a) )
=> ! [F,A_2] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co302546470etprod(nat,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),hAPP(fun(int,nat),fun(int,X_a),hAPP(fun(nat,X_a),fun(fun(int,nat),fun(int,X_a)),combb(nat,X_a,int),F),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) ) ).
fof(fact_3942_transfer__int__nat__sum__prod_I2_J,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& comm_monoid_mult(X_a) )
=> ! [F,A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),F),A_2) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co302546470etprod(nat,X_a),hAPP(fun(nat,int),fun(nat,X_a),hAPP(fun(int,X_a),fun(fun(nat,int),fun(nat,X_a)),combb(int,X_a,nat),F),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)) ) ) ).
fof(fact_3943_setprod__timesf,axiom,
! [X_b,X_a] :
( comm_monoid_mult(X_a)
=> ! [F,G,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,X_a)),combs(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),F)),G)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),G),A_2)) ) ).
fof(fact_3944_setprod__1,axiom,
! [X_b,X_a] :
( comm_monoid_mult(X_a)
=> ! [A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),one_one(X_a))),A_2) = one_one(X_a) ) ).
fof(fact_3945_abs__setprod,axiom,
! [X_b,X_a] :
( linordered_field(X_a)
=> ! [F,A_2] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2) ) ).
fof(fact_3946_setprod__zero__iff,axiom,
! [X_a,X_b] :
( ( comm_semiring_1(X_b)
& no_zero_divisors(X_b) )
=> ! [F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2) = zero_zero(X_b)
<=> ? [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
& hAPP(X_a,X_b,F,X_2) = zero_zero(X_b) ) ) ) ) ).
fof(fact_3947_setprod__zero,axiom,
! [X_a,X_b] :
( comm_semiring_1(X_b)
=> ! [F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ? [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
& hAPP(X_a,X_b,F,X_2) = zero_zero(X_b) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2) = zero_zero(X_b) ) ) ) ).
fof(fact_3948_setprod_Oinfinite,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [G,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2) = one_one(X_b) ) ) ).
fof(fact_3949_setprod__infinite,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [F,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2) = one_one(X_b) ) ) ).
fof(fact_3950_dvd__setprod,axiom,
! [X_a,X_b] :
( comm_semiring_1(X_b)
=> ! [F,I,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(X_a,X_b,F,I)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2))) ) ) ) ).
fof(fact_3951_setprod_Odistrib,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [G,H,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),H)),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),H),A_2)) ) ) ).
fof(fact_3952_setprod__delta_H,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [Ba,Aa,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),Aa))),Ba)),one_one(X_b))),S) = hAPP(X_a,X_b,Ba,Aa) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),Aa))),Ba)),one_one(X_b))),S) = one_one(X_b) ) ) ) ) ).
fof(fact_3953_setprod__delta,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [Ba,Aa,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),Aa))),Ba)),one_one(X_b))),S) = hAPP(X_a,X_b,Ba,Aa) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),Aa))),Ba)),one_one(X_b))),S) = one_one(X_b) ) ) ) ) ).
fof(fact_3954_setprod__dividef,axiom,
! [X_a,X_b] :
( field_inverse_zero(X_b)
=> ! [F,G,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),F)),G)),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2)) ) ) ).
fof(fact_3955_setprod__dvd__setprod__subset,axiom,
! [X_a,X_b] :
( ( comm_monoid_mult(X_b)
& dvd(X_b) )
=> ! [F,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),B_1))) ) ) ) ).
fof(fact_3956_setprod__constant,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [Y_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),Y_1)),A_2) = hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),Y_1),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)) ) ) ).
fof(fact_3957_setprod__mono__one__right,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [F,S,T_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_4))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_4),S)))
=> hAPP(X_a,X_b,F,X_2) = one_one(X_b) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),T_4) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),S) ) ) ) ) ).
fof(fact_3958_setprod__mono__one__left,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [F,S,T_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_4))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_4),S)))
=> hAPP(X_a,X_b,F,X_2) = one_one(X_b) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),S) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),T_4) ) ) ) ) ).
fof(fact_3959_transfer__nat__int__sum__prod__cong_I2_J,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& comm_monoid_mult(X_a) )
=> ! [F,G,A_2,B_1] :
( A_2 = B_1
=> ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1))
=> ( ! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
=> hAPP(int,X_a,F,X_2) = hAPP(int,X_a,G,X_2) )
=> hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co302546470etprod(int,X_a),G),B_1) ) ) ) ) ).
fof(fact_3960_setprod__pos__nat,axiom,
! [X_a,F,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),S))
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(X_a,nat,F,X_2))) )
=> hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),S))) ) ) ).
fof(fact_3961_imageE,axiom,
! [X_a,X_b,Ba,F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Ba),hAPP(fun(X_b,bool),fun(X_a,bool),hAPP(fun(X_b,X_a),fun(fun(X_b,bool),fun(X_a,bool)),image(X_b,X_a),F),A_2)))
=> ~ ! [X_2] :
( ti(X_a,Ba) = hAPP(X_b,X_a,F,X_2)
=> ~ hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),A_2)) ) ) ).
fof(fact_3962_setprod__dvd__setprod__subset2,axiom,
! [X_a,X_b] :
( comm_semiring_1(X_b)
=> ! [F,G,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,G,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),B_1))) ) ) ) ) ).
fof(fact_3963_arctan__series,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X_1)),one_one(real)))
=> hAPP(real,real,arctan,X_1) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),X_1)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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_3964_DERIV__arctan__series,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),X_1)),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,fun(nat,real)),fun(real,real),hAPP(fun(fun(nat,real),real),fun(fun(real,fun(nat,real)),fun(real,real)),combb(fun(nat,real),real,real),suminf(real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))))),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(real,fun(nat,real)),hAPP(fun(real,fun(fun(nat,nat),fun(nat,real))),fun(fun(nat,nat),fun(real,fun(nat,real))),combc(real,fun(nat,nat),fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real))),hAPP(fun(fun(nat,real),fun(fun(nat,nat),fun(nat,real))),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real)))),combb(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),real),combb(nat,real,nat)),power_power(real))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))))),X_1),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),X_1)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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_3965_suminf__zero,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& t2_space(X_a) )
=> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),zero_zero(X_a))) = zero_zero(X_a) ) ).
fof(fact_3966_sin__def,axiom,
! [X_1] : hAPP(real,real,sin,X_1) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1))) ).
fof(fact_3967_lemma__sin__ext,axiom,
! [X_2] : hAPP(real,real,sin,X_2) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),X_2))) ).
fof(fact_3968_cos__def,axiom,
! [X_1] : hAPP(real,real,cos,X_1) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1))) ).
fof(fact_3969_lemma__cos__ext,axiom,
! [X_2] : hAPP(real,real,cos,X_2) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),X_2))) ).
fof(fact_3970_powser__zero,axiom,
! [X_a] :
( real_n2089651433ebra_1(X_a)
=> ! [F] : hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),zero_zero(X_a)))) = hAPP(nat,X_a,F,zero_zero(nat)) ) ).
fof(fact_3971_lemma__sin__minus,axiom,
! [X_1] : hAPP(real,real,uminus_uminus(real),hAPP(real,real,sin,X_1)) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1)))) ).
fof(fact_3972_exp__first__two__terms,axiom,
! [X_1] : hAPP(real,real,exp(real),X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),one_one(real)),X_1)),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),X_1)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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_3973_pi__series,axiom,
hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),one_one(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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_3974_ln__series,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))
=> hAPP(real,real,ln,X_1) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X_1),one_one(real)))),suc))) ) ) ).
fof(fact_3975_of__int__num,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [K] :
( ( K = zero_zero(int)
=> hAPP(int,X_a,ring_1_of_int(X_a),K) = zero_zero(X_a) )
& ( K != zero_zero(int)
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> hAPP(int,X_a,ring_1_of_int(X_a),K) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(int,X_a,ring_1_of_int(X_a),hAPP(int,int,uminus_uminus(int),K))) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> hAPP(int,X_a,ring_1_of_int(X_a),K) = hAPP(product_prod(int,int),X_a,hAPP(fun(int,fun(int,X_a)),fun(product_prod(int,int),X_a),product_prod_case(int,int,X_a),hAPP(fun(int,X_a),fun(int,fun(int,X_a)),hAPP(fun(int,fun(X_a,fun(int,X_a))),fun(fun(int,X_a),fun(int,fun(int,X_a))),combs(int,X_a,fun(int,X_a)),hAPP(fun(int,fun(int,fun(X_a,X_a))),fun(int,fun(X_a,fun(int,X_a))),hAPP(fun(fun(int,fun(X_a,X_a)),fun(X_a,fun(int,X_a))),fun(fun(int,fun(int,fun(X_a,X_a))),fun(int,fun(X_a,fun(int,X_a)))),combb(fun(int,fun(X_a,X_a)),fun(X_a,fun(int,X_a)),int),combc(int,X_a,X_a)),hAPP(fun(int,X_a),fun(int,fun(int,fun(X_a,X_a))),hAPP(fun(X_a,fun(int,fun(X_a,X_a))),fun(fun(int,X_a),fun(int,fun(int,fun(X_a,X_a)))),combb(X_a,fun(int,fun(X_a,X_a)),int),hAPP(fun(int,fun(X_a,fun(X_a,X_a))),fun(X_a,fun(int,fun(X_a,X_a))),combc(int,X_a,fun(X_a,X_a)),hAPP(fun(int,bool),fun(int,fun(X_a,fun(X_a,X_a))),hAPP(fun(bool,fun(X_a,fun(X_a,X_a))),fun(fun(int,bool),fun(int,fun(X_a,fun(X_a,X_a)))),combb(bool,fun(X_a,fun(X_a,X_a)),int),if(X_a)),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))),hAPP(fun(int,X_a),fun(int,X_a),hAPP(fun(int,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,X_a)),combs(int,X_a,X_a),hAPP(fun(int,X_a),fun(int,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),int),plus_plus(X_a)),ring_1_of_int(X_a))),ring_1_of_int(X_a))))),hAPP(X_a,fun(int,X_a),hAPP(fun(int,fun(X_a,X_a)),fun(X_a,fun(int,X_a)),combc(int,X_a,X_a),hAPP(fun(int,X_a),fun(int,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),int),plus_plus(X_a)),hAPP(fun(int,X_a),fun(int,X_a),hAPP(fun(int,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,X_a)),combs(int,X_a,X_a),hAPP(fun(int,X_a),fun(int,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(int,X_a),fun(int,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),int),plus_plus(X_a)),ring_1_of_int(X_a))),ring_1_of_int(X_a)))),one_one(X_a)))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))) ) ) ) ) ) ).
fof(fact_3976_summable__arctan__series,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X_1)),one_one(real)))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),X_1)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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_3977_summable__zero,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),zero_zero(X_a)))) ) ).
fof(fact_3978_pair__imageI,axiom,
! [X_c,X_a,X_b,F,Aa,Ba,A_2] :
( hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Aa),Ba)),A_2))
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),F,Aa),Ba)),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_c,bool),hAPP(fun(product_prod(X_a,X_b),X_c),fun(fun(product_prod(X_a,X_b),bool),fun(X_c,bool)),image(product_prod(X_a,X_b),X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),F)),A_2))) ) ).
fof(fact_3979_suminf__add,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_3,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_3))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_3)),hAPP(fun(nat,X_a),X_a,suminf(X_a),Y_3)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_3)),Y_3)) ) ) ) ).
fof(fact_3980_suminf__diff,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_3,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_3))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_3)),hAPP(fun(nat,X_a),X_a,suminf(X_a),Y_3)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_3)),Y_3)) ) ) ) ).
fof(fact_3981_suminf__divide,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [C_1,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),F)),C_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)),C_1) ) ) ).
fof(fact_3982_divide_Osuminf,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_3)),Y_1) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_3)),Y_1)) ) ) ).
fof(fact_3983_mult__left_Osuminf,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_3)),Y_1) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_3)),Y_1)) ) ) ).
fof(fact_3984_mult__right_Osuminf,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_3)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1)),X_3)) ) ) ).
fof(fact_3985_suminf__mult2,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [C_1,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)),C_1) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),C_1)) ) ) ).
fof(fact_3986_suminf__mult,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [C_1,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1)),F)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)) ) ) ).
fof(fact_3987_suminf__minus,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_3)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_3)) ) ) ).
fof(fact_3988_prod__case__beta,axiom,
! [X_b,X_c,X_a,F,Pa] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F),Pa) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F,hAPP(product_prod(X_b,X_c),X_b,product_fst(X_b,X_c),Pa)),hAPP(product_prod(X_b,X_c),X_c,product_snd(X_b,X_c),Pa)) ).
fof(fact_3989_internal__split__def,axiom,
! [X_a,X_b,X_c] : produc1605651328_split(X_a,X_b,X_c) = product_prod_case(X_a,X_b,X_c) ).
fof(fact_3990_Product__Type_Oprod_Osimps_I2_J,axiom,
! [X_b,X_c,X_a,F1,Aa,Ba] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F1),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),Aa),Ba)) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F1,Aa),Ba) ).
fof(fact_3991_split__conv,axiom,
! [X_b,X_c,X_a,F,Aa,Ba] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),Aa),Ba)) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),F,Aa),Ba) ).
fof(fact_3992_summable__comparison__test,axiom,
! [X_a] :
( banach(X_a)
=> ! [F,G] :
( ? [N_3] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_2))),hAPP(nat,real,G,N_2))) )
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),G))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) ) ) ) ).
fof(fact_3993_split__curry,axiom,
! [X_a,X_b,X_c,F] : hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),hAPP(fun(product_prod(X_a,X_b),X_c),fun(X_a,fun(X_b,X_c)),product_curry(X_a,X_b,X_c),F)) = ti(fun(product_prod(X_a,X_b),X_c),F) ).
fof(fact_3994_curry__split,axiom,
! [X_a,X_b,X_c,F] : hAPP(fun(product_prod(X_a,X_b),X_c),fun(X_a,fun(X_b,X_c)),product_curry(X_a,X_b,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),F)) = ti(fun(X_a,fun(X_b,X_c)),F) ).
fof(fact_3995_split__weak__cong,axiom,
! [X_c,X_a,X_b,C_1,Pa,Qa] :
( ti(product_prod(X_a,X_b),Pa) = ti(product_prod(X_a,X_b),Qa)
=> hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),C_1),Pa) = hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),C_1),Qa) ) ).
fof(fact_3996_powser__insidea,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [Z_1,F,X_1] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Z_1)),hAPP(X_a,real,norm_norm(X_a),X_1)))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1))))) ) ) ) ).
fof(fact_3997_summable__rabs__cancel,axiom,
! [F] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) ) ).
fof(fact_3998_summable__minus,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_3))) ) ) ).
fof(fact_3999_summable__mult,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [C_1,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1)),F))) ) ) ).
fof(fact_4000_summable__mult2,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [C_1,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),C_1))) ) ) ).
fof(fact_4001_mult__right_Osummable,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1)),X_3))) ) ) ).
fof(fact_4002_mult__left_Osummable,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_3)),Y_1))) ) ) ).
fof(fact_4003_summable__norm__cancel,axiom,
! [X_a] :
( banach(X_a)
=> ! [F] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F)))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) ) ) ).
fof(fact_4004_split__twice,axiom,
! [X_b,X_c,X_a,X_d,X_e,F,G,Pa] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F),hAPP(product_prod(X_d,X_e),product_prod(X_b,X_c),hAPP(fun(X_d,fun(X_e,product_prod(X_b,X_c))),fun(product_prod(X_d,X_e),product_prod(X_b,X_c)),product_prod_case(X_d,X_e,product_prod(X_b,X_c)),G),Pa)) = hAPP(product_prod(X_d,X_e),X_a,hAPP(fun(X_d,fun(X_e,X_a)),fun(product_prod(X_d,X_e),X_a),product_prod_case(X_d,X_e,X_a),hAPP(fun(X_d,fun(X_e,product_prod(X_b,X_c))),fun(X_d,fun(X_e,X_a)),hAPP(fun(fun(X_e,product_prod(X_b,X_c)),fun(X_e,X_a)),fun(fun(X_d,fun(X_e,product_prod(X_b,X_c))),fun(X_d,fun(X_e,X_a))),combb(fun(X_e,product_prod(X_b,X_c)),fun(X_e,X_a),X_d),hAPP(fun(product_prod(X_b,X_c),X_a),fun(fun(X_e,product_prod(X_b,X_c)),fun(X_e,X_a)),combb(product_prod(X_b,X_c),X_a,X_e),hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),F))),G)),Pa) ).
fof(fact_4005_snd__def,axiom,
! [X_b,X_a,Pa] : hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),Pa) = hAPP(product_prod(X_b,X_a),X_a,hAPP(fun(X_b,fun(X_a,X_a)),fun(product_prod(X_b,X_a),X_a),product_prod_case(X_b,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_b,fun(X_a,X_a)),combk(fun(X_a,X_a),X_b),combi(X_a))),Pa) ).
fof(fact_4006_fst__def,axiom,
! [X_a,X_b,Pa] : hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Pa) = hAPP(product_prod(X_a,X_b),X_a,hAPP(fun(X_a,fun(X_b,X_a)),fun(product_prod(X_a,X_b),X_a),product_prod_case(X_a,X_b,X_a),combk(X_a,X_b)),Pa) ).
fof(fact_4007_of__real_Osummable,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_3))) ) ) ).
fof(fact_4008_divide_Osummable,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_3)),Y_1))) ) ) ).
fof(fact_4009_summable__divide,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [C_1,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),F)),C_1))) ) ) ).
fof(fact_4010_split__eta,axiom,
! [X_a,X_b,X_c,F] : hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,X_c)),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,X_c)),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,X_c))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,X_c),X_a),hAPP(fun(product_prod(X_a,X_b),X_c),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,X_c)),combb(product_prod(X_a,X_b),X_c,X_b),F)),product_Pair(X_a,X_b))) = ti(fun(product_prod(X_a,X_b),X_c),F) ).
fof(fact_4011_summable__diff,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_3,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_3)),Y_3))) ) ) ) ).
fof(fact_4012_summable__ignore__initial__segment,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [K,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))) ) ) ).
fof(fact_4013_summable__add,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_3,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),Y_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_3)),Y_3))) ) ) ) ).
fof(fact_4014_split__def,axiom,
! [X_a,X_b,X_c,X_2,Xa] : hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),X_2),Xa) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),X_2,hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),Xa)),hAPP(product_prod(X_a,X_b),X_b,product_snd(X_a,X_b),Xa)) ).
fof(fact_4015_split__comp__eq,axiom,
! [X_a,X_b,X_c,X_d,F,G] : hAPP(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),X_c),hAPP(fun(product_prod(X_a,X_b),fun(X_b,X_c)),fun(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),X_c)),combs(product_prod(X_a,X_b),X_b,X_c),hAPP(fun(product_prod(X_a,X_b),X_d),fun(product_prod(X_a,X_b),fun(X_b,X_c)),hAPP(fun(X_d,fun(X_b,X_c)),fun(fun(product_prod(X_a,X_b),X_d),fun(product_prod(X_a,X_b),fun(X_b,X_c))),combb(X_d,fun(X_b,X_c),product_prod(X_a,X_b)),F),hAPP(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),X_d),hAPP(fun(X_a,X_d),fun(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),X_d)),combb(X_a,X_d,product_prod(X_a,X_b)),G),product_fst(X_a,X_b)))),product_snd(X_a,X_b)) = hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),hAPP(fun(X_a,X_d),fun(X_a,fun(X_b,X_c)),hAPP(fun(X_d,fun(X_b,X_c)),fun(fun(X_a,X_d),fun(X_a,fun(X_b,X_c))),combb(X_d,fun(X_b,X_c),X_a),F),G)) ).
fof(fact_4016_split__beta,axiom,
! [X_b,X_c,X_a,P_1,Z_1] : hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),P_1),Z_1) = hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),P_1,hAPP(product_prod(X_b,X_c),X_b,product_fst(X_b,X_c),Z_1)),hAPP(product_prod(X_b,X_c),X_c,product_snd(X_b,X_c),Z_1)) ).
fof(fact_4017_summable__norm__comparison__test,axiom,
! [X_a] :
( banach(X_a)
=> ! [F,G] :
( ? [N_3] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_2))),hAPP(nat,real,G,N_2))) )
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),G))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F))) ) ) ) ).
fof(fact_4018_summable__rabs__comparison__test,axiom,
! [F,G] :
( ? [N_3] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(nat,real,F,N_2))),hAPP(nat,real,G,N_2))) )
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),G))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F))) ) ) ).
fof(fact_4019_summable__rabs,axiom,
! [F] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(fun(nat,real),real,suminf(real),F))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)))) ) ).
fof(fact_4020_of__real_Osuminf,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),X_3))
=> hAPP(real,X_a,of_real(X_a),hAPP(fun(nat,real),real,suminf(real),X_3)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_3)) ) ) ).
fof(fact_4021_summable__sin,axiom,
! [X_1] : hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1)))) ).
fof(fact_4022_summable__cos,axiom,
! [X_1] : hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1)))) ).
fof(fact_4023_summable__norm,axiom,
! [X_a] :
( banach(X_a)
=> ! [F] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),F))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),F)))) ) ) ).
fof(fact_4024_powser__inside,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [Z_1,F,X_1] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),Z_1)),hAPP(X_a,real,norm_norm(X_a),X_1)))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1)))) ) ) ) ).
fof(fact_4025_complete__algebra__summable__geometric,axiom,
! [X_a] :
( ( real_n2089651433ebra_1(X_a)
& banach(X_a) )
=> ! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X_1)),one_one(real)))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))) ) ) ).
fof(fact_4026_summable__geometric,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X_1)),one_one(real)))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))) ) ) ).
fof(fact_4027_rat__floor__code,axiom,
! [Pa] : hAPP(rat,int,archim2085082626_floor(rat),Pa) = hAPP(product_prod(int,int),int,hAPP(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_prod_case(int,int,int),div_div(int)),hAPP(rat,product_prod(int,int),quotient_of,Pa)) ).
fof(fact_4028_rat__uminus__code,axiom,
! [Pa] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,uminus_uminus(rat),Pa)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),uminus_uminus(int))),hAPP(rat,product_prod(int,int),quotient_of,Pa)) ).
fof(fact_4029_summable__exp,axiom,
! [X_1] : hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),X_1)))) ).
fof(fact_4030_rat__abs__code,axiom,
! [Pa] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,abs_abs(rat),Pa)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),abs_abs(int))),hAPP(rat,product_prod(int,int),quotient_of,Pa)) ).
fof(fact_4031_divmod__int__rel__def,axiom,
! [Aa,Ba] : hAPP(int,fun(product_prod(int,int),bool),hAPP(int,fun(int,fun(product_prod(int,int),bool)),divmod_int_rel,Aa),Ba) = hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,bool),fun(int,fun(int,bool)),hAPP(fun(int,fun(fun(int,bool),fun(int,bool))),fun(fun(int,bool),fun(int,fun(int,bool))),combc(int,fun(int,bool),fun(int,bool)),hAPP(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool))),hAPP(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),int),combs(int,bool,bool)),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool))),hAPP(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),int),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj)),hAPP(fun(int,fun(int,int)),fun(int,fun(int,bool)),hAPP(fun(fun(int,int),fun(int,bool)),fun(fun(int,fun(int,int)),fun(int,fun(int,bool))),combb(fun(int,int),fun(int,bool),int),hAPP(fun(int,bool),fun(fun(int,int),fun(int,bool)),combb(int,bool,int),hAPP(int,fun(int,bool),fequal(int),Aa))),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),plus_plus(int)),hAPP(int,fun(int,int),times_times(int),Ba)))))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),combb(bool,bool,int),hAPP(bool,fun(bool,bool),fimplies,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),Ba))))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),combb(bool,bool,int),hAPP(bool,fun(bool,bool),fimplies,hAPP(bool,bool,fNot,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Ba)))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),Ba))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less_eq(int)),zero_zero(int))))))) ).
fof(fact_4032_sum2sq__def,axiom,
twoSqu1094008024sum2sq = hAPP(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_prod_case(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(fun(int,int),fun(int,int))),fun(fun(int,int),fun(int,fun(int,int))),combc(int,fun(int,int),fun(int,int)),hAPP(fun(int,fun(int,int)),fun(int,fun(fun(int,int),fun(int,int))),hAPP(fun(fun(int,int),fun(fun(int,int),fun(int,int))),fun(fun(int,fun(int,int)),fun(int,fun(fun(int,int),fun(int,int)))),combb(fun(int,int),fun(fun(int,int),fun(int,int)),int),combb(int,int,int)),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),plus_plus(int)),hAPP(nat,fun(int,int),hAPP(fun(int,fun(nat,int)),fun(nat,fun(int,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(fun(int,fun(nat,int)),fun(nat,fun(int,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_4033_rat__inverse__code,axiom,
! [Pa] : hAPP(rat,product_prod(int,int),quotient_of,hAPP(rat,rat,inverse_inverse(rat),Pa)) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(product_prod(int,int),product_prod(int,int)),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(product_prod(int,int),product_prod(int,int)),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combb(product_prod(int,int),product_prod(int,int),int)),hAPP(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combc(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int)))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combs(int,int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(int,product_prod(int,int)))),fun(int,fun(int,fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(int,product_prod(int,int)))),fun(int,fun(int,fun(int,product_prod(int,int))))),combb(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),int),combc(int,int,product_prod(int,int))),hAPP(fun(int,fun(int,int)),fun(int,fun(int,fun(int,product_prod(int,int)))),hAPP(fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),fun(fun(int,fun(int,int)),fun(int,fun(int,fun(int,product_prod(int,int))))),combb(fun(int,int),fun(int,fun(int,product_prod(int,int))),int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int))),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),int),times_times(int)),sgn_sgn(int))))),abs_abs(int)))),hAPP(rat,product_prod(int,int),quotient_of,Pa)) ).
fof(fact_4034_exp__tail__after__first__two__terms__summable,axiom,
! [X_1] : hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),X_1)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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_4035_adjust__def,axiom,
! [Ba] : hAPP(int,fun(product_prod(int,int),product_prod(int,int)),adjust,Ba) = hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combs(int,product_prod(int,int),product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(int,fun(product_prod(int,int),product_prod(int,int))),int),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,product_prod(int,int)),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combs(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(fun(int,int),fun(int,bool),hAPP(fun(int,bool),fun(fun(int,int),fun(int,bool)),combb(int,bool,int),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int))),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int)),Ba))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combc(int,fun(int,int),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int))),int),combb(int,product_prod(int,int),int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),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(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int)),Ba))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),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_4036_DERIV__power__series_H,axiom,
! [X0,F,R_3] :
( ! [X_2] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_2),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),hAPP(real,real,uminus_uminus(real),R_3)),R_3)))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))),hAPP(real,fun(nat,real),power_power(real),X_2)))) )
=> ( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X0),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),hAPP(real,real,uminus_uminus(real),R_3)),R_3)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,fun(nat,real)),fun(real,real),hAPP(fun(fun(nat,real),real),fun(fun(real,fun(nat,real)),fun(real,real)),combb(fun(nat,real),real,real),suminf(real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F))),hAPP(fun(nat,nat),fun(real,fun(nat,real)),hAPP(fun(real,fun(fun(nat,nat),fun(nat,real))),fun(fun(nat,nat),fun(real,fun(nat,real))),combc(real,fun(nat,nat),fun(nat,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real))),hAPP(fun(fun(nat,real),fun(fun(nat,nat),fun(nat,real))),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,nat),fun(nat,real)))),combb(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),real),combb(nat,real,nat)),power_power(real))),suc)))),X0),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))),hAPP(real,fun(nat,real),power_power(real),X0))))) ) ) ) ).
fof(fact_4037_Nitpick_OFrac__def,axiom,
frac_1 = hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),hAPP(fun(fun(int,bool),fun(int,bool)),fun(fun(int,fun(int,bool)),fun(int,fun(int,bool))),combb(fun(int,bool),fun(int,bool),int),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int))))),hAPP(int,fun(int,fun(int,bool)),hAPP(fun(int,fun(int,fun(int,bool))),fun(int,fun(int,fun(int,bool))),combc(int,int,fun(int,bool)),hAPP(fun(int,fun(int,fun(int,bool))),fun(int,fun(int,fun(int,bool))),hAPP(fun(fun(int,fun(int,bool)),fun(int,fun(int,bool))),fun(fun(int,fun(int,fun(int,bool))),fun(int,fun(int,fun(int,bool)))),combb(fun(int,fun(int,bool)),fun(int,fun(int,bool)),int),combc(int,int,bool)),hAPP(fun(int,fun(int,int)),fun(int,fun(int,fun(int,bool))),hAPP(fun(fun(int,int),fun(int,fun(int,bool))),fun(fun(int,fun(int,int)),fun(int,fun(int,fun(int,bool)))),combb(fun(int,int),fun(int,fun(int,bool)),int),hAPP(fun(int,fun(int,bool)),fun(fun(int,int),fun(int,fun(int,bool))),combb(int,fun(int,bool),int),fequal(int))),int_gcd))),one_one(int)))) ).
fof(fact_4038_mem__splitI,axiom,
! [X_a,X_c,X_b,Z_1,C_1,Aa,Ba] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z_1),hAPP(X_c,fun(X_a,bool),hAPP(X_b,fun(X_c,fun(X_a,bool)),C_1,Aa),Ba)))
=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z_1),hAPP(product_prod(X_b,X_c),fun(X_a,bool),hAPP(fun(X_b,fun(X_c,fun(X_a,bool))),fun(product_prod(X_b,X_c),fun(X_a,bool)),product_prod_case(X_b,X_c,fun(X_a,bool)),C_1),hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),Aa),Ba)))) ) ).
fof(fact_4039_prod__caseI,axiom,
! [X_b,X_a,F1,Aa,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),F1,Aa),Ba))
=> hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),F1),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Aa),Ba))) ) ).
fof(fact_4040_splitI,axiom,
! [X_b,X_a,F,Aa,Ba] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),F,Aa),Ba))
=> hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),F),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Aa),Ba))) ) ).
fof(fact_4041_cnj_Osuminf,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_3))
=> hAPP(complex,complex,cnj,hAPP(fun(nat,complex),complex,suminf(complex),X_3)) = hAPP(fun(nat,complex),complex,suminf(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_3)) ) ).
fof(fact_4042_splitD,axiom,
! [X_a,X_b,F,Aa,Ba] :
( hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),F),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Aa),Ba)))
=> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),F,Aa),Ba)) ) ).
fof(fact_4043_splitD_H,axiom,
! [X_c,X_a,X_b,R_3,Aa,Ba,C_1] :
( hBOOL(hAPP(X_c,bool,hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),R_3),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),Aa),Ba)),C_1))
=> hBOOL(hAPP(X_c,bool,hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),R_3,Aa),Ba),C_1)) ) ).
fof(fact_4044_cnj_Osummable,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_3))
=> hBOOL(hAPP(fun(nat,complex),bool,summable(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_3))) ) ).
fof(fact_4045_split__part,axiom,
! [X_a,X_b,P_1,Q_1,X_2] :
( hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),combb(bool,bool,X_b),hAPP(bool,fun(bool,bool),fconj,P_1))),Q_1)),X_2))
<=> ( hBOOL(P_1)
& hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),Q_1),X_2)) ) ) ).
fof(fact_4046_Re_Osummable,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_3))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_3))) ) ).
fof(fact_4047_Im_Osummable,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_3))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_3))) ) ).
fof(fact_4048_Re_Osuminf,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_3))
=> hAPP(complex,real,re,hAPP(fun(nat,complex),complex,suminf(complex),X_3)) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_3)) ) ).
fof(fact_4049_Im_Osuminf,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,complex),bool,summable(complex),X_3))
=> hAPP(complex,real,im,hAPP(fun(nat,complex),complex,suminf(complex),X_3)) = hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_3)) ) ).
fof(fact_4050_finite__psubset__def,axiom,
! [X_a] : finite_psubset(X_a) = hAPP(fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),collect(product_prod(fun(X_a,bool),fun(X_a,bool))),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(product_prod(fun(X_a,bool),fun(X_a,bool)),bool),product_prod_case(fun(X_a,bool),fun(X_a,bool),bool),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool))),combc(fun(X_a,bool),fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),hAPP(fun(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)))),combb(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),fun(X_a,bool)),combs(fun(X_a,bool),bool,bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),hAPP(fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool)))),combb(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool)),fun(X_a,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),combb(bool,fun(bool,bool),fun(X_a,bool)),fconj)),ord_less(fun(X_a,bool))))),finite_finite_1(X_a)))) ).
fof(fact_4051_ratrel__def,axiom,
ratrel = hAPP(fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),collect(product_prod(product_prod(int,int),product_prod(int,int))),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(product_prod(int,int),product_prod(int,int)),bool),product_prod_case(product_prod(int,int),product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),fun(product_prod(int,int),bool)),hAPP(fun(product_prod(int,int),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool))),fun(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),fun(product_prod(int,int),bool))),combs(product_prod(int,int),fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),hAPP(fun(product_prod(int,int),fun(bool,bool)),fun(product_prod(int,int),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool))),hAPP(fun(fun(bool,bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool))),fun(fun(product_prod(int,int),fun(bool,bool)),fun(product_prod(int,int),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)))),combb(fun(bool,bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),product_prod(int,int)),combb(bool,bool,product_prod(int,int))),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool))),combb(bool,fun(bool,bool),product_prod(int,int)),fconj),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),hAPP(fun(bool,bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combb(bool,bool,product_prod(int,int)),fNot),hAPP(int,fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(int,bool)),fun(int,fun(product_prod(int,int),bool)),combc(product_prod(int,int),int,bool),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),combb(int,fun(int,bool),product_prod(int,int)),fequal(int)),product_snd(int,int))),zero_zero(int)))))),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),fun(product_prod(int,int),bool)),hAPP(fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),fun(fun(product_prod(int,int),fun(product_prod(int,int),bool)),fun(product_prod(int,int),fun(product_prod(int,int),bool))),combb(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),product_prod(int,int)),hAPP(fun(product_prod(int,int),fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combs(product_prod(int,int),bool,bool),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool))),combb(bool,fun(bool,bool),product_prod(int,int)),fconj),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),hAPP(fun(bool,bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combb(bool,bool,product_prod(int,int)),fNot),hAPP(int,fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(int,bool)),fun(int,fun(product_prod(int,int),bool)),combc(product_prod(int,int),int,bool),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),combb(int,fun(int,bool),product_prod(int,int)),fequal(int)),product_snd(int,int))),zero_zero(int)))))),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),int)),fun(product_prod(int,int),fun(product_prod(int,int),bool)),hAPP(fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool))),fun(fun(product_prod(int,int),fun(product_prod(int,int),int)),fun(product_prod(int,int),fun(product_prod(int,int),bool))),combs(product_prod(int,int),fun(product_prod(int,int),int),fun(product_prod(int,int),bool)),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),fun(int,bool))),fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool))),hAPP(fun(fun(product_prod(int,int),fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool))),fun(fun(product_prod(int,int),fun(product_prod(int,int),fun(int,bool))),fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool)))),combb(fun(product_prod(int,int),fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool)),product_prod(int,int)),combs(product_prod(int,int),int,bool)),hAPP(fun(product_prod(int,int),fun(product_prod(int,int),int)),fun(product_prod(int,int),fun(product_prod(int,int),fun(int,bool))),hAPP(fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),fun(fun(product_prod(int,int),fun(product_prod(int,int),int)),fun(product_prod(int,int),fun(product_prod(int,int),fun(int,bool)))),combb(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool)),product_prod(int,int)),hAPP(fun(int,fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),combb(int,fun(int,bool),product_prod(int,int)),fequal(int))),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(product_prod(int,int),int)),hAPP(fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int))),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(product_prod(int,int),int))),combc(product_prod(int,int),fun(product_prod(int,int),int),fun(product_prod(int,int),int)),hAPP(fun(product_prod(int,int),fun(int,int)),fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int))),hAPP(fun(fun(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int))),fun(fun(product_prod(int,int),fun(int,int)),fun(product_prod(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int)))),combb(fun(int,int),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int)),product_prod(int,int)),combb(int,int,product_prod(int,int))),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int))),combb(int,fun(int,int),product_prod(int,int)),times_times(int)),product_fst(int,int)))),product_snd(int,int))))),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(product_prod(int,int),int)),hAPP(fun(int,fun(product_prod(int,int),int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(product_prod(int,int),int))),combb(int,fun(product_prod(int,int),int),product_prod(int,int)),hAPP(fun(product_prod(int,int),fun(int,int)),fun(int,fun(product_prod(int,int),int)),combc(product_prod(int,int),int,int),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int))),combb(int,fun(int,int),product_prod(int,int)),times_times(int)),product_fst(int,int)))),product_snd(int,int))))))) ).
fof(fact_4052_divmod__nat__if,axiom,
! [M,N] :
( ( ( N = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) )
=> hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M) )
& ( ~ ( N = zero_zero(nat)
| hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N)) )
=> hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = hAPP(product_prod(nat,nat),product_prod(nat,nat),hAPP(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_prod_case(nat,nat,product_prod(nat,nat)),hAPP(fun(nat,nat),fun(nat,fun(nat,product_prod(nat,nat))),hAPP(fun(nat,fun(nat,product_prod(nat,nat))),fun(fun(nat,nat),fun(nat,fun(nat,product_prod(nat,nat)))),combb(nat,fun(nat,product_prod(nat,nat)),nat),product_Pair(nat,nat)),suc)),hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ).
fof(fact_4053_Bolzano__bisect_Osimps_I2_J,axiom,
! [P_1,Aa,Ba,N] : hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),hAPP(nat,nat,suc,N)) = hAPP(product_prod(real,real),product_prod(real,real),hAPP(fun(real,fun(real,product_prod(real,real))),fun(product_prod(real,real),product_prod(real,real)),product_prod_case(real,real,product_prod(real,real)),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,product_prod(real,real)))),combs(real,fun(real,product_prod(real,real)),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real)))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,fun(product_prod(real,real),product_prod(real,real))),fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real)))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real))))),combb(fun(real,fun(product_prod(real,real),product_prod(real,real))),fun(fun(real,product_prod(real,real)),fun(real,product_prod(real,real))),real),combs(real,product_prod(real,real),product_prod(real,real))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real)))),hAPP(fun(real,fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real))))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,fun(product_prod(real,real),product_prod(real,real))))),combs(real,fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real)))),hAPP(fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real))))),hAPP(fun(fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real))))),fun(fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),fun(real,fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real)))))),combb(fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),fun(fun(real,product_prod(real,real)),fun(real,fun(product_prod(real,real),product_prod(real,real)))),real),combs(real,product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),hAPP(fun(fun(real,bool),fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),fun(fun(real,fun(real,bool)),fun(real,fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))))),combb(fun(real,bool),fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),real),hAPP(fun(bool,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real)))),fun(fun(real,bool),fun(real,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))))),combb(bool,fun(product_prod(real,real),fun(product_prod(real,real),product_prod(real,real))),real),if(product_prod(real,real)))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,bool)),hAPP(fun(fun(real,product_prod(real,real)),fun(real,bool)),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(real,bool))),combb(fun(real,product_prod(real,real)),fun(real,bool),real),hAPP(fun(product_prod(real,real),bool),fun(fun(real,product_prod(real,real)),fun(real,bool)),combb(product_prod(real,real),bool,real),P_1)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real)))),combs(real,fun(real,real),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real))))),combb(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real))),real),combb(real,product_prod(real,real),real)),product_Pair(real,real))),hAPP(real,fun(real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),combc(real,real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,fun(real,real)),fun(real,fun(real,real))),fun(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real)))),combb(fun(real,fun(real,real)),fun(real,fun(real,real)),real),combc(real,real,real)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,real),fun(real,fun(real,real))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real)))),combb(fun(real,real),fun(real,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real))),plus_plus(real)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(real,real),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,real),fun(real,fun(real,product_prod(real,real)))),combc(real,fun(real,real),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,fun(real,product_prod(real,real)))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,fun(real,product_prod(real,real))),fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,fun(real,product_prod(real,real)))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real))))),combb(fun(real,fun(real,product_prod(real,real))),fun(fun(real,real),fun(real,product_prod(real,real))),real),combs(real,real,product_prod(real,real))),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,product_prod(real,real)))),hAPP(fun(fun(real,real),fun(real,fun(real,product_prod(real,real)))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,product_prod(real,real))))),combb(fun(real,real),fun(real,fun(real,product_prod(real,real))),real),hAPP(fun(real,fun(real,product_prod(real,real))),fun(fun(real,real),fun(real,fun(real,product_prod(real,real)))),combb(real,fun(real,product_prod(real,real)),real),product_Pair(real,real))),hAPP(real,fun(real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),combc(real,real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,fun(real,real)),fun(real,fun(real,real))),fun(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real)))),combb(fun(real,fun(real,real)),fun(real,fun(real,real)),real),combc(real,real,real)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,real),fun(real,fun(real,real))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real)))),combb(fun(real,real),fun(real,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real))),plus_plus(real)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),combi(real))))),hAPP(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real))),hAPP(fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,real)),fun(real,fun(real,product_prod(real,real)))),combs(real,fun(real,real),fun(real,product_prod(real,real))),hAPP(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real)))),hAPP(fun(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real)))),fun(fun(real,fun(real,product_prod(real,real))),fun(real,fun(fun(real,real),fun(real,product_prod(real,real))))),combb(fun(real,product_prod(real,real)),fun(fun(real,real),fun(real,product_prod(real,real))),real),combb(real,product_prod(real,real),real)),product_Pair(real,real))),hAPP(real,fun(real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),combc(real,real,fun(real,real)),hAPP(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,fun(real,real)),fun(real,fun(real,real))),fun(fun(real,fun(real,fun(real,real))),fun(real,fun(real,fun(real,real)))),combb(fun(real,fun(real,real)),fun(real,fun(real,real)),real),combc(real,real,real)),hAPP(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real))),hAPP(fun(fun(real,real),fun(real,fun(real,real))),fun(fun(real,fun(real,real)),fun(real,fun(real,fun(real,real)))),combb(fun(real,real),fun(real,fun(real,real)),real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real))),plus_plus(real)))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N)) ).
fof(fact_4054_ratio__test,axiom,
! [X_a] :
( banach(X_a)
=> ! [F,N_4,C_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),C_1),one_one(real)))
=> ( ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_4),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_2)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C_1),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_2))))) )
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) ) ) ) ).
fof(fact_4055_ratio__test__lemma2,axiom,
! [X_a] :
( banach(X_a)
=> ! [F,C_1,N_4] :
( ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_4),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N_2)))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),C_1),hAPP(X_a,real,norm_norm(X_a),hAPP(nat,X_a,F,N_2))))) )
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),C_1))
| hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) ) ) ) ).
fof(fact_4056_suminf__exist__split,axiom,
! [Aa,R_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),Aa))
=> ? [N_3] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),Aa),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),N_2))))),R_1)) ) ) ) ).
fof(fact_4057_mem__splitI2,axiom,
! [X_c,X_a,X_b,Z_1,C_1,Pa] :
( ! [A_3,B_3] :
( ti(product_prod(X_a,X_b),Pa) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_3)
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),Z_1),hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),C_1,A_3),B_3))) )
=> hBOOL(hAPP(fun(X_c,bool),bool,hAPP(X_c,fun(fun(X_c,bool),bool),member(X_c),Z_1),hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),C_1),Pa))) ) ).
fof(fact_4058_splitE_H,axiom,
! [X_c,X_a,X_b,C_1,Pa,Z_1] :
( hBOOL(hAPP(X_c,bool,hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),C_1),Pa),Z_1))
=> ~ ! [X_2,Y_2] :
( ti(product_prod(X_a,X_b),Pa) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_2),Y_2)
=> ~ hBOOL(hAPP(X_c,bool,hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),C_1,X_2),Y_2),Z_1)) ) ) ).
fof(fact_4059_mem__splitE,axiom,
! [X_a,X_b,X_c,Z_1,C_1,Pa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z_1),hAPP(product_prod(X_b,X_c),fun(X_a,bool),hAPP(fun(X_b,fun(X_c,fun(X_a,bool))),fun(product_prod(X_b,X_c),fun(X_a,bool)),product_prod_case(X_b,X_c,fun(X_a,bool)),C_1),Pa)))
=> ~ ! [X_2,Y_2] :
( ti(product_prod(X_b,X_c),Pa) = hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),X_2),Y_2)
=> ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Z_1),hAPP(X_c,fun(X_a,bool),hAPP(X_b,fun(X_c,fun(X_a,bool)),C_1,X_2),Y_2))) ) ) ).
fof(fact_4060_splitI2_H,axiom,
! [X_c,X_b,X_a,C_1,X_1,Pa] :
( ! [A_3,B_3] :
( hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_3) = ti(product_prod(X_a,X_b),Pa)
=> hBOOL(hAPP(X_c,bool,hAPP(X_b,fun(X_c,bool),hAPP(X_a,fun(X_b,fun(X_c,bool)),C_1,A_3),B_3),X_1)) )
=> hBOOL(hAPP(X_c,bool,hAPP(product_prod(X_a,X_b),fun(X_c,bool),hAPP(fun(X_a,fun(X_b,fun(X_c,bool))),fun(product_prod(X_a,X_b),fun(X_c,bool)),product_prod_case(X_a,X_b,fun(X_c,bool)),C_1),Pa),X_1)) ) ).
fof(fact_4061_splitI2,axiom,
! [X_a,X_b,C_1,Pa] :
( ! [A_3,B_3] :
( ti(product_prod(X_a,X_b),Pa) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),A_3),B_3)
=> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),C_1,A_3),B_3)) )
=> hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),C_1),Pa)) ) ).
fof(fact_4062_splitE,axiom,
! [X_a,X_b,C_1,Pa] :
( hBOOL(hAPP(product_prod(X_a,X_b),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),C_1),Pa))
=> ~ ! [X_2,Y_2] :
( ti(product_prod(X_a,X_b),Pa) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_2),Y_2)
=> ~ hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),C_1,X_2),Y_2)) ) ) ).
fof(fact_4063_divmod__int__pdivmod,axiom,
! [L,K] :
( ( K = zero_zero(int)
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)) )
& ( K != zero_zero(int)
=> ( ( L = zero_zero(int)
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K) )
& ( L != zero_zero(int)
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,int),fun(product_prod(int,int),product_prod(int,int)),product_apsnd(int,int,int),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,sgn_sgn(int),L))),hAPP(product_prod(int,int),product_prod(int,int),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),hAPP(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),if(product_prod(int,int)),hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),L)),hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))),hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),L),zero_zero(int))),hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),K),zero_zero(int))))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)),hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combs(int,product_prod(int,int),product_prod(int,int))),hAPP(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int))))),combb(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int))),int),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combc(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))),hAPP(int,fun(int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),combc(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),uminus_uminus(int))),zero_zero(int))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combc(int,fun(int,int),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int))),int),combb(int,product_prod(int,int),int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),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))))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)))) ) ) ) ) ).
fof(fact_4064_apsnd__conv,axiom,
! [X_a,X_c,X_b,F,X_1,Y_1] : hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),F),hAPP(X_c,product_prod(X_a,X_c),hAPP(X_a,fun(X_c,product_prod(X_a,X_c)),product_Pair(X_a,X_c),X_1),Y_1)) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),hAPP(X_c,X_b,F,Y_1)) ).
fof(fact_4065_apsnd__eq__conv,axiom,
! [X_a,X_c,X_b,F,X_1,G] :
( hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),F),X_1) = hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),G),X_1)
<=> hAPP(X_c,X_b,F,hAPP(product_prod(X_a,X_c),X_c,product_snd(X_a,X_c),X_1)) = hAPP(X_c,X_b,G,hAPP(product_prod(X_a,X_c),X_c,product_snd(X_a,X_c),X_1)) ) ).
fof(fact_4066_snd__apsnd,axiom,
! [X_b,X_a,X_c,F,X_1] : hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),hAPP(product_prod(X_b,X_c),product_prod(X_b,X_a),hAPP(fun(X_c,X_a),fun(product_prod(X_b,X_c),product_prod(X_b,X_a)),product_apsnd(X_c,X_a,X_b),F),X_1)) = hAPP(X_c,X_a,F,hAPP(product_prod(X_b,X_c),X_c,product_snd(X_b,X_c),X_1)) ).
fof(fact_4067_fst__apsnd,axiom,
! [X_a,X_b,X_c,F,X_1] : hAPP(product_prod(X_a,X_b),X_a,product_fst(X_a,X_b),hAPP(product_prod(X_a,X_c),product_prod(X_a,X_b),hAPP(fun(X_c,X_b),fun(product_prod(X_a,X_c),product_prod(X_a,X_b)),product_apsnd(X_c,X_b,X_a),F),X_1)) = hAPP(product_prod(X_a,X_c),X_a,product_fst(X_a,X_c),X_1) ).
fof(fact_4068_negateSnd__def,axiom,
negateSnd = hAPP(fun(int,int),fun(product_prod(int,int),product_prod(int,int)),product_apsnd(int,int,int),uminus_uminus(int)) ).
fof(fact_4069_divmod__int__code,axiom,
! [L,K] :
( ( K = zero_zero(int)
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),zero_zero(int)) )
& ( K != zero_zero(int)
=> ( ( L = zero_zero(int)
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K) )
& ( L != zero_zero(int)
=> hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),divmod_int,K),L) = hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,int),fun(product_prod(int,int),product_prod(int,int)),product_apsnd(int,int,int),hAPP(int,fun(int,int),times_times(int),hAPP(int,int,sgn_sgn(int),L))),hAPP(product_prod(int,int),product_prod(int,int),hAPP(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),hAPP(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),if(product_prod(int,int)),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),hAPP(int,int,sgn_sgn(int),K)),hAPP(int,int,sgn_sgn(int),L))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)),hAPP(product_prod(int,int),product_prod(int,int),hAPP(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_prod_case(int,int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int)))),combs(int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(int,fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))))),combb(fun(int,fun(product_prod(int,int),product_prod(int,int))),fun(fun(int,product_prod(int,int)),fun(int,product_prod(int,int))),int),combs(int,product_prod(int,int),product_prod(int,int))),hAPP(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,product_prod(int,int)),fun(int,fun(int,fun(product_prod(int,int),product_prod(int,int))))),combb(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int))),int),hAPP(fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(product_prod(int,int),fun(int,fun(product_prod(int,int),product_prod(int,int)))),combc(int,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),hAPP(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),hAPP(fun(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),fun(fun(int,bool),fun(int,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))),combb(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),int),if(product_prod(int,int))),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))),hAPP(int,fun(int,product_prod(int,int)),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,product_prod(int,int))),combc(int,int,product_prod(int,int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),uminus_uminus(int))),zero_zero(int))))),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combc(int,fun(int,int),fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int)))),hAPP(fun(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int)))),fun(fun(int,fun(int,product_prod(int,int))),fun(int,fun(fun(int,int),fun(int,product_prod(int,int))))),combb(fun(int,product_prod(int,int)),fun(fun(int,int),fun(int,product_prod(int,int))),int),combb(int,product_prod(int,int),int)),hAPP(fun(int,int),fun(int,fun(int,product_prod(int,int))),hAPP(fun(int,fun(int,product_prod(int,int))),fun(fun(int,int),fun(int,fun(int,product_prod(int,int)))),combb(int,fun(int,product_prod(int,int)),int),product_Pair(int,int)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),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))))),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),pdivmod,K),L)))) ) ) ) ) ).
fof(fact_4070_summable__le2,axiom,
! [F,G] :
( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(nat,real,F,N_2))),hAPP(nat,real,G,N_2)))
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),G))
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),F)),hAPP(fun(nat,real),real,suminf(real),G))) ) ) ) ).
fof(fact_4071_suminf__gt__zero,axiom,
! [F] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,F,N_2)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).
fof(fact_4072_suminf__ge__zero,axiom,
! [F] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,N_2)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).
fof(fact_4073_suminf__0__le,axiom,
! [F] :
( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,N_2)))
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).
fof(fact_4074_int__ge__less__than__def,axiom,
! [D_1] : hAPP(int,fun(product_prod(int,int),bool),int_ge_less_than,D_1) = hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),collect(product_prod(int,int)),hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),hAPP(fun(int,fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(int,bool))),combs(int,fun(int,bool),fun(int,bool)),hAPP(fun(int,fun(bool,bool)),fun(int,fun(fun(int,bool),fun(int,bool))),hAPP(fun(fun(bool,bool),fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(bool,bool)),fun(int,fun(fun(int,bool),fun(int,bool)))),combb(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),int),combb(bool,bool,int)),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),D_1)))),ord_less(int)))) ).
fof(fact_4075_int__ge__less__than2__def,axiom,
! [D_1] : hAPP(int,fun(product_prod(int,int),bool),int_ge_less_than2,D_1) = hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),collect(product_prod(int,int)),hAPP(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_prod_case(int,int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),hAPP(fun(fun(int,bool),fun(int,bool)),fun(fun(int,fun(int,bool)),fun(int,fun(int,bool))),combb(fun(int,bool),fun(int,bool),int),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),D_1)))),ord_less(int)))) ).
fof(fact_4076_sin__paired,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),X_1)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),one_one(nat))))),hAPP(real,real,sin,X_1))) ).
fof(fact_4077_of__real_OCauchy,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [X_3] :
( hBOOL(hAPP(fun(nat,real),bool,cauchy(real),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_3))) ) ) ).
fof(fact_4078_sums__zero,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),zero_zero(X_a))),zero_zero(X_a))) ) ).
fof(fact_4079_sums__minus,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [X_3,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_3),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_3)),hAPP(X_a,X_a,uminus_uminus(X_a),Aa))) ) ) ).
fof(fact_4080_sums__mult,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [C_1,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1)),F)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),C_1),Aa))) ) ) ).
fof(fact_4081_sums__mult2,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [C_1,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),F)),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),C_1))) ) ) ).
fof(fact_4082_mult__right_Osums,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X_1,X_3,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_3),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1)),X_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Aa))) ) ) ).
fof(fact_4083_mult__left_Osums,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y_1,X_3,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_3),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_3)),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Y_1))) ) ) ).
fof(fact_4084_of__real_Osums,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [X_3,Aa] :
( hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),X_3),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_3)),hAPP(real,X_a,of_real(X_a),Aa))) ) ) ).
fof(fact_4085_divide_Osums,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_1,X_3,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_3),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_3)),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Y_1))) ) ) ).
fof(fact_4086_sums__divide,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [C_1,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),F)),C_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),C_1))) ) ) ).
fof(fact_4087_sums__diff,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_3,Ba,X_3,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_3),Aa))
=> ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),Y_3),Ba))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_3)),Y_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba))) ) ) ) ).
fof(fact_4088_sums__add,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_3,Ba,X_3,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_3),Aa))
=> ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),Y_3),Ba))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_3)),Y_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Ba))) ) ) ) ).
fof(fact_4089_sums__unique,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& t2_space(X_a) )
=> ! [F,S_3] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),S_3))
=> ti(X_a,S_3) = hAPP(fun(nat,X_a),X_a,suminf(X_a),F) ) ) ).
fof(fact_4090_sums__summable,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> ! [F,L] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),L))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F)) ) ) ).
fof(fact_4091_summable__def,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> ! [F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
<=> ? [X1] : hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),X1)) ) ) ).
fof(fact_4092_sums__iff,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& t2_space(X_a) )
=> ! [F,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),X_1))
<=> ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
& hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = ti(X_a,X_1) ) ) ) ).
fof(fact_4093_summable__sums,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& t2_space(X_a) )
=> ! [F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),hAPP(fun(nat,X_a),X_a,suminf(X_a),F))) ) ) ).
fof(fact_4094_sums__Suc__imp,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> ! [S_3,F] :
( hAPP(nat,X_a,F,zero_zero(nat)) = zero_zero(X_a)
=> ( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),S_3))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),S_3)) ) ) ) ).
fof(fact_4095_sums__Suc,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [F,L] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),L),hAPP(nat,X_a,F,zero_zero(nat))))) ) ) ).
fof(fact_4096_sin__converges,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),sin_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(real,real,sin,X_1))) ).
fof(fact_4097_cos__converges,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),cos_coeff)),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(real,real,cos,X_1))) ).
fof(fact_4098_geometric__sums,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X_1)),one_one(real)))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),one_one(X_a)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),one_one(X_a)),X_1)))) ) ) ).
fof(fact_4099_power__half__series,axiom,
hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),suc)),one_one(real))) ).
fof(fact_4100_mult__left_OCauchy,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_3)),Y_1))) ) ) ).
fof(fact_4101_mult__right_OCauchy,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1)),X_3))) ) ) ).
fof(fact_4102_divide_OCauchy,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_3)),Y_1))) ) ) ).
fof(fact_4103_Ln_Oaux2,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))) ).
fof(fact_4104_sums__if_H,axiom,
! [G,X_1] :
( hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),G),X_1))
=> hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),G),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),one_one(nat)))),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),X_1)) ) ).
fof(fact_4105_sums__if,axiom,
! [F,Y_1,G,X_1] :
( hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),G),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),F),Y_1))
=> hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),G),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(real,fun(real,real),plus_plus(real),X_1),Y_1))) ) ) ).
fof(fact_4106_cos__paired,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),X_1)),hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,real,cos,X_1))) ).
fof(fact_4107_diffs__equiv,axiom,
! [X_a] :
( ( real_normed_vector(X_a)
& ring_1(X_a) )
=> ! [C_1,X_1] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C_1))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),semiring_1_of_nat(X_a))),C_1))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat)))))),hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C_1))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))))) ) ) ).
fof(fact_4108_nat__case__add__eq__if,axiom,
! [X_a,Aa,F,V_1,N] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),Aa),F),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_1)),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),Aa),F),N)),hAPP(nat,X_a,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),N))) ).
fof(fact_4109_cnj_Osums,axiom,
! [X_3,Aa] :
( hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),X_3),Aa))
=> hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_3)),hAPP(complex,complex,cnj,Aa))) ) ).
fof(fact_4110_nat__case__Suc,axiom,
! [X_a,F1,F2,Nat] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),F1),F2),hAPP(nat,nat,suc,Nat)) = hAPP(nat,X_a,F2,Nat) ).
fof(fact_4111_diffs__of__real,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [F,X_2] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),F)),X_2) = hAPP(real,X_a,of_real(X_a),hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),F),X_2)) ) ).
fof(fact_4112_nat__case__0,axiom,
! [X_a,F1,F2] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),F1),F2),zero_zero(nat)) = ti(X_a,F1) ).
fof(fact_4113_diffs__minus,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [C_1,X_2] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),C_1)),X_2) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C_1),X_2)) ) ).
fof(fact_4114_Re_Osums,axiom,
! [X_3,Aa] :
( hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),X_3),Aa))
=> hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_3)),hAPP(complex,real,re,Aa))) ) ).
fof(fact_4115_Im_Osums,axiom,
! [X_3,Aa] :
( hBOOL(hAPP(complex,bool,hAPP(fun(nat,complex),fun(complex,bool),sums(complex),X_3),Aa))
=> hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_3)),hAPP(complex,real,im,Aa))) ) ).
fof(fact_4116_sin__fdiffs2,axiom,
! [N] : hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),sin_coeff),N) = hAPP(nat,real,cos_coeff,N) ).
fof(fact_4117_sin__fdiffs,axiom,
hAPP(fun(nat,real),fun(nat,real),diffs(real),sin_coeff) = cos_coeff ).
fof(fact_4118_diffs__def,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [C_1,X_2] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C_1),X_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,suc,X_2))),hAPP(nat,X_a,C_1,hAPP(nat,nat,suc,X_2))) ) ).
fof(fact_4119_exp__fdiffs,axiom,
! [X_2] : hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),X_2) = hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),X_2))) ).
fof(fact_4120_less__eq__nat_Osimps_I2_J,axiom,
! [M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,suc,M)),N))
<=> hBOOL(hAPP(nat,bool,hAPP(fun(nat,bool),fun(nat,bool),hAPP(bool,fun(fun(nat,bool),fun(nat,bool)),nat_case(bool),fFalse),hAPP(nat,fun(nat,bool),ord_less_eq(nat),M)),N)) ) ).
fof(fact_4121_cos__fdiffs,axiom,
! [X_2] : hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),cos_coeff),X_2) = hAPP(real,real,uminus_uminus(real),hAPP(nat,real,sin_coeff,X_2)) ).
fof(fact_4122_cos__fdiffs2,axiom,
! [N] : hAPP(nat,real,hAPP(fun(nat,real),fun(nat,real),diffs(real),cos_coeff),N) = hAPP(real,real,uminus_uminus(real),hAPP(nat,real,sin_coeff,N)) ).
fof(fact_4123_diff__Suc,axiom,
! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),zero_zero(nat)),combi(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M),N)) ).
fof(fact_4124_termdiffs,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X_1,C_1,K_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),C_1)),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3))))
=> ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C_1))),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3))))
=> ( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C_1)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X_1)),hAPP(X_a,real,norm_norm(X_a),K_3)))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,X_a),hAPP(fun(fun(nat,X_a),X_a),fun(fun(X_a,fun(nat,X_a)),fun(X_a,X_a)),combb(fun(nat,X_a),X_a,X_a),suminf(X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),C_1))),power_power(X_a)))),X_1),hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C_1))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))))) ) ) ) ) ) ).
fof(fact_4125_nat__case__number__of,axiom,
! [X_a,Aa,F,V_1] : hAPP(nat,X_a,hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(X_a,fun(fun(nat,X_a),fun(nat,X_a)),nat_case(X_a),Aa),F),hAPP(int,nat,number_number_of(nat),V_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),Aa),hAPP(nat,X_a,F,hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1))))) ).
fof(fact_4126_exE__realizer,axiom,
! [X_c,X_b,X_a,Q_1,F,P_1,Pa] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,hAPP(product_prod(X_b,X_a),X_a,product_snd(X_b,X_a),Pa)),hAPP(product_prod(X_b,X_a),X_b,product_fst(X_b,X_a),Pa)))
=> ( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,Y_2),X_2))
=> hBOOL(hAPP(X_c,bool,Q_1,hAPP(X_a,X_c,hAPP(X_b,fun(X_a,X_c),F,X_2),Y_2))) )
=> hBOOL(hAPP(X_c,bool,Q_1,hAPP(product_prod(X_b,X_a),X_c,hAPP(fun(X_b,fun(X_a,X_c)),fun(product_prod(X_b,X_a),X_c),product_prod_case(X_b,X_a,X_c),F),Pa))) ) ) ).
fof(fact_4127_finite__conv__nat__seg__image,axiom,
! [X_a,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
<=> ? [N_2,F_3] : ti(fun(X_a,bool),A_2) = hAPP(fun(nat,bool),fun(X_a,bool),hAPP(fun(nat,X_a),fun(fun(nat,bool),fun(X_a,bool)),image(nat,X_a),F_3),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),N_2))) ) ).
fof(fact_4128_setprod__coprime__nat,axiom,
! [X_a,F,X_1,A_2] :
( ! [I_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(X_a,nat,F,I_2)),X_1) = one_one(nat) )
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),F),A_2)),X_1) = one_one(nat) ) ).
fof(fact_4129_summable__le,axiom,
! [F,G] :
( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_2)),hAPP(nat,real,G,N_2)))
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),G))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),F)),hAPP(fun(nat,real),real,suminf(real),G))) ) ) ) ).
fof(fact_4130_monoI1,axiom,
! [X_a] :
( order(X_a)
=> ! [X_3] :
( ! [M_3,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_3,M_3)),hAPP(nat,X_a,X_3,N_2))) )
=> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_3)) ) ) ).
fof(fact_4131_monoI2,axiom,
! [X_a] :
( order(X_a)
=> ! [X_3] :
( ! [M_3,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_3,N_2)),hAPP(nat,X_a,X_3,M_3))) )
=> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_3)) ) ) ).
fof(fact_4132_arctan__def,axiom,
! [Y_1] : hAPP(real,real,arctan,Y_1) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),tan)),Y_1)))) ).
fof(fact_4133_arcsin__def,axiom,
! [Y_1] : hAPP(real,real,arcsin,Y_1) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,uminus_uminus(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),sin)),Y_1)))) ).
fof(fact_4134_ln__def,axiom,
! [X_1] : hAPP(real,real,ln,X_1) = hAPP(fun(real,bool),real,the(real),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),exp(real))),X_1)) ).
fof(fact_4135_suminf__def,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> ! [F] : hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = hAPP(fun(X_a,bool),X_a,the(X_a),hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F)) ) ).
fof(fact_4136_root__def,axiom,
! [N,X_1] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),N))),X_1))) )
& ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> ( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),zero_zero(real)))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1) = hAPP(real,real,uminus_uminus(real),hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),N))),hAPP(real,real,uminus_uminus(real),X_1))))) )
& ( ~ hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),zero_zero(real)))
=> hAPP(real,real,hAPP(nat,fun(real,real),root,N),X_1) = zero_zero(real) ) ) ) ) ).
fof(fact_4137_arccos__def,axiom,
! [Y_1] : hAPP(real,real,arccos,Y_1) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),pi))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),cos)),Y_1)))) ).
fof(fact_4138_pi__def,axiom,
pi = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),cos)),zero_zero(real)))))) ).
fof(fact_4139_pi__half,axiom,
hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))) = hAPP(fun(real,bool),real,the(real),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)))),hAPP(fun(real,bool),fun(real,bool),hAPP(fun(real,fun(bool,bool)),fun(fun(real,bool),fun(real,bool)),combs(real,bool,bool),hAPP(fun(real,bool),fun(real,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(real,bool),fun(real,fun(bool,bool))),combb(bool,fun(bool,bool),real),fconj),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),ord_less_eq(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))))),hAPP(real,fun(real,bool),hAPP(fun(real,fun(real,bool)),fun(real,fun(real,bool)),combc(real,real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(real,real),fun(real,fun(real,bool))),combb(real,fun(real,bool),real),fequal(real)),cos)),zero_zero(real))))) ).
fof(fact_4140_mono__SucI2,axiom,
! [X_a] :
( order(X_a)
=> ! [X_3] :
( ! [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_3,hAPP(nat,nat,suc,N_2))),hAPP(nat,X_a,X_3,N_2)))
=> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_3)) ) ) ).
fof(fact_4141_mono__SucI1,axiom,
! [X_a] :
( order(X_a)
=> ! [X_3] :
( ! [N_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,X_3,N_2)),hAPP(nat,X_a,X_3,hAPP(nat,nat,suc,N_2))))
=> hBOOL(hAPP(fun(nat,X_a),bool,monoseq(X_a),X_3)) ) ) ).
fof(fact_4142_split__paired__The,axiom,
! [X_a,X_b,P_1] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),P_1) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool),X_a),hAPP(fun(product_prod(X_a,X_b),bool),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),combb(product_prod(X_a,X_b),bool,X_b),P_1)),product_Pair(X_a,X_b)))) ).
fof(fact_4143_The__split__eq,axiom,
! [X_a,X_b,X_1,Y_1] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_b,bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_a),combb(bool,bool,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(X_a,fun(X_a,bool),fequal(X_a),X_1)))),hAPP(X_b,fun(X_b,bool),fequal(X_b),Y_1)))) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1) ).
fof(fact_4144_divmod__nat__def,axiom,
! [M,N] : hAPP(nat,product_prod(nat,nat),hAPP(nat,fun(nat,product_prod(nat,nat)),divmod_nat,M),N) = hAPP(fun(product_prod(nat,nat),bool),product_prod(nat,nat),the(product_prod(nat,nat)),hAPP(nat,fun(product_prod(nat,nat),bool),hAPP(nat,fun(nat,fun(product_prod(nat,nat),bool)),divmod_nat_rel,M),N)) ).
fof(fact_4145_The__split,axiom,
! [X_a,X_b,P_1] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),P_1)) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),the(product_prod(X_a,X_b)),hAPP(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool),hAPP(fun(product_prod(X_a,X_b),fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool)),combs(product_prod(X_a,X_b),X_b,bool),hAPP(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool))),combb(X_a,fun(X_b,bool),product_prod(X_a,X_b)),P_1),product_fst(X_a,X_b))),product_snd(X_a,X_b))) ).
fof(fact_4146_floor__real__def,axiom,
! [X_1] : hAPP(real,int,archim2085082626_floor(real),X_1) = hAPP(fun(int,bool),int,the(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(real,fun(int,bool),hAPP(fun(int,fun(real,bool)),fun(real,fun(int,bool)),combc(int,real,bool),hAPP(fun(int,real),fun(int,fun(real,bool)),hAPP(fun(real,fun(real,bool)),fun(fun(int,real),fun(int,fun(real,bool))),combb(real,fun(real,bool),int),ord_less_eq(real)),ring_1_of_int(real))),X_1))),hAPP(fun(int,real),fun(int,bool),hAPP(fun(real,bool),fun(fun(int,real),fun(int,bool)),combb(real,bool,int),hAPP(real,fun(real,bool),ord_less(real),X_1)),hAPP(fun(int,int),fun(int,real),hAPP(fun(int,real),fun(fun(int,int),fun(int,real)),combb(int,real,int),ring_1_of_int(real)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),plus_plus(int)),one_one(int)))))) ).
fof(fact_4147_floor__rat__def,axiom,
! [X_1] : hAPP(rat,int,archim2085082626_floor(rat),X_1) = hAPP(fun(int,bool),int,the(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(rat,fun(int,bool),hAPP(fun(int,fun(rat,bool)),fun(rat,fun(int,bool)),combc(int,rat,bool),hAPP(fun(int,rat),fun(int,fun(rat,bool)),hAPP(fun(rat,fun(rat,bool)),fun(fun(int,rat),fun(int,fun(rat,bool))),combb(rat,fun(rat,bool),int),ord_less_eq(rat)),ring_1_of_int(rat))),X_1))),hAPP(fun(int,rat),fun(int,bool),hAPP(fun(rat,bool),fun(fun(int,rat),fun(int,bool)),combb(rat,bool,int),hAPP(rat,fun(rat,bool),ord_less(rat),X_1)),hAPP(fun(int,int),fun(int,rat),hAPP(fun(int,rat),fun(fun(int,int),fun(int,rat)),combb(int,rat,int),ring_1_of_int(rat)),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),plus_plus(int)),one_one(int)))))) ).
fof(fact_4148_setprod_Oneutral,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [G,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(X_a,X_b,G,X_2) = one_one(X_b) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2) = one_one(X_b) ) ) ) ).
fof(fact_4149_setprod__pos,axiom,
! [X_a,X_b] :
( linordered_semidom(X_b)
=> ! [F,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less(X_b),zero_zero(X_b)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2))) ) ) ).
fof(fact_4150_setprod__mono,axiom,
! [X_a,X_b] :
( linordered_semidom(X_b)
=> ! [G,F,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_2)))
& hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,G,X_2))) ) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2))) ) ) ).
fof(fact_4151_setprod__nonneg,axiom,
! [X_a,X_b] :
( linordered_semidom(X_b)
=> ! [F,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2))) ) ) ).
fof(fact_4152_nat__rec__add__eq__if,axiom,
! [X_a,Aa,F,V_1,N] : hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),Aa),F),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,number_number_of(nat),V_1)),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),Aa),F),N)),hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),N)),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),Aa),F),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),N)))) ).
fof(fact_4153_RRset2norRR__def,axiom,
! [Aa,A_2,M] :
( ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M))
& hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),M))
& hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),A_2)) )
=> hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),M),Aa) = hAPP(fun(int,bool),int,hilbert_Eps(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa)),M))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(fun(int,bool),bool)),fun(fun(int,bool),fun(int,bool)),combc(int,fun(int,bool),bool),member(int)),hAPP(int,fun(int,bool),norRRset,M)))) )
& ( ~ ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M))
& hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),M))
& hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),A_2)) )
=> hAPP(int,int,hAPP(int,fun(int,int),hAPP(fun(int,bool),fun(int,fun(int,int)),rRset2norRR,A_2),M),Aa) = zero_zero(int) ) ) ).
fof(fact_4154_nat__rec__Suc,axiom,
! [X_a,F1,F2,Nat] : hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),F1),F2),hAPP(nat,nat,suc,Nat)) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),F2,Nat),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),F1),F2),Nat)) ).
fof(fact_4155_nat__rec__0,axiom,
! [X_a,F1,F2] : hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),F1),F2),zero_zero(nat)) = ti(X_a,F1) ).
fof(fact_4156_aux__some,axiom,
! [Aa,A_2,M] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),one_one(int)),M))
=> ( hBOOL(hAPP(int,bool,hAPP(fun(int,bool),fun(int,bool),is_RRset,A_2),M))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Aa),A_2))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa),hAPP(fun(int,bool),int,hilbert_Eps(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa)),M))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(fun(int,bool),bool)),fun(fun(int,bool),fun(int,bool)),combc(int,fun(int,bool),bool),member(int)),hAPP(int,fun(int,bool),norRRset,M))))),M))
& hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(fun(int,bool),int,hilbert_Eps(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,Aa)),M))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(fun(int,bool),bool)),fun(fun(int,bool),fun(int,bool)),combc(int,fun(int,bool),bool),member(int)),hAPP(int,fun(int,bool),norRRset,M))))),hAPP(int,fun(int,bool),norRRset,M))) ) ) ) ) ).
fof(fact_4157_nat__rec__number__of,axiom,
! [X_a,Aa,F,V_1] : hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),Aa),F),hAPP(int,nat,number_number_of(nat),V_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),Aa),hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),F,hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),Aa),F),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))))) ).
fof(fact_4158_semiring__1__class_Oof__nat__code,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [N] : hAPP(nat,X_a,semiring_1_of_nat(X_a),N) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),plus_plus(X_a)),one_one(X_a))),N),zero_zero(X_a)) ) ).
fof(fact_4159_Nitpick_Oprod__def,axiom,
! [X_b,X_a,A_2,B_1] : hAPP(fun(X_b,bool),fun(product_prod(X_a,X_b),bool),hAPP(fun(X_a,bool),fun(fun(X_b,bool),fun(product_prod(X_a,X_b),bool)),prod(X_a,X_b),A_2),B_1) = hAPP(fun(product_prod(X_a,X_b),bool),fun(product_prod(X_a,X_b),bool),collect(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_b,bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_a),combb(bool,bool,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2)))),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),B_1)))) ).
fof(fact_4160_of__nat__aux_Osimps_I1_J,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [Inc,I] : hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),Inc),zero_zero(nat)),I) = ti(X_a,I) ) ).
fof(fact_4161_of__nat__aux_Osimps_I2_J,axiom,
! [X_a] :
( semiring_1(X_a)
=> ! [Inc,N,I] : hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),Inc),hAPP(nat,nat,suc,N)),I) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(nat,fun(X_a,X_a)),semiri532925092at_aux(X_a),Inc),N),hAPP(X_a,X_a,Inc,I)) ) ).
fof(fact_4162_Maclaurin__sin__bound,axiom,
! [X_1,N] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,sin,X_1)),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,abs_abs(real),X_1)),N)))) ).
fof(fact_4163_finite__less__ub,axiom,
! [U_1,F] :
( ! [N_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_2),hAPP(nat,nat,F,N_2)))
=> hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(fun(nat,bool),fun(nat,bool),collect(nat),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),ord_less_eq(nat)),F)),U_1)))) ) ).
fof(fact_4164_finite__atLeastLessThan,axiom,
! [L,U_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),L),U_1))) ).
fof(fact_4165_setsum__abs,axiom,
! [X_b,X_a] :
( ordere142940540dd_abs(X_a)
=> ! [F,A_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,abs_abs(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2))),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2))) ) ).
fof(fact_4166_setsum__abs__ge__zero,axiom,
! [X_b,X_a] :
( ordere142940540dd_abs(X_a)
=> ! [F,A_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2))) ) ).
fof(fact_4167_sums__split__initial__segment,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [K,F,S_3] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),S_3))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),S_3),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))))) ) ) ).
fof(fact_4168_suminf__split__initial__segment,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [K,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))),hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)))) ) ) ).
fof(fact_4169_suminf__minus__initial__segment,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [K,S_3,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> ( hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = ti(X_a,S_3)
=> hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),S_3),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) ) ) ) ).
fof(fact_4170_of__nat__setsum,axiom,
! [X_b,X_a] :
( semiring_1(X_a)
=> ! [F,A_2] : hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,nat),fun(X_b,X_a),hAPP(fun(nat,X_a),fun(fun(X_b,nat),fun(X_b,X_a)),combb(nat,X_a,X_b),semiring_1_of_nat(X_a)),F)),A_2) ) ).
fof(fact_4171_of__int__setsum,axiom,
! [X_b,X_a] :
( ring_1(X_a)
=> ! [F,A_2] : hAPP(int,X_a,ring_1_of_int(X_a),hAPP(fun(X_b,bool),int,hAPP(fun(X_b,int),fun(fun(X_b,bool),int),big_co1399186613setsum(X_b,int),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,int),fun(X_b,X_a),hAPP(fun(int,X_a),fun(fun(X_b,int),fun(X_b,X_a)),combb(int,X_a,X_b),ring_1_of_int(X_a)),F)),A_2) ) ).
fof(fact_4172_sumr__diff__mult__const2,axiom,
! [X_a] :
( ring_1(X_a)
=> ! [F,N,R_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),R_1)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),R_1)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)) ) ).
fof(fact_4173_lemma__realpow__diff__sumr,axiom,
! [X_a] :
( ( monoid_mult(X_a)
& comm_semiring_0(X_a) )
=> ! [X_1,Y_1,N] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,N))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Y_1),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1)),hAPP(nat,fun(nat,nat),minus_minus(nat),N)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,N)))) ) ).
fof(fact_4174_lemma__realpow__rev__sumr,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& power(X_a) )
=> ! [X_1,Y_1,N] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1)),hAPP(nat,fun(nat,nat),minus_minus(nat),N)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,N))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1)),hAPP(nat,fun(nat,nat),minus_minus(nat),N)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,N))) ) ).
fof(fact_4175_sumr__diff__mult__const,axiom,
! [F,N,R_1] : hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),R_1)) = hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),F)),R_1)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)) ).
fof(fact_4176_sumr__group,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,K,N] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combs(nat,nat,fun(nat,bool)),hAPP(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool)))),combb(nat,fun(nat,fun(nat,bool)),nat),ord_atLeastLessThan(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),K)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),K))) ) ).
fof(fact_4177_norm__setsum,axiom,
! [X_b,X_a] :
( real_normed_vector(X_a)
=> ! [F,A_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2))),hAPP(fun(X_b,bool),real,hAPP(fun(X_b,real),fun(fun(X_b,bool),real),big_co1399186613setsum(X_b,real),hAPP(fun(X_b,X_a),fun(X_b,real),hAPP(fun(X_a,real),fun(fun(X_b,X_a),fun(X_b,real)),combb(X_a,real,X_b),norm_norm(X_a)),F)),A_2))) ) ).
fof(fact_4178_setsum__shift__bounds__nat__ivl,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,M,K,N] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),N)) ) ).
fof(fact_4179_geometric__sum,axiom,
! [X_a] :
( field(X_a)
=> ! [N,X_1] :
( ti(X_a,X_1) != one_one(X_a)
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),N)),one_one(X_a))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),one_one(X_a))) ) ) ).
fof(fact_4180_sumr__offset3,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,N,K] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) ) ).
fof(fact_4181_sumr__offset4,axiom,
! [K,N_2,F_3] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N_2),K))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N_2))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) ).
fof(fact_4182_sumr__offset2,axiom,
! [K,N,F_3] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),F_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)) = hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K)))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F_3),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) ).
fof(fact_4183_sumr__offset,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [F,K,N] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K)))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))) ) ).
fof(fact_4184_divide_Osetsum,axiom,
! [X_b,X_a] :
( real_normed_field(X_a)
=> ! [G,A_2,Y_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)),Y_1) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),inverse_divide(X_a)),G)),Y_1)),A_2) ) ).
fof(fact_4185_setsum__restrict__set_H_H,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,P_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),P_1))) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),P_1)),F)),zero_zero(X_b))),A_2) ) ) ).
fof(fact_4186_setsum__restrict__set_H,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1)))) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),B_1))),F)),zero_zero(X_b))),A_2) ) ) ).
fof(fact_4187_setsum__shift__bounds__Suc__ivl,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,M,N] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),N)) ) ).
fof(fact_4188_setsum__setsum__restrict,axiom,
! [X_b,X_a,X_c] :
( comm_monoid_add(X_c)
=> ! [F,R_3,T_4,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),T_4))
=> hAPP(fun(X_a,bool),X_c,hAPP(fun(X_a,X_c),fun(fun(X_a,bool),X_c),big_co1399186613setsum(X_a,X_c),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,X_c),hAPP(fun(X_a,fun(fun(X_b,bool),X_c)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,X_c)),combs(X_a,fun(X_b,bool),X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_a,fun(fun(X_b,bool),X_c)),hAPP(fun(fun(X_b,X_c),fun(fun(X_b,bool),X_c)),fun(fun(X_a,fun(X_b,X_c)),fun(X_a,fun(fun(X_b,bool),X_c))),combb(fun(X_b,X_c),fun(fun(X_b,bool),X_c),X_a),big_co1399186613setsum(X_b,X_c)),F)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),collect(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),T_4)))),R_3)))),S) = hAPP(fun(X_b,bool),X_c,hAPP(fun(X_b,X_c),fun(fun(X_b,bool),X_c),big_co1399186613setsum(X_b,X_c),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,X_c),hAPP(fun(X_b,fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,X_c)),combs(X_b,fun(X_a,bool),X_c),hAPP(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c)),hAPP(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c))),combb(fun(X_a,X_c),fun(fun(X_a,bool),X_c),X_b),big_co1399186613setsum(X_a,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),F))),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_b),collect(X_a)),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_b),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S)))),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),R_3))))),T_4) ) ) ) ).
fof(fact_4189_of__real_Osetsum,axiom,
! [X_b,X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [G,A_2] : hAPP(real,X_a,of_real(X_a),hAPP(fun(X_b,bool),real,hAPP(fun(X_b,real),fun(fun(X_b,bool),real),big_co1399186613setsum(X_b,real),G),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,real),fun(X_b,X_a),hAPP(fun(real,X_a),fun(fun(X_b,real),fun(X_b,X_a)),combb(real,X_a,X_b),of_real(X_a)),G)),A_2) ) ).
fof(fact_4190_mult__left_Osetsum,axiom,
! [X_b,X_a] :
( real_normed_algebra(X_a)
=> ! [G,A_2,Y_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)),Y_1) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),G)),Y_1)),A_2) ) ).
fof(fact_4191_mult__right_Osetsum,axiom,
! [X_b,X_a] :
( real_normed_algebra(X_a)
=> ! [X_1,G,A_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1)),G)),A_2) ) ).
fof(fact_4192_sumr__one__lb__realpow__zero,axiom,
! [F,N] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),F)),hAPP(real,fun(nat,real),power_power(real),zero_zero(real)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,zero_zero(nat))),N)) = zero_zero(real) ).
fof(fact_4193_all__nat__less__eq,axiom,
! [P_1,N] :
( ! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
=> hBOOL(hAPP(nat,bool,P_1,M_3)) )
<=> ! [X_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)))
=> hBOOL(hAPP(nat,bool,P_1,X_2)) ) ) ).
fof(fact_4194_ex__nat__less__eq,axiom,
! [P_1,N] :
( ? [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
& hBOOL(hAPP(nat,bool,P_1,M_3)) )
<=> ? [X_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)))
& hBOOL(hAPP(nat,bool,P_1,X_2)) ) ) ).
fof(fact_4195_atLeastLessThan__eq__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [C_1,D_1,Aa,Ba] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_1))
=> ( hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),Aa),Ba) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C_1),D_1)
<=> ( ti(X_a,Aa) = ti(X_a,C_1)
& ti(X_a,Ba) = ti(X_a,D_1) ) ) ) ) ) ).
fof(fact_4196_atLeastLessThan__inj_I1_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),Aa),Ba) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C_1),D_1)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_1))
=> ti(X_a,Aa) = ti(X_a,C_1) ) ) ) ) ).
fof(fact_4197_atLeastLessThan__inj_I2_J,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),Aa),Ba) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C_1),D_1)
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),D_1))
=> ti(X_a,Ba) = ti(X_a,D_1) ) ) ) ) ).
fof(fact_4198_setsum__shift__lb__Suc0__0__upt,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [K,F] :
( hAPP(nat,X_a,F,zero_zero(nat)) = zero_zero(X_a)
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,zero_zero(nat))),K)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K)) ) ) ).
fof(fact_4199_setsum__diff__nat__ivl,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [F,Pa,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),Pa))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),Pa))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),N))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),N),Pa)) ) ) ) ).
fof(fact_4200_setsum__add__nat__ivl,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,Pa,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),Pa))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),N))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),N),Pa))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),Pa)) ) ) ) ).
fof(fact_4201_card__atLeastLessThan,axiom,
! [L,U_1] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),L),U_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),U_1),L) ).
fof(fact_4202_atLeastLessThan__subset__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),Aa),Ba)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C_1),D_1)))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),Aa))
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),D_1)) ) ) ) ) ).
fof(fact_4203_setsum__head__upt__Suc,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M),N))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,F,M)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,M)),N))) ) ) ).
fof(fact_4204_setsum__op__ivl__Suc,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,N,M] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),hAPP(nat,nat,suc,N))) = zero_zero(X_a) )
& ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),N),M))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),hAPP(nat,nat,suc,N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),N))),hAPP(nat,X_a,F,N)) ) ) ) ).
fof(fact_4205_ivl__diff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [M,I,N] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),I),N))
=> hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),I),M)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),I),N)) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),N),M) ) ) ).
fof(fact_4206_sumr__geometric,axiom,
! [N,X_1] :
( X_1 != one_one(real)
=> hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),power_power(real),X_1)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)) = hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N)),one_one(real))),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X_1),one_one(real))) ) ).
fof(fact_4207_setsum_Oinfinite,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [G,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2) = zero_zero(X_b) ) ) ).
fof(fact_4208_setsum__infinite,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2) = zero_zero(X_b) ) ) ).
fof(fact_4209_abs__setsum__abs,axiom,
! [X_b,X_a] :
( ordere142940540dd_abs(X_a)
=> ! [F,A_2] : hAPP(X_a,X_a,abs_abs(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),abs_abs(X_a)),F)),A_2) ) ).
fof(fact_4210_setsum__negf,axiom,
! [X_b,X_a] :
( ab_group_add(X_a)
=> ! [F,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),uminus_uminus(X_a)),F)),A_2) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)) ) ).
fof(fact_4211_setsum__right__distrib,axiom,
! [X_b,X_a] :
( semiring_0(X_a)
=> ! [R_1,F,A_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),R_1),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),hAPP(X_a,fun(X_a,X_a),times_times(X_a),R_1)),F)),A_2) ) ).
fof(fact_4212_setsum__left__distrib,axiom,
! [X_b,X_a] :
( semiring_0(X_a)
=> ! [F,A_2,R_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),R_1) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),F)),R_1)),A_2) ) ).
fof(fact_4213_setsum__product,axiom,
! [X_b,X_c,X_a] :
( semiring_0(X_a)
=> ! [F,A_2,G,B_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),hAPP(fun(X_c,bool),X_a,hAPP(fun(X_c,X_a),fun(fun(X_c,bool),X_a),big_co1399186613setsum(X_c,X_a),G),B_1)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_c,bool),fun(X_b,X_a),hAPP(fun(X_b,fun(fun(X_c,bool),X_a)),fun(fun(X_c,bool),fun(X_b,X_a)),combc(X_b,fun(X_c,bool),X_a),hAPP(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a)),hAPP(fun(fun(X_c,X_a),fun(fun(X_c,bool),X_a)),fun(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a))),combb(fun(X_c,X_a),fun(fun(X_c,bool),X_a),X_b),big_co1399186613setsum(X_c,X_a)),hAPP(fun(X_c,X_a),fun(X_b,fun(X_c,X_a)),hAPP(fun(X_b,fun(fun(X_c,X_a),fun(X_c,X_a))),fun(fun(X_c,X_a),fun(X_b,fun(X_c,X_a))),combc(X_b,fun(X_c,X_a),fun(X_c,X_a)),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_b,fun(fun(X_c,X_a),fun(X_c,X_a))),hAPP(fun(fun(X_a,X_a),fun(fun(X_c,X_a),fun(X_c,X_a))),fun(fun(X_b,fun(X_a,X_a)),fun(X_b,fun(fun(X_c,X_a),fun(X_c,X_a)))),combb(fun(X_a,X_a),fun(fun(X_c,X_a),fun(X_c,X_a)),X_b),combb(X_a,X_a,X_c)),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),times_times(X_a)),F))),G))),B_1)),A_2) ) ).
fof(fact_4214_setsum__commute,axiom,
! [X_b,X_c,X_a] :
( comm_monoid_add(X_a)
=> ! [F,B_1,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_c,bool),fun(X_b,X_a),hAPP(fun(X_b,fun(fun(X_c,bool),X_a)),fun(fun(X_c,bool),fun(X_b,X_a)),combc(X_b,fun(X_c,bool),X_a),hAPP(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a)),hAPP(fun(fun(X_c,X_a),fun(fun(X_c,bool),X_a)),fun(fun(X_b,fun(X_c,X_a)),fun(X_b,fun(fun(X_c,bool),X_a))),combb(fun(X_c,X_a),fun(fun(X_c,bool),X_a),X_b),big_co1399186613setsum(X_c,X_a)),F)),B_1)),A_2) = hAPP(fun(X_c,bool),X_a,hAPP(fun(X_c,X_a),fun(fun(X_c,bool),X_a),big_co1399186613setsum(X_c,X_a),hAPP(fun(X_b,bool),fun(X_c,X_a),hAPP(fun(X_c,fun(fun(X_b,bool),X_a)),fun(fun(X_b,bool),fun(X_c,X_a)),combc(X_c,fun(X_b,bool),X_a),hAPP(fun(X_c,fun(X_b,X_a)),fun(X_c,fun(fun(X_b,bool),X_a)),hAPP(fun(fun(X_b,X_a),fun(fun(X_b,bool),X_a)),fun(fun(X_c,fun(X_b,X_a)),fun(X_c,fun(fun(X_b,bool),X_a))),combb(fun(X_b,X_a),fun(fun(X_b,bool),X_a),X_c),big_co1399186613setsum(X_b,X_a)),hAPP(fun(X_b,fun(X_c,X_a)),fun(X_c,fun(X_b,X_a)),combc(X_b,X_c,X_a),F))),A_2)),B_1) ) ).
fof(fact_4215_setsum__delta_H,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [Ba,Aa,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),Aa))),Ba)),zero_zero(X_b))),S) = hAPP(X_a,X_b,Ba,Aa) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),fequal(X_a),Aa))),Ba)),zero_zero(X_b))),S) = zero_zero(X_b) ) ) ) ) ).
fof(fact_4216_setsum__delta,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [Ba,Aa,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ( ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),Aa))),Ba)),zero_zero(X_b))),S) = hAPP(X_a,X_b,Ba,Aa) )
& ( ~ hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Aa),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_a,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combs(X_a,X_b,fun(X_b,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b))),hAPP(fun(bool,fun(X_b,fun(X_b,X_b))),fun(fun(X_a,bool),fun(X_a,fun(X_b,fun(X_b,X_b)))),combb(bool,fun(X_b,fun(X_b,X_b)),X_a),if(X_b)),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),Aa))),Ba)),zero_zero(X_b))),S) = zero_zero(X_b) ) ) ) ) ).
fof(fact_4217_setsum__divide__distrib,axiom,
! [X_b,X_a] :
( field(X_a)
=> ! [F,A_2,R_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),R_1) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),inverse_divide(X_a)),F)),R_1)),A_2) ) ).
fof(fact_4218_setsum__0,axiom,
! [X_b,X_a] :
( comm_monoid_add(X_a)
=> ! [A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),zero_zero(X_a))),A_2) = zero_zero(X_a) ) ).
fof(fact_4219_setsum_Odistrib,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [G,H,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),G)),H)),A_2) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),H),A_2)) ) ) ).
fof(fact_4220_setsum__subtractf,axiom,
! [X_b,X_a] :
( ab_group_add(X_a)
=> ! [F,G,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,X_a)),combs(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),minus_minus(X_a)),F)),G)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)) ) ).
fof(fact_4221_setsum__addf,axiom,
! [X_b,X_a] :
( comm_monoid_add(X_a)
=> ! [F,G,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,X_a)),combs(X_b,X_a,X_a),hAPP(fun(X_b,X_a),fun(X_b,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_b,X_a),fun(X_b,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_b),plus_plus(X_a)),F)),G)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),F),A_2)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)) ) ).
fof(fact_4222_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L,U_1] : hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,L)),U_1) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_gr676655987ssThan(nat),L),U_1) ).
fof(fact_4223_image__add__atLeastLessThan,axiom,
! [K,I,J] : hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K)) ).
fof(fact_4224_setsum__image__gen,axiom,
! [X_c,X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [G,F,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),S) = hAPP(fun(X_c,bool),X_b,hAPP(fun(X_c,X_b),fun(fun(X_c,bool),X_b),big_co1399186613setsum(X_c,X_b),hAPP(fun(X_c,fun(X_a,bool)),fun(X_c,X_b),hAPP(fun(fun(X_a,bool),X_b),fun(fun(X_c,fun(X_a,bool)),fun(X_c,X_b)),combb(fun(X_a,bool),X_b,X_c),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G)),hAPP(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_c),collect(X_a)),hAPP(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_c,fun(X_a,bool)),fun(X_c,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_c),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S)))),hAPP(fun(X_a,fun(X_c,bool)),fun(X_c,fun(X_a,bool)),combc(X_a,X_c,bool),hAPP(fun(X_a,X_c),fun(X_a,fun(X_c,bool)),hAPP(fun(X_c,fun(X_c,bool)),fun(fun(X_a,X_c),fun(X_a,fun(X_c,bool))),combb(X_c,fun(X_c,bool),X_a),fequal(X_c)),F)))))),hAPP(fun(X_a,bool),fun(X_c,bool),hAPP(fun(X_a,X_c),fun(fun(X_a,bool),fun(X_c,bool)),image(X_a,X_c),F),S)) ) ) ).
fof(fact_4225_image__Suc__atLeastLessThan,axiom,
! [I,J] : hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),suc),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),hAPP(nat,nat,suc,I)),hAPP(nat,nat,suc,J)) ).
fof(fact_4226_lemma__termdiff1,axiom,
! [X_a] :
( ( monoid_mult(X_a)
& comm_ring(X_a) )
=> ! [Z_1,H,M] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),H))),hAPP(nat,fun(nat,nat),minus_minus(nat),M)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),M))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),M)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1))),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),H))),hAPP(nat,fun(nat,nat),minus_minus(nat),M)))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1)),hAPP(nat,fun(nat,nat),minus_minus(nat),M))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),M)) ) ).
fof(fact_4227_lemma__realpow__diff__sumr2,axiom,
! [X_a] :
( ( monoid_mult(X_a)
& comm_ring(X_a) )
=> ! [X_1,N,Y_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1),hAPP(nat,nat,suc,N))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1),hAPP(nat,nat,suc,N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_1),Y_1)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1)),hAPP(nat,fun(nat,nat),minus_minus(nat),N)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,N)))) ) ).
fof(fact_4228_setsum__constant,axiom,
! [X_b,X_a] :
( semiring_1(X_a)
=> ! [Y_1,A_2] : hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),Y_1)),A_2) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(fun(X_b,bool),nat,finite_card(X_b),A_2))),Y_1) ) ).
fof(fact_4229_real__of__card,axiom,
! [X_a,A_2] : hAPP(nat,real,real_1(nat),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(real,fun(X_a,real),combk(real,X_a),one_one(real))),A_2) ).
fof(fact_4230_Maclaurin__zero,axiom,
! [Diff,N,X_1] :
( X_1 = zero_zero(real)
=> ( N != zero_zero(nat)
=> hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)) = hAPP(real,real,hAPP(nat,fun(real,real),Diff,zero_zero(nat)),zero_zero(real)) ) ) ).
fof(fact_4231_sums__group,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [K,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combs(nat,nat,fun(nat,bool)),hAPP(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(fun(nat,nat),fun(nat,fun(nat,fun(nat,bool)))),combb(nat,fun(nat,fun(nat,bool)),nat),ord_atLeastLessThan(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),K))),K)))),hAPP(fun(nat,X_a),X_a,suminf(X_a),F))) ) ) ) ).
fof(fact_4232_subset__card__intvl__is__intvl,axiom,
! [A_2,K] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),A_2),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),hAPP(fun(nat,bool),nat,finite_card(nat),A_2)))))
=> A_2 = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),K),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),K),hAPP(fun(nat,bool),nat,finite_card(nat),A_2))) ) ).
fof(fact_4233_setsum__diff,axiom,
! [X_a,X_b] :
( ab_group_add(X_b)
=> ! [F,B_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) = hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),B_1)) ) ) ) ).
fof(fact_4234_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),Aa),Ba)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C_1),D_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),D_1)) ) ) ) ) ).
fof(fact_4235_sum__split__even__odd,axiom,
! [F,G,N] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),F)),G)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,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(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),G),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),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(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))) ).
fof(fact_4236_Maclaurin__bi__le__lemma,axiom,
! [X_a] :
( zero(X_a)
=> ! [Diff,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> hAPP(X_a,real,hAPP(nat,fun(X_a,real),Diff,zero_zero(nat)),zero_zero(X_a)) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(X_a,fun(nat,real),hAPP(fun(nat,fun(X_a,real)),fun(X_a,fun(nat,real)),combc(nat,X_a,real),Diff),zero_zero(X_a)))),hAPP(real,fun(nat,real),power_power(real),zero_zero(real))))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,hAPP(nat,fun(X_a,real),Diff,N),zero_zero(X_a))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),zero_zero(real)),N))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))) ) ) ).
fof(fact_4237_sumr__minus__one__realpow__zero,axiom,
! [N] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N))) = zero_zero(real) ).
fof(fact_4238_lemma__termdiff2,axiom,
! [X_a] :
( field(X_a)
=> ! [Z_1,N,H] :
( ti(X_a,H) != zero_zero(X_a)
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),H)),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),N))),H)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),H),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Z_1),H)))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Z_1)),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat))))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,zero_zero(nat)))))) ) ) ).
fof(fact_4239_lemma__STAR__cos2,axiom,
! [N] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),zero_zero(real))))),zero_zero(real))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),one_one(nat)),N)) = zero_zero(real) ).
fof(fact_4240_sumr__cos__zero__one,axiom,
! [N] : hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),zero_zero(real)))),hAPP(real,fun(nat,real),power_power(real),zero_zero(real)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,suc,N))) = one_one(real) ).
fof(fact_4241_Maclaurin__sin__expansion2,axiom,
! [N,X_1] :
? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_5)),hAPP(real,real,abs_abs(real),X_1)))
& hAPP(real,real,sin,X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),T_5),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,real_1(nat),N))),pi)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N))) ) ).
fof(fact_4242_Maclaurin__cos__expansion,axiom,
! [N,X_1] :
? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_5)),hAPP(real,real,abs_abs(real),X_1)))
& hAPP(real,real,cos,X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(fun(nat,real),fun(nat,fun(real,real))),combs(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),zero_zero(real)))),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,cos,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),T_5),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,real_1(nat),N))),pi)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N))) ) ).
fof(fact_4243_finite__atLeastLessThan__int,axiom,
! [L,U_1] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),U_1))) ).
fof(fact_4244_int__setsum,axiom,
! [X_a,F,A_2] : hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2)) = hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2) ).
fof(fact_4245_transfer__nat__int__sum__prod2_I1_J,axiom,
! [X_a,F,A_2] : hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) = hAPP(int,nat,nat_1,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(fun(X_a,nat),fun(X_a,int),hAPP(fun(nat,int),fun(fun(X_a,nat),fun(X_a,int)),combb(nat,int,X_a),semiring_1_of_nat(int)),F)),A_2)) ).
fof(fact_4246_real__of__int__setsum,axiom,
! [X_a,F,A_2] : hAPP(int,real,real_1(int),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,int),fun(X_a,real),hAPP(fun(int,real),fun(fun(X_a,int),fun(X_a,real)),combb(int,real,X_a),real_1(int)),F)),A_2) ).
fof(fact_4247_real__of__nat__setsum,axiom,
! [X_a,F,A_2] : hAPP(nat,real,real_1(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,nat),fun(X_a,real),hAPP(fun(nat,real),fun(fun(X_a,nat),fun(X_a,real)),combb(nat,real,X_a),real_1(nat)),F)),A_2) ).
fof(fact_4248_cnj_Osetsum,axiom,
! [X_a,G,A_2] : hAPP(complex,complex,cnj,hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),G),A_2)) = hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),A_2) ).
fof(fact_4249_Re_Osetsum,axiom,
! [X_a,G,A_2] : hAPP(complex,real,re,hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),G),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),A_2) ).
fof(fact_4250_Im_Osetsum,axiom,
! [X_a,G,A_2] : hAPP(complex,real,im,hAPP(fun(X_a,bool),complex,hAPP(fun(X_a,complex),fun(fun(X_a,bool),complex),big_co1399186613setsum(X_a,complex),G),A_2)) = hAPP(fun(X_a,bool),real,hAPP(fun(X_a,real),fun(fun(X_a,bool),real),big_co1399186613setsum(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),A_2) ).
fof(fact_4251_setsum__const__mult,axiom,
! [X_a,C_1,F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(fun(X_a,int),fun(X_a,int),hAPP(fun(int,int),fun(fun(X_a,int),fun(X_a,int)),combb(int,int,X_a),hAPP(int,fun(int,int),times_times(int),C_1)),F)),A_2) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),A_2)) ) ).
fof(fact_4252_finite__atLeastZeroLessThan__int,axiom,
! [U_1] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),U_1))) ).
fof(fact_4253_setsum__eq__0__iff,axiom,
! [X_a,F,F_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),F_1))
=> ( hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),F_1) = zero_zero(nat)
<=> ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),F_1))
=> hAPP(X_a,nat,F,X_2) = zero_zero(nat) ) ) ) ).
fof(fact_4254_card__eq__setsum,axiom,
! [X_a,A_2] : hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2) = hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(nat,fun(X_a,nat),combk(nat,X_a),one_one(nat))),A_2) ).
fof(fact_4255_Sup__atLeastLessThan,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_1),X_1))
=> hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastLessThan(real),Y_1),X_1)) = X_1 ) ).
fof(fact_4256_setsum__eq__Suc0__iff,axiom,
! [X_a,F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) = hAPP(nat,nat,suc,zero_zero(nat))
<=> ? [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
& hAPP(X_a,nat,F,X_2) = hAPP(nat,nat,suc,zero_zero(nat))
& ! [Xa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2))
=> ( ti(X_a,X_2) != ti(X_a,Xa)
=> hAPP(X_a,nat,F,Xa) = zero_zero(nat) ) ) ) ) ) ).
fof(fact_4257_setsum__eq__1__iff,axiom,
! [X_a,F,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) = one_one(nat)
<=> ? [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
& hAPP(X_a,nat,F,X_2) = one_one(nat)
& ! [Xa] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),Xa),A_2))
=> ( ti(X_a,X_2) != ti(X_a,Xa)
=> hAPP(X_a,nat,F,Xa) = zero_zero(nat) ) ) ) ) ) ).
fof(fact_4258_card__atLeastZeroLessThan__int,axiom,
! [U_1] : hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),U_1)) = hAPP(int,nat,nat_1,U_1) ).
fof(fact_4259_card__atLeastLessThan__int,axiom,
! [L,U_1] : hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),U_1)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U_1),L)) ).
fof(fact_4260_setsum__diff__nat,axiom,
! [X_a,F,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),B_1),A_2))
=> hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),A_2),B_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2)),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),B_1)) ) ) ).
fof(fact_4261_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L,U_1] : hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),L),one_one(int))),U_1) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_gr676655987ssThan(int),L),U_1) ).
fof(fact_4262_setsum__const,axiom,
! [X_a,C_1,X_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),X_3))
=> hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),hAPP(int,fun(X_a,int),combk(int,X_a),C_1)),X_3) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),C_1),hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,finite_card(X_a),X_3))) ) ).
fof(fact_4263_setsum__const2,axiom,
! [X_a,C_1,X_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),X_3))
=> hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(nat,fun(X_a,nat),combk(nat,X_a),C_1)),X_3)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),C_1)),hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,finite_card(X_a),X_3))) ) ).
fof(fact_4264_transfer__nat__int__sum__prod_I1_J,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& comm_monoid_mult(X_a) )
=> ! [F,A_2] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),hAPP(fun(int,nat),fun(int,X_a),hAPP(fun(nat,X_a),fun(fun(int,nat),fun(int,X_a)),combb(nat,X_a,int),F),nat_1)),hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),A_2)) ) ).
fof(fact_4265_image__add__int__atLeastLessThan,axiom,
! [L,U_1] : hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,int),fun(fun(int,bool),fun(int,bool)),image(int,int),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),plus_plus(int)),L)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U_1),L))) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),U_1) ).
fof(fact_4266_transfer__int__nat__sum__prod_I1_J,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& comm_monoid_mult(X_a) )
=> ! [F,A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),F),A_2) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,int),fun(nat,X_a),hAPP(fun(int,X_a),fun(fun(nat,int),fun(nat,X_a)),combb(int,X_a,nat),F),semiring_1_of_nat(int))),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2)) ) ) ).
fof(fact_4267_Maclaurin__sin__expansion,axiom,
! [X_1,N] :
? [T_5] : hAPP(real,real,sin,X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(real,fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,fun(real,real))),fun(real,fun(nat,fun(real,real))),combc(nat,real,fun(real,real)),hAPP(fun(nat,bool),fun(nat,fun(real,fun(real,real))),hAPP(fun(bool,fun(real,fun(real,real))),fun(fun(nat,bool),fun(nat,fun(real,fun(real,real)))),combb(bool,fun(real,fun(real,real)),nat),if(real)),even_odd_even(nat))),zero_zero(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),div_div(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,sin,hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),T_5),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),one_one(real)),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))),hAPP(nat,real,real_1(nat),N))),pi)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N))) ).
fof(fact_4268_Maclaurin__lemma2,axiom,
! [B_1,K,Diff,H,N] :
( ! [M_3,T_5] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_5),H)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),T_5),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),T_5))) )
=> ( N = hAPP(nat,nat,suc,K)
=> ! [M_3,T_5] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_5),H)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),minus_minus(real)),hAPP(nat,fun(real,real),Diff,M_3))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),plus_plus(real)),hAPP(fun(nat,bool),fun(real,real),hAPP(fun(real,fun(fun(nat,bool),real)),fun(fun(nat,bool),fun(real,real)),combc(real,fun(nat,bool),real),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real)),hAPP(fun(fun(nat,real),fun(fun(nat,bool),real)),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real))),combb(fun(nat,real),fun(fun(nat,bool),real),real),big_co1399186613setsum(nat,real)),hAPP(fun(real,fun(nat,real)),fun(real,fun(nat,real)),hAPP(fun(fun(nat,real),fun(nat,real)),fun(fun(real,fun(nat,real)),fun(real,fun(nat,real))),combb(fun(nat,real),fun(nat,real),real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,nat),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,real)),fun(fun(nat,nat),fun(nat,fun(real,real))),combb(nat,fun(real,real),nat),Diff),hAPP(nat,fun(nat,nat),plus_plus(nat),M_3))),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))))),power_power(real)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),M_3))))),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,real),fun(fun(real,real),fun(real,real)),combb(real,real,real),hAPP(real,fun(real,real),times_times(real),B_1)),hAPP(real,fun(real,real),hAPP(fun(real,fun(real,real)),fun(real,fun(real,real)),combc(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),hAPP(nat,fun(real,real),hAPP(fun(real,fun(nat,real)),fun(nat,fun(real,real)),combc(real,nat,real),power_power(real)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),M_3)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),M_3)))))))),T_5),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),T_5)),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,nat),fun(nat,fun(real,real)),hAPP(fun(nat,fun(real,real)),fun(fun(nat,nat),fun(nat,fun(real,real))),combb(nat,fun(real,real),nat),Diff),hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,suc,M_3)))),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),T_5))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,M_3))))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B_1),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),T_5),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,M_3)))),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),hAPP(nat,nat,suc,M_3)))))))))) ) ) ) ).
fof(fact_4269_Maclaurin__exp__lt,axiom,
! [N,X_1] :
( X_1 != zero_zero(real)
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),T_5)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),T_5)),hAPP(real,real,abs_abs(real),X_1)))
& hAPP(real,real,exp(real),X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,exp(real),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N))) ) ) ) ).
fof(fact_4270_Maclaurin__exp__le,axiom,
! [N,X_1] :
? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_5)),hAPP(real,real,abs_abs(real),X_1)))
& hAPP(real,real,exp(real),X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,exp(real),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N))) ) ).
fof(fact_4271_sumr__pos__lt__pair,axiom,
! [K,F] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
=> ( ! [D_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,hAPP(real,fun(real,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_3)))),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_3)),one_one(nat)))))))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),K))),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).
fof(fact_4272_Maclaurin__lemma,axiom,
! [F,J,N,H] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
=> ? [B_5] : hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),J)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),B_5),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),N)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N))))) ) ).
fof(fact_4273_real__setsum__nat__ivl__bounded2,axiom,
! [X_a] :
( linordered_semidom(X_a)
=> ! [K,F,K_3,N] :
( ! [P_5] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),P_5),N))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,P_5)),K_3)) )
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),zero_zero(X_a)),K_3))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),K)))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),K_3))) ) ) ) ).
fof(fact_4274_series__pos__less,axiom,
! [N,F] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
=> ( ! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,F,M_3))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).
fof(fact_4275_series__pos__le,axiom,
! [N,F] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
=> ( ! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M_3))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,M_3))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(fun(nat,real),real,suminf(real),F))) ) ) ).
fof(fact_4276_Maclaurin__all__lt__objl,axiom,
! [N,X_1,Diff,F] :
( ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
& ! [M_3,X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),X_2),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),X_2)))
& X_1 != zero_zero(real)
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) )
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),T_5)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),T_5)),hAPP(real,real,abs_abs(real),X_1)))
& hAPP(real,real,F,X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N))) ) ) ).
fof(fact_4277_Maclaurin__all__lt,axiom,
! [X_1,N,Diff,F] :
( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( X_1 != zero_zero(real)
=> ( ! [M_3,X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),X_2),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),X_2)))
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),T_5)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),T_5)),hAPP(real,real,abs_abs(real),X_1)))
& hAPP(real,real,F,X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N))) ) ) ) ) ) ).
fof(fact_4278_Maclaurin__bi__le,axiom,
! [X_1,N,Diff,F] :
( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
=> ( ! [M_3,T_5] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_5)),hAPP(real,real,abs_abs(real),X_1))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),T_5),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),T_5))) )
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_5)),hAPP(real,real,abs_abs(real),X_1)))
& hAPP(real,real,F,X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N))) ) ) ) ).
fof(fact_4279_Maclaurin__minus,axiom,
! [Diff,F,N,H] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),zero_zero(real)))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
=> ( ! [M_3,T_5] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),H),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_5),zero_zero(real))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),T_5),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),T_5))) )
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_5),zero_zero(real)))
& hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ).
fof(fact_4280_Maclaurin2,axiom,
! [N,Diff,F,H] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
=> ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
=> ( ! [M_3,T_5] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_5),H)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),T_5),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),T_5))) )
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_5),H))
& hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ).
fof(fact_4281_Maclaurin__objl,axiom,
! [Diff,F,N,H] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
& hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
& ! [M_3,T_5] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_5),H)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),T_5),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),T_5))) ) )
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_5),H))
& hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),N))) ) ) ).
fof(fact_4282_Maclaurin2__objl,axiom,
! [N,Diff,F,H] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
& hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
& ! [M_3,T_5] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_5),H)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),T_5),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),T_5))) ) )
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_5),H))
& hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),N))) ) ) ).
fof(fact_4283_Maclaurin,axiom,
! [Diff,F,N,H] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
=> ( ! [M_3,T_5] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_5),H)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),T_5),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),T_5))) )
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_5),H))
& hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ).
fof(fact_4284_Maclaurin__minus__objl,axiom,
! [Diff,F,N,H] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),zero_zero(real)))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
& hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
& ! [M_3,T_5] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),M_3),N))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),H),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),T_5),zero_zero(real))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),T_5),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),T_5))) ) )
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),H),T_5))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),T_5),zero_zero(real)))
& hAPP(real,real,F,H) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),H))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),H),N))) ) ) ).
fof(fact_4285_Maclaurin__all__le__objl,axiom,
! [N,X_1,Diff,F] :
( ( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
& ! [M_3,X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),X_2),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),X_2))) )
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_5)),hAPP(real,real,abs_abs(real),X_1)))
& hAPP(real,real,F,X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N))) ) ) ).
fof(fact_4286_Maclaurin__all__le,axiom,
! [N,X_1,Diff,F] :
( hAPP(nat,fun(real,real),Diff,zero_zero(nat)) = F
=> ( ! [M_3,X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),Diff,M_3)),X_2),hAPP(real,real,hAPP(nat,fun(real,real),Diff,hAPP(nat,nat,suc,M_3)),X_2)))
=> ? [T_5] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),T_5)),hAPP(real,real,abs_abs(real),X_1)))
& hAPP(real,real,F,X_1) = hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),inverse_divide(real)),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),Diff),zero_zero(real)))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(real,fun(nat,real),power_power(real),X_1))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),hAPP(real,real,hAPP(nat,fun(real,real),Diff,N),T_5)),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,real,hAPP(real,fun(nat,real),power_power(real),X_1),N))) ) ) ) ).
fof(fact_4287_summable__Cauchy,axiom,
! [X_a] :
( banach(X_a)
=> ! [F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
<=> ! [E] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
=> ? [N_3] :
! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),M_3))
=> ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M_3),N_2)))),E)) ) ) ) ) ).
fof(fact_4288_real__setsum__nat__ivl__bounded,axiom,
! [F,K_3,N] :
( ! [P_5] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),P_5),N))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,P_5)),K_3)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,real_1(nat),N)),K_3))) ) ).
fof(fact_4289_DERIV__sumr,axiom,
! [F,X_1,F_2,N,M] :
( ! [R_2] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),R_2))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),R_2),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),N))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(nat,fun(real,real),F,R_2)),X_1),hAPP(real,real,hAPP(nat,fun(real,real),F_2,R_2),X_1))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),hAPP(fun(nat,bool),fun(real,real),hAPP(fun(real,fun(fun(nat,bool),real)),fun(fun(nat,bool),fun(real,real)),combc(real,fun(nat,bool),real),hAPP(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real)),hAPP(fun(fun(nat,real),fun(fun(nat,bool),real)),fun(fun(real,fun(nat,real)),fun(real,fun(fun(nat,bool),real))),combb(fun(nat,real),fun(fun(nat,bool),real),real),big_co1399186613setsum(nat,real)),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),F))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),N))),X_1),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),F_2),X_1)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),M),N)))) ) ).
fof(fact_4290_suminf__le,axiom,
! [X_1,F] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N_2))),X_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),F)),X_1)) ) ) ).
fof(fact_4291_series__zero,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& t2_space(X_a) )
=> ! [F,N] :
( ! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M_3))
=> hAPP(nat,X_a,F,M_3) = zero_zero(X_a) )
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)))) ) ) ).
fof(fact_4292_setsum__mono3,axiom,
! [X_a,X_b] :
( ( comm_monoid_add(X_b)
& ordere779506340up_add(X_b) )
=> ! [F,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),A_2)))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),B_1))) ) ) ) ) ).
fof(fact_4293_setsum__mono2,axiom,
! [X_a,X_b] :
( ordere216010020id_add(X_b)
=> ! [F,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),B_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> ( ! [B_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),B_3),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),B_1),A_2)))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,B_3))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),B_1))) ) ) ) ) ).
fof(fact_4294_setsum__mono__zero__left,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,S,T_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_4))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_4),S)))
=> hAPP(X_a,X_b,F,X_2) = zero_zero(X_b) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),T_4) ) ) ) ) ).
fof(fact_4295_setsum__mono__zero__right,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,S,T_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_4))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_4),S)))
=> hAPP(X_a,X_b,F,X_2) = zero_zero(X_b) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),T_4) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S) ) ) ) ) ).
fof(fact_4296_setsum__bounded,axiom,
! [X_a,X_b] :
( ( ordere779506340up_add(X_b)
& semiring_1(X_b) )
=> ! [F,K_3,A_2] :
( ! [I_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,I_2)),K_3)) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),hAPP(nat,X_b,semiring_1_of_nat(X_b),hAPP(fun(X_a,bool),nat,finite_card(X_a),A_2))),K_3))) ) ) ).
fof(fact_4297_transfer__nat__int__sum__prod__cong_I1_J,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& comm_monoid_mult(X_a) )
=> ! [F,G,A_2,B_1] :
( A_2 = B_1
=> ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,B_1))
=> ( ! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
=> hAPP(int,X_a,F,X_2) = hAPP(int,X_a,G,X_2) )
=> hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),F),A_2) = hAPP(fun(int,bool),X_a,hAPP(fun(int,X_a),fun(fun(int,bool),X_a),big_co1399186613setsum(int,X_a),G),B_1) ) ) ) ) ).
fof(fact_4298_setsum__multicount,axiom,
! [X_b,X_a,R_3,K,T_4,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),T_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),T_4))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S))),hAPP(X_b,fun(X_a,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),R_3),X_2)))) = K )
=> hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,nat),hAPP(fun(fun(X_b,bool),nat),fun(fun(X_a,fun(X_b,bool)),fun(X_a,nat)),combb(fun(X_b,bool),nat,X_a),finite_card(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),collect(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),T_4)))),R_3)))),S) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),K),hAPP(fun(X_b,bool),nat,finite_card(X_b),T_4)) ) ) ) ).
fof(fact_4299_setsum__nonneg__leq__bound,axiom,
! [X_a,X_b] :
( ordered_ab_group_add(X_b)
=> ! [I,B_1,F,S_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S_3))
=> ( ! [I_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),S_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,I_2))) )
=> ( hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S_3) = B_1
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),S_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,I)),B_1)) ) ) ) ) ) ).
fof(fact_4300_setsum__nonneg__0,axiom,
! [X_a,X_b] :
( ordered_ab_group_add(X_b)
=> ! [I,F,S_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S_3))
=> ( ! [I_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),S_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,I_2))) )
=> ( hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S_3) = zero_zero(X_b)
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),S_3))
=> hAPP(X_a,X_b,F,I) = zero_zero(X_b) ) ) ) ) ) ).
fof(fact_4301_setsum__SucD,axiom,
! [X_a,F,A_2,N] :
( hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),F),A_2) = hAPP(nat,nat,suc,N)
=> ? [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
& hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),hAPP(X_a,nat,F,X_2))) ) ) ).
fof(fact_4302_setsum__ivl__cong,axiom,
! [X_b,X_a] :
( ( ord(X_a)
& comm_monoid_add(X_b) )
=> ! [F,G,Ba,D_1,Aa,C_1] :
( ti(X_a,Aa) = ti(X_a,C_1)
=> ( ti(X_a,Ba) = ti(X_a,D_1)
=> ( ! [X_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),X_2))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_2),D_1))
=> hAPP(X_a,X_b,F,X_2) = hAPP(X_a,X_b,G,X_2) ) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),Aa),Ba)) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C_1),D_1)) ) ) ) ) ).
fof(fact_4303_setsum__multicount__gen,axiom,
! [X_b,X_a,R_3,K,T_3,S_3] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S_3))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),T_3))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),T_3))
=> hAPP(fun(X_a,bool),nat,finite_card(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S_3))),hAPP(X_b,fun(X_a,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),R_3),X_2)))) = hAPP(X_b,nat,K,X_2) )
=> hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,nat),hAPP(fun(fun(X_b,bool),nat),fun(fun(X_a,fun(X_b,bool)),fun(X_a,nat)),combb(fun(X_b,bool),nat,X_a),finite_card(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),collect(X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,bool),fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),fun(X_a,fun(X_b,bool))),combb(fun(X_b,bool),fun(X_b,bool),X_a),hAPP(fun(X_b,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,bool)),combs(X_b,bool,bool),hAPP(fun(X_b,bool),fun(X_b,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_b,bool),fun(X_b,fun(bool,bool))),combb(bool,fun(bool,bool),X_b),fconj),hAPP(fun(X_b,bool),fun(X_b,bool),hAPP(fun(X_b,fun(fun(X_b,bool),bool)),fun(fun(X_b,bool),fun(X_b,bool)),combc(X_b,fun(X_b,bool),bool),member(X_b)),T_3)))),R_3)))),S_3) = hAPP(fun(X_b,bool),nat,hAPP(fun(X_b,nat),fun(fun(X_b,bool),nat),big_co1399186613setsum(X_b,nat),K),T_3) ) ) ) ).
fof(fact_4304_DERIV__setsum,axiom,
! [X_a,X_b] :
( real_normed_field(X_b)
=> ! [F,X_1,F_2,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),S))
=> ( ! [N_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),N_2),S))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,X_b),fun(X_b,fun(X_b,bool)),deriv(X_b),hAPP(X_a,fun(X_b,X_b),hAPP(fun(X_b,fun(X_a,X_b)),fun(X_a,fun(X_b,X_b)),combc(X_b,X_a,X_b),F),N_2)),X_1),hAPP(X_a,X_b,hAPP(X_b,fun(X_a,X_b),F_2,X_1),N_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),hAPP(fun(X_b,X_b),fun(X_b,fun(X_b,bool)),deriv(X_b),hAPP(fun(X_a,bool),fun(X_b,X_b),hAPP(fun(X_b,fun(fun(X_a,bool),X_b)),fun(fun(X_a,bool),fun(X_b,X_b)),combc(X_b,fun(X_a,bool),X_b),hAPP(fun(X_b,fun(X_a,X_b)),fun(X_b,fun(fun(X_a,bool),X_b)),hAPP(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),fun(fun(X_b,fun(X_a,X_b)),fun(X_b,fun(fun(X_a,bool),X_b))),combb(fun(X_a,X_b),fun(fun(X_a,bool),X_b),X_b),big_co1399186613setsum(X_a,X_b)),F)),S)),X_1),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),hAPP(X_b,fun(X_a,X_b),F_2,X_1)),S))) ) ) ) ).
fof(fact_4305_transfer__nat__int__sum__prod__closure_I1_J,axiom,
! [F,A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> ( ! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,F,X_2))) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co1399186613setsum(int,int),F),A_2))) ) ) ).
fof(fact_4306_pos__summable,axiom,
! [X_1,F] :
( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,F,N_2)))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N_2))),X_1))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),F)) ) ) ).
fof(fact_4307_setsum_Oneutral,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [G,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(X_a,X_b,G,X_2) = zero_zero(X_b) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2) = zero_zero(X_b) ) ) ) ).
fof(fact_4308_setsum__nonpos,axiom,
! [X_a,X_b] :
( ( comm_monoid_add(X_b)
& ordere779506340up_add(X_b) )
=> ! [F,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,X_2)),zero_zero(X_b))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2)),zero_zero(X_b))) ) ) ).
fof(fact_4309_setsum__nonneg,axiom,
! [X_a,X_b] :
( ( comm_monoid_add(X_b)
& ordere779506340up_add(X_b) )
=> ! [F,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(X_a,X_b,F,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),zero_zero(X_b)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2))) ) ) ).
fof(fact_4310_setsum__UNION__zero,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,S] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,finite_finite_1(fun(X_a,bool)),S))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_2),S))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),X_2)) )
=> ( ! [T1] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),T1),S))
=> ! [T2] :
( hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),T2),S))
=> ( ti(fun(X_a,bool),T1) != ti(fun(X_a,bool),T2)
=> ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),T1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),T2))
=> hAPP(X_a,X_b,F,X_2) = zero_zero(X_b) ) ) ) ) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),hAPP(fun(fun(X_a,bool),bool),fun(X_a,bool),complete_Sup_Sup(fun(X_a,bool)),S)) = hAPP(fun(fun(X_a,bool),bool),X_b,hAPP(fun(fun(X_a,bool),X_b),fun(fun(fun(X_a,bool),bool),X_b),big_co1399186613setsum(fun(X_a,bool),X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F)),S) ) ) ) ) ).
fof(fact_4311_arith__series__int,axiom,
! [Aa,D_1,N] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,number_number_of(int),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),hAPP(fun(nat,bool),int,hAPP(fun(nat,int),fun(fun(nat,bool),int),big_co1399186613setsum(nat,int),hAPP(fun(nat,int),fun(nat,int),hAPP(fun(int,int),fun(fun(nat,int),fun(nat,int)),combb(int,int,nat),hAPP(int,fun(int,int),plus_plus(int),Aa)),hAPP(int,fun(nat,int),hAPP(fun(nat,fun(int,int)),fun(int,fun(nat,int)),combc(nat,int,int),hAPP(fun(nat,int),fun(nat,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(nat,int),fun(nat,fun(int,int))),combb(int,fun(int,int),nat),times_times(int)),semiring_1_of_nat(int))),D_1))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N))) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),N)),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Aa),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),D_1)))) ).
fof(fact_4312_BseqI2,axiom,
! [K_3,K,F] :
( ! [N_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),K),hAPP(nat,real,F,N_2)))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_2)),K_3)) )
=> hBOOL(hAPP(fun(nat,real),bool,bseq(real),F)) ) ).
fof(fact_4313_finite__lessThan,axiom,
! [K] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),ord_lessThan(nat),K))) ).
fof(fact_4314_lessThan__iff,axiom,
! [X_a] :
( ord(X_a)
=> ! [I,K] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),K)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),I),K)) ) ) ).
fof(fact_4315_card__lessThan,axiom,
! [U_1] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),ord_lessThan(nat),U_1)) = U_1 ).
fof(fact_4316_lessThan__eq__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),X_1) = hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Y_1)
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ).
fof(fact_4317_atLeast0LessThan,axiom,
! [N] : hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N) = hAPP(nat,fun(nat,bool),ord_lessThan(nat),N) ).
fof(fact_4318_finite__nat__iff__bounded,axiom,
! [S] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S))
<=> ? [K_1] : hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),S),hAPP(nat,fun(nat,bool),ord_lessThan(nat),K_1))) ) ).
fof(fact_4319_lessThan__subset__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),X_1)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),Y_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1)) ) ) ).
fof(fact_4320_lessThan__strict__subset__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [M,N] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),M)),hAPP(X_a,fun(X_a,bool),ord_lessThan(X_a),N)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),M),N)) ) ) ).
fof(fact_4321_setsum__lessThan__Suc,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,N] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),ord_lessThan(nat),hAPP(nat,nat,suc,N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N))),hAPP(nat,X_a,F,N)) ) ).
fof(fact_4322_arith__series__nat,axiom,
! [Aa,D_1,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,suc,hAPP(nat,nat,suc,zero_zero(nat)))),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),hAPP(nat,fun(nat,nat),plus_plus(nat),Aa)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),D_1))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N))) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),N),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Aa),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),Aa),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),D_1)))) ).
fof(fact_4323_image__atLeastZeroLessThan__int,axiom,
! [U_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),U_1))
=> hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),zero_zero(int)),U_1) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),hAPP(int,nat,nat_1,U_1))) ) ).
fof(fact_4324_arith__series__general,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [Aa,D_1,N] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa)),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),semiring_1_of_nat(X_a))),D_1))),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),D_1)))) ) ).
fof(fact_4325_finite__nat__bounded,axiom,
! [S] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S))
=> ? [K_1] : hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),S),hAPP(nat,fun(nat,bool),ord_lessThan(nat),K_1))) ) ).
fof(fact_4326_sum__diff__distrib,axiom,
! [N,Q_1,P_1] :
( ! [X_2] : hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),hAPP(nat,nat,Q_1,X_2)),hAPP(nat,nat,P_1,X_2)))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),P_1),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N))),hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),Q_1),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N))) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,nat)),combs(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),minus_minus(nat)),P_1)),Q_1)),hAPP(nat,fun(nat,bool),ord_lessThan(nat),N)) ) ).
fof(fact_4327_Sup__lessThan,axiom,
! [X_1] : hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),ord_lessThan(real),X_1)) = X_1 ).
fof(fact_4328_suminf__eq__lim,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& metric_space(X_a) )
=> ! [F] : hAPP(fun(nat,X_a),X_a,suminf(X_a),F) = hAPP(fun(nat,X_a),X_a,lim(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))) ) ).
fof(fact_4329_cond__split__eta,axiom,
! [X_b,X_c,X_a,F,G] :
( ! [X_2,Y_2] : hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),F,X_2),Y_2) = hAPP(product_prod(X_a,X_b),X_c,G,hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_2),Y_2))
=> hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),F) = ti(fun(product_prod(X_a,X_b),X_c),G) ) ).
fof(fact_4330_def__nat__rec__Suc,axiom,
! [X_a,N,F,C_1,H] :
( ! [N_2] : hAPP(nat,X_a,F,N_2) = hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),C_1),H),N_2)
=> hAPP(nat,X_a,F,hAPP(nat,nat,suc,N)) = hAPP(X_a,X_a,hAPP(nat,fun(X_a,X_a),H,N),hAPP(nat,X_a,F,N)) ) ).
fof(fact_4331_pigeonhole__infinite,axiom,
! [X_b,X_a,F,A_2] :
( ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_a,bool),fun(X_b,bool),hAPP(fun(X_a,X_b),fun(fun(X_a,bool),fun(X_b,bool)),image(X_a,X_b),F),A_2)))
=> ? [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
& ~ hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2))),hAPP(X_b,fun(X_a,bool),hAPP(fun(X_a,fun(X_b,bool)),fun(X_b,fun(X_a,bool)),combc(X_a,X_b,bool),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,bool))),combb(X_b,fun(X_b,bool),X_a),fequal(X_b)),F)),hAPP(X_a,X_b,F,X_2)))))) ) ) ) ).
fof(fact_4332_def__nat__rec__0,axiom,
! [X_a,F,C_1,H] :
( ! [N_2] : hAPP(nat,X_a,F,N_2) = hAPP(nat,X_a,hAPP(fun(nat,fun(X_a,X_a)),fun(nat,X_a),hAPP(X_a,fun(fun(nat,fun(X_a,X_a)),fun(nat,X_a)),nat_rec(X_a),C_1),H),N_2)
=> hAPP(nat,X_a,F,zero_zero(nat)) = ti(X_a,C_1) ) ).
fof(fact_4333_setsum__same__function__zcong,axiom,
! [X_a,F,G,M,S] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),S))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(X_a,int,F,X_2)),hAPP(X_a,int,G,X_2)),M)) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),S)),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),G),S)),M)) ) ).
fof(fact_4334_transfer__nat__int__set__cong,axiom,
! [P_2,P_1] :
( ! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_2))
=> ( hBOOL(hAPP(int,bool,P_1,X_2))
<=> hBOOL(hAPP(int,bool,P_2,X_2)) ) )
=> hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_1)) = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),hAPP(fun(int,bool),fun(int,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)))),P_2)) ) ).
fof(fact_4335_pred__subset__eq2,axiom,
! [X_a,X_b,R_3,S] :
( hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)))),combb(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),X_a),combc(X_b,fun(product_prod(X_a,X_b),bool),bool)),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),X_a),hAPP(fun(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),combb(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),X_b),member(product_prod(X_a,X_b)))),product_Pair(X_a,X_b)))),R_3)),hAPP(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(product_prod(X_a,X_b),bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),hAPP(fun(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool))),fun(fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(X_a,fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)))),combb(fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(product_prod(X_a,X_b),bool),fun(X_b,bool)),X_a),combc(X_b,fun(product_prod(X_a,X_b),bool),bool)),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool)),X_a),hAPP(fun(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool)),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,fun(fun(product_prod(X_a,X_b),bool),bool))),combb(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),X_b),member(product_prod(X_a,X_b)))),product_Pair(X_a,X_b)))),S)))
<=> hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(fun(product_prod(X_a,X_b),bool),fun(fun(product_prod(X_a,X_b),bool),bool),ord_less_eq(fun(product_prod(X_a,X_b),bool)),R_3),S)) ) ).
fof(fact_4336_rev__predicate1D,axiom,
! [X_a,Q_1,P_1,X_1] :
( hBOOL(hAPP(X_a,bool,P_1,X_1))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),P_1),Q_1))
=> hBOOL(hAPP(X_a,bool,Q_1,X_1)) ) ) ).
fof(fact_4337_predicate1D,axiom,
! [X_a,X_1,P_1,Q_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),P_1),Q_1))
=> ( hBOOL(hAPP(X_a,bool,P_1,X_1))
=> hBOOL(hAPP(X_a,bool,Q_1,X_1)) ) ) ).
fof(fact_4338_pred__equals__eq2,axiom,
! [X_a,X_b,S,R_3] :
( ! [X_2,Xa] :
( hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_2),Xa)),R_3))
<=> hBOOL(hAPP(fun(product_prod(X_a,X_b),bool),bool,hAPP(product_prod(X_a,X_b),fun(fun(product_prod(X_a,X_b),bool),bool),member(product_prod(X_a,X_b)),hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_2),Xa)),S)) )
<=> ti(fun(product_prod(X_a,X_b),bool),R_3) = ti(fun(product_prod(X_a,X_b),bool),S) ) ).
fof(fact_4339_pred__subset__eq,axiom,
! [X_a,R_3,S] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),R_3)),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),S)))
<=> hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),R_3),S)) ) ).
fof(fact_4340_rev__predicate2D,axiom,
! [X_b,X_a,Q_1,P_1,X_1,Y_1] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,X_1),Y_1))
=> ( hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),P_1),Q_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),Q_1,X_1),Y_1)) ) ) ).
fof(fact_4341_predicate2D,axiom,
! [X_a,X_b,X_1,Y_1,P_1,Q_1] :
( hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),P_1),Q_1))
=> ( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,X_1),Y_1))
=> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),Q_1,X_1),Y_1)) ) ) ).
fof(fact_4342_predicate2I,axiom,
! [X_b,X_a,Q_1,P_1] :
( ! [X_2,Y_2] :
( hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),P_1,X_2),Y_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_a,fun(X_b,bool),Q_1,X_2),Y_2)) )
=> hBOOL(hAPP(fun(X_a,fun(X_b,bool)),bool,hAPP(fun(X_a,fun(X_b,bool)),fun(fun(X_a,fun(X_b,bool)),bool),ord_less_eq(fun(X_a,fun(X_b,bool))),P_1),Q_1)) ) ).
fof(fact_4343_setsum__mono,axiom,
! [X_a,X_b] :
( ( comm_monoid_add(X_b)
& ordere779506340up_add(X_b) )
=> ! [F,G,K_3] :
( ! [I_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),K_3))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(X_a,X_b,F,I_2)),hAPP(X_a,X_b,G,I_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),K_3)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),K_3))) ) ) ).
fof(fact_4344_normalize__def,axiom,
! [P_4] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),P_4)))
=> hAPP(product_prod(int,int),product_prod(int,int),normalize,P_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_4)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_4)),hAPP(product_prod(int,int),int,product_snd(int,int),P_4)))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_snd(int,int),P_4)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_4)),hAPP(product_prod(int,int),int,product_snd(int,int),P_4)))) )
& ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),P_4)))
=> ( ( hAPP(product_prod(int,int),int,product_snd(int,int),P_4) = zero_zero(int)
=> hAPP(product_prod(int,int),product_prod(int,int),normalize,P_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) )
& ( hAPP(product_prod(int,int),int,product_snd(int,int),P_4) != zero_zero(int)
=> hAPP(product_prod(int,int),product_prod(int,int),normalize,P_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_4)),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_4)),hAPP(product_prod(int,int),int,product_snd(int,int),P_4))))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),hAPP(product_prod(int,int),int,product_snd(int,int),P_4)),hAPP(int,int,uminus_uminus(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),P_4)),hAPP(product_prod(int,int),int,product_snd(int,int),P_4))))) ) ) ) ) ).
fof(fact_4345_setsum__mono__zero__cong__right,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [G,F,S,T_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_4))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_4),S)))
=> hAPP(X_a,X_b,F,X_2) = zero_zero(X_b) )
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),S))
=> hAPP(X_a,X_b,F,X_2) = hAPP(X_a,X_b,G,X_2) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),T_4) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),S) ) ) ) ) ) ).
fof(fact_4346_gcd__dvd1__int,axiom,
! [X,Y] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)),X)) ).
fof(fact_4347_gcd__dvd2__int,axiom,
! [X,Y] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)),Y)) ).
fof(fact_4348_coprime__exp2__int,axiom,
! [N_1,M_1,A_1,B] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) = 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_1),N_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),M_1)) = one_one(int) ) ).
fof(fact_4349_invertible__coprime__int,axiom,
! [X,Y,M_1] :
( hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),X),Y)),M_1) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),M_1) = one_one(int) ) ).
fof(fact_4350_coprime__dvd__mult__int,axiom,
! [M_1,K_4,N_1] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K_4),N_1) = one_one(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),N_1)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M_1)) ) ) ).
fof(fact_4351_coprime__dvd__mult__iff__int,axiom,
! [M,K,N] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K),N) = one_one(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M),N)))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),M)) ) ) ).
fof(fact_4352_divides__mult__int,axiom,
! [N_1,M_1,R_4] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),M_1),R_4))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),N_1),R_4))
=> ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),N_1) = one_one(int)
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),M_1),N_1)),R_4)) ) ) ) ).
fof(fact_4353_gcd__coprime__int,axiom,
! [B_4,A_5,A_1,B] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) != zero_zero(int)
=> ( A_1 = 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_1),B))
=> ( B = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_4),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_5),B_4) = one_one(int) ) ) ) ).
fof(fact_4354_transfer__int__nat__gcd_I1_J,axiom,
! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(nat,int,semiring_1_of_nat(int),X)),hAPP(nat,int,semiring_1_of_nat(int),Y)) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) ).
fof(fact_4355_gcd__add__mult__int,axiom,
! [M_1,K_4,N_1] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M_1)),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),N_1) ).
fof(fact_4356_gcd__add2__int,axiom,
! [M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M_1),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),N_1) ).
fof(fact_4357_gcd__add1__int,axiom,
! [M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),M_1),N_1)),N_1) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),N_1) ).
fof(fact_4358_coprime__minus__one__int,axiom,
! [N_1] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),N_1),one_one(int))),N_1) = one_one(int) ).
fof(fact_4359_coprime__plus__one__int,axiom,
! [N_1] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),N_1),one_one(int))),N_1) = one_one(int) ).
fof(fact_4360_coprime__exp__int,axiom,
! [N_1,D,A_1] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D),A_1) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N_1)) = one_one(int) ) ).
fof(fact_4361_coprime__divisors__nat,axiom,
! [E_1,B,D,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E_1),B))
=> ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D),E_1) = one_one(int) ) ) ) ).
fof(fact_4362_gcd__1__int,axiom,
! [M_1] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),one_one(int)) = one_one(int) ).
fof(fact_4363_gcd__zero__int,axiom,
! [M,N] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N) = zero_zero(int)
<=> ( M = zero_zero(int)
& N = zero_zero(int) ) ) ).
fof(fact_4364_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,hAPP(int,fun(int,int),div_mod(int),X),Y)) ).
fof(fact_4365_gcd__greatest__iff__int,axiom,
! [K,M,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N)))
<=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),M))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K),N)) ) ) ).
fof(fact_4366_gcd__greatest__int,axiom,
! [N_1,K_4,M_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),M_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),N_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),K_4),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),N_1))) ) ) ).
fof(fact_4367_dvd__gcd__D1__int,axiom,
! [I_1,M_1,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),N_1)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),M_1)) ) ).
fof(fact_4368_dvd__gcd__D2__int,axiom,
! [I_1,M_1,N_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),N_1)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),I_1),N_1)) ) ).
fof(fact_4369_gcd__assoc__int,axiom,
! [A_1,B,C] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),C) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),C)) ).
fof(fact_4370_gcd__int_Oleft__commute,axiom,
! [B,A_1,C] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),C)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),C)) ).
fof(fact_4371_gcd__commute__int,axiom,
! [A_1,B] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),B),A_1) ).
fof(fact_4372_gcd__exp__int,axiom,
! [A_1,N_1,B] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),A_1),N_1)),hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),B),N_1)) = hAPP(nat,int,hAPP(int,fun(nat,int),power_power(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),N_1) ).
fof(fact_4373_gcd__mult__distrib__int,axiom,
! [K_4,M_1,N_1] : hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),K_4)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),N_1)) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M_1)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),N_1)) ).
fof(fact_4374_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_4375_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_4376_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_4377_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_4378_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_4379_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_4380_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_4381_gcd__proj2__if__dvd__int,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_4382_gcd__proj1__if__dvd__int,axiom,
! [X,Y] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_4383_gcd__pos__int,axiom,
! [M,N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M),N)))
<=> ( M != zero_zero(int)
| N != zero_zero(int) ) ) ).
fof(fact_4384_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_4385_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_4386_coprime__rmult__int,axiom,
! [D,A_1,B] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D),B) = one_one(int) ) ).
fof(fact_4387_coprime__lmult__int,axiom,
! [D,A_1,B] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D),A_1) = one_one(int) ) ).
fof(fact_4388_coprime__mult__int,axiom,
! [B,D,A_1] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D),A_1) = one_one(int)
=> ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D),B) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),A_1),B)) = one_one(int) ) ) ).
fof(fact_4389_gcd__mult__cancel__int,axiom,
! [M_1,K_4,N_1] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),K_4),N_1) = one_one(int)
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_4),M_1)),N_1) = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),M_1),N_1) ) ).
fof(fact_4390_coprime__mul__eq__int,axiom,
! [D_1,Aa,Ba] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),Aa),Ba)) = one_one(int)
<=> ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_1),Aa) = one_one(int)
& hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),D_1),Ba) = one_one(int) ) ) ).
fof(fact_4391_Fract__coprime,axiom,
! [A_1,B] : hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(int,int,hAPP(int,fun(int,int),div_div(int),A_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))) = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_1),B) ).
fof(fact_4392_div__gcd__coprime__int,axiom,
! [B,A_1] :
( ( A_1 != zero_zero(int)
| B != 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_1),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))),hAPP(int,int,hAPP(int,fun(int,int),div_div(int),B),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))) = one_one(int) ) ).
fof(fact_4393_gcd__non__0__int,axiom,
! [X,Y] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,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,hAPP(int,fun(int,int),div_mod(int),X),Y)) ) ).
fof(fact_4394_transfer__nat__int__gcd__closures_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y))) ) ) ).
fof(fact_4395_gcd__ge__0__int,axiom,
! [X,Y] : hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y))) ).
fof(fact_4396_gcd__le1__int,axiom,
! [B,A_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),A_1))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),A_1)) ) ).
fof(fact_4397_gcd__le2__int,axiom,
! [A_1,B] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B)),B)) ) ).
fof(fact_4398_gcd__unique__int,axiom,
! [Ba,Aa,D_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D_1))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),Aa))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),Ba))
& ! [E] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E),Aa))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E),Ba)) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),E),D_1)) ) )
<=> D_1 = hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Aa),Ba) ) ).
fof(fact_4399_gcd__cases__int,axiom,
! [P_1,Y_1,X_1] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X_1),Y_1))) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y_1),zero_zero(int)))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X_1),hAPP(int,int,uminus_uminus(int),Y_1)))) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y_1))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,uminus_uminus(int),X_1)),Y_1))) ) )
=> ( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_1),zero_zero(int)))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),Y_1),zero_zero(int)))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(int,int,uminus_uminus(int),X_1)),hAPP(int,int,uminus_uminus(int),Y_1)))) ) )
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X_1),Y_1))) ) ) ) ) ).
fof(fact_4400_coprime__crossproduct__int,axiom,
! [Ba,C_1,Aa,D_1] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Aa),D_1) = one_one(int)
=> ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Ba),C_1) = one_one(int)
=> ( hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),Aa)),hAPP(int,int,abs_abs(int),C_1)) = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(int,int,abs_abs(int),Ba)),hAPP(int,int,abs_abs(int),D_1))
<=> ( hAPP(int,int,abs_abs(int),Aa) = hAPP(int,int,abs_abs(int),Ba)
& hAPP(int,int,abs_abs(int),C_1) = hAPP(int,int,abs_abs(int),D_1) ) ) ) ) ).
fof(fact_4401_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(bool,fun(int,fun(int,int)),if(int),hAPP(int,bool,hAPP(int,fun(int,bool),fequal(int),L),zero_zero(int))),K),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),L),hAPP(int,int,hAPP(int,fun(int,int),div_mod(int),hAPP(int,int,abs_abs(int),K)),hAPP(int,int,abs_abs(int),L))))) ).
fof(fact_4402_coprime__common__divisor__int,axiom,
! [X,A_1,B] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) = one_one(int)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),X),A_1))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),X),B))
=> hAPP(int,int,abs_abs(int),X) = one_one(int) ) ) ) ).
fof(fact_4403_quotient__of__coprime,axiom,
! [R_4,P_4,Q_2] :
( hAPP(rat,product_prod(int,int),quotient_of,R_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),Q_2)
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),P_4),Q_2) = one_one(int) ) ).
fof(fact_4404_normalize__coprime,axiom,
! [R_4,P_4,Q_2] :
( hAPP(product_prod(int,int),product_prod(int,int),normalize,R_4) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),Q_2)
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),P_4),Q_2) = one_one(int) ) ).
fof(fact_4405_coprime__int,axiom,
! [Aa,Ba] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),Aa),Ba) = one_one(int)
<=> ! [D_3] :
( ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D_3))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_3),Aa))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_3),Ba)) )
<=> D_3 = one_one(int) ) ) ).
fof(fact_4406_transfer__nat__int__gcd_I1_J,axiom,
! [Y,X] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(int,nat,nat_1,X)),hAPP(int,nat,nat_1,Y)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y)) ) ) ).
fof(fact_4407_gcd__int__def,axiom,
! [X,Y] : hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(nat,nat,hAPP(nat,fun(nat,nat),gcd_gcd(nat),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),X))),hAPP(int,nat,nat_1,hAPP(int,int,abs_abs(int),Y)))) ).
fof(fact_4408_normalize__stable,axiom,
! [P_4,Q_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),Q_2))
=> ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),P_4),Q_2) = one_one(int)
=> hAPP(product_prod(int,int),product_prod(int,int),normalize,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),Q_2)) = hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),P_4),Q_2) ) ) ).
fof(fact_4409_quotient__of__def,axiom,
! [X_1] : hAPP(rat,product_prod(int,int),quotient_of,X_1) = hAPP(fun(product_prod(int,int),bool),product_prod(int,int),the(product_prod(int,int)),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combs(product_prod(int,int),bool,bool),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool))),combb(bool,fun(bool,bool),product_prod(int,int)),fconj),hAPP(fun(product_prod(int,int),rat),fun(product_prod(int,int),bool),hAPP(fun(rat,bool),fun(fun(product_prod(int,int),rat),fun(product_prod(int,int),bool)),combb(rat,bool,product_prod(int,int)),hAPP(rat,fun(rat,bool),fequal(rat),X_1)),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),rat),hAPP(fun(product_prod(int,int),fun(int,rat)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),rat)),combs(product_prod(int,int),int,rat),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,rat)),hAPP(fun(int,fun(int,rat)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,rat))),combb(int,fun(int,rat),product_prod(int,int)),fract),product_fst(int,int))),product_snd(int,int))))),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),combs(product_prod(int,int),bool,bool),hAPP(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),fun(bool,bool))),combb(bool,fun(bool,bool),product_prod(int,int)),fconj),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),bool),hAPP(fun(int,bool),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),bool)),combb(int,bool,product_prod(int,int)),hAPP(int,fun(int,bool),ord_less(int),zero_zero(int))),product_snd(int,int)))),hAPP(int,fun(product_prod(int,int),bool),hAPP(fun(product_prod(int,int),fun(int,bool)),fun(int,fun(product_prod(int,int),bool)),combc(product_prod(int,int),int,bool),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,bool))),combb(int,fun(int,bool),product_prod(int,int)),fequal(int)),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),int),hAPP(fun(product_prod(int,int),fun(int,int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),int)),combs(product_prod(int,int),int,int),hAPP(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(product_prod(int,int),int),fun(product_prod(int,int),fun(int,int))),combb(int,fun(int,int),product_prod(int,int)),gcd_gcd(int)),product_fst(int,int))),product_snd(int,int)))),one_one(int))))) ).
fof(fact_4410_quotient__of__unique,axiom,
! [R_4] :
? [X_2] :
( R_4 = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(product_prod(int,int),int,product_fst(int,int),X_2)),hAPP(product_prod(int,int),int,product_snd(int,int),X_2))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),X_2)))
& hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),X_2)),hAPP(product_prod(int,int),int,product_snd(int,int),X_2)) = one_one(int)
& ! [Y_2] :
( ( R_4 = hAPP(int,rat,hAPP(int,fun(int,rat),fract,hAPP(product_prod(int,int),int,product_fst(int,int),Y_2)),hAPP(product_prod(int,int),int,product_snd(int,int),Y_2))
& hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),hAPP(product_prod(int,int),int,product_snd(int,int),Y_2)))
& hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(product_prod(int,int),int,product_fst(int,int),Y_2)),hAPP(product_prod(int,int),int,product_snd(int,int),Y_2)) = one_one(int) )
=> Y_2 = X_2 ) ) ).
fof(fact_4411_Rat__cases__nonzero,axiom,
! [Q_2] :
( ! [A_3,B_3] :
( Q_2 = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_3)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3))
=> ( A_3 != zero_zero(int)
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_3),B_3) != one_one(int) ) ) )
=> Q_2 = zero_zero(rat) ) ).
fof(fact_4412_Rat__induct,axiom,
! [Qa,P_1] :
( ! [A_3,B_3] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3))
=> ( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_3),B_3) = one_one(int)
=> hBOOL(hAPP(rat,bool,P_1,hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_3))) ) )
=> hBOOL(hAPP(rat,bool,P_1,Qa)) ) ).
fof(fact_4413_Rat__cases,axiom,
! [Q_2] :
~ ! [A_3,B_3] :
( Q_2 = hAPP(int,rat,hAPP(int,fun(int,rat),fract,A_3),B_3)
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),B_3))
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_3),B_3) != one_one(int) ) ) ).
fof(fact_4414_setprod__coprime__int,axiom,
! [X_a,F,X_1,A_2] :
( ! [I_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2))
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(X_a,int,F,I_2)),X_1) = one_one(int) )
=> hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),A_2)),X_1) = one_one(int) ) ).
fof(fact_4415_setsum__mono__zero__cong__left,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,G,S,T_4] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),T_4))
=> ( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),S),T_4))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),minus_minus(fun(X_a,bool)),T_4),S)))
=> hAPP(X_a,X_b,G,X_2) = zero_zero(X_b) )
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),S))
=> hAPP(X_a,X_b,F,X_2) = hAPP(X_a,X_b,G,X_2) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),S) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),T_4) ) ) ) ) ) ).
fof(fact_4416_card__setsum__aux,axiom,
! [N,S] :
( hBOOL(hAPP(fun(fun(int,bool),bool),bool,finite_finite_1(fun(int,bool)),S))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_2),S))
=> hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),X_2)) )
=> ( ! [X_2] :
( hBOOL(hAPP(fun(fun(int,bool),bool),bool,hAPP(fun(int,bool),fun(fun(fun(int,bool),bool),bool),member(fun(int,bool)),X_2),S))
=> hAPP(fun(int,bool),nat,finite_card(int),X_2) = N )
=> hAPP(fun(fun(int,bool),bool),nat,hAPP(fun(fun(int,bool),nat),fun(fun(fun(int,bool),bool),nat),big_co1399186613setsum(fun(int,bool),nat),finite_card(int)),S) = hAPP(fun(fun(int,bool),bool),nat,hAPP(fun(fun(int,bool),nat),fun(fun(fun(int,bool),bool),nat),big_co1399186613setsum(fun(int,bool),nat),hAPP(nat,fun(fun(int,bool),nat),combk(nat,fun(int,bool)),N)),S) ) ) ) ).
fof(fact_4417_setprod__same__function__zcong,axiom,
! [X_a,F,G,M,S] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),S))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(X_a,int,F,X_2)),hAPP(X_a,int,G,X_2)),M)) )
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),S)),hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),G),S)),M)) ) ).
fof(fact_4418_Nitpick_Ofrac__def,axiom,
! [X_a,A_1,B] : hAPP(int,X_a,hAPP(int,fun(int,X_a),frac(X_a),A_1),B) = hAPP(product_prod(int,int),X_a,abs_Frac(X_a),hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),norm_frac,A_1),B)) ).
fof(fact_4419_zOdd__def,axiom,
zOdd = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,fun(int,bool)),fun(int,bool),hAPP(fun(fun(int,bool),bool),fun(fun(int,fun(int,bool)),fun(int,bool)),combb(fun(int,bool),bool,int),fEx(int)),hAPP(fun(int,int),fun(int,fun(int,bool)),hAPP(fun(int,fun(fun(int,int),fun(int,bool))),fun(fun(int,int),fun(int,fun(int,bool))),combc(int,fun(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool))),hAPP(fun(fun(int,bool),fun(fun(int,int),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool)))),combb(fun(int,bool),fun(fun(int,int),fun(int,bool)),int),combb(int,bool,int)),fequal(int))),hAPP(int,fun(int,int),hAPP(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),hAPP(fun(int,int),fun(int,fun(int,int)),hAPP(fun(int,fun(int,int)),fun(fun(int,int),fun(int,fun(int,int))),combb(int,fun(int,int),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_4420_finite__Collect__bounded__ex,axiom,
! [X_b,X_a,Q_1,P_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)))
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,bool),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_b,fun(X_a,bool)),fun(X_b,bool)),combb(fun(X_a,bool),bool,X_b),fEx(X_a)),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,bool))),combb(fun(X_a,bool),fun(X_a,bool),X_b),hAPP(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),combs(X_a,bool,bool),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),P_1))),Q_1)))))
<=> ! [Y_2] :
( hBOOL(hAPP(X_a,bool,P_1,Y_2))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(X_a,fun(X_b,bool),hAPP(fun(X_b,fun(X_a,bool)),fun(X_a,fun(X_b,bool)),combc(X_b,X_a,bool),Q_1),Y_2)))) ) ) ) ).
fof(fact_4421_finite__image__set,axiom,
! [X_b,X_a,F,P_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),collect(X_a),P_1)))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),hAPP(fun(X_b,bool),fun(X_b,bool),collect(X_b),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,bool),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_b,fun(X_a,bool)),fun(X_b,bool)),combb(fun(X_a,bool),bool,X_b),fEx(X_a)),hAPP(fun(X_a,bool),fun(X_b,fun(X_a,bool)),hAPP(fun(X_b,fun(fun(X_a,bool),fun(X_a,bool))),fun(fun(X_a,bool),fun(X_b,fun(X_a,bool))),combc(X_b,fun(X_a,bool),fun(X_a,bool)),hAPP(fun(X_b,fun(X_a,fun(bool,bool))),fun(X_b,fun(fun(X_a,bool),fun(X_a,bool))),hAPP(fun(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool))),fun(fun(X_b,fun(X_a,fun(bool,bool))),fun(X_b,fun(fun(X_a,bool),fun(X_a,bool)))),combb(fun(X_a,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,bool)),X_b),combs(X_a,bool,bool)),hAPP(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,fun(bool,bool))),hAPP(fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),fun(fun(X_b,fun(X_a,bool)),fun(X_b,fun(X_a,fun(bool,bool)))),combb(fun(X_a,bool),fun(X_a,fun(bool,bool)),X_b),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj)),hAPP(fun(X_a,X_b),fun(X_b,fun(X_a,bool)),hAPP(fun(X_b,fun(fun(X_a,X_b),fun(X_a,bool))),fun(fun(X_a,X_b),fun(X_b,fun(X_a,bool))),combc(X_b,fun(X_a,X_b),fun(X_a,bool)),hAPP(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_a,X_b),fun(X_a,bool))),hAPP(fun(fun(X_b,bool),fun(fun(X_a,X_b),fun(X_a,bool))),fun(fun(X_b,fun(X_b,bool)),fun(X_b,fun(fun(X_a,X_b),fun(X_a,bool)))),combb(fun(X_b,bool),fun(fun(X_a,X_b),fun(X_a,bool)),X_b),combb(X_b,bool,X_a)),fequal(X_b))),F)))),P_1))))) ) ).
fof(fact_4422_Rats__eq__int__div__int,axiom,
field_char_0_Rats(real) = hAPP(fun(real,bool),fun(real,bool),collect(real),hAPP(fun(real,fun(int,bool)),fun(real,bool),hAPP(fun(fun(int,bool),bool),fun(fun(real,fun(int,bool)),fun(real,bool)),combb(fun(int,bool),bool,real),fEx(int)),hAPP(fun(real,fun(int,fun(int,bool))),fun(real,fun(int,bool)),hAPP(fun(fun(int,fun(int,bool)),fun(int,bool)),fun(fun(real,fun(int,fun(int,bool))),fun(real,fun(int,bool))),combb(fun(int,fun(int,bool)),fun(int,bool),real),hAPP(fun(fun(int,bool),bool),fun(fun(int,fun(int,bool)),fun(int,bool)),combb(fun(int,bool),bool,int),fEx(int))),hAPP(fun(int,bool),fun(real,fun(int,fun(int,bool))),hAPP(fun(real,fun(fun(int,bool),fun(int,fun(int,bool)))),fun(fun(int,bool),fun(real,fun(int,fun(int,bool)))),combc(real,fun(int,bool),fun(int,fun(int,bool))),hAPP(fun(real,fun(int,fun(fun(int,bool),fun(int,bool)))),fun(real,fun(fun(int,bool),fun(int,fun(int,bool)))),hAPP(fun(fun(int,fun(fun(int,bool),fun(int,bool))),fun(fun(int,bool),fun(int,fun(int,bool)))),fun(fun(real,fun(int,fun(fun(int,bool),fun(int,bool)))),fun(real,fun(fun(int,bool),fun(int,fun(int,bool))))),combb(fun(int,fun(fun(int,bool),fun(int,bool))),fun(fun(int,bool),fun(int,fun(int,bool))),real),combc(int,fun(int,bool),fun(int,bool))),hAPP(fun(real,fun(int,fun(int,fun(bool,bool)))),fun(real,fun(int,fun(fun(int,bool),fun(int,bool)))),hAPP(fun(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool)))),fun(fun(real,fun(int,fun(int,fun(bool,bool)))),fun(real,fun(int,fun(fun(int,bool),fun(int,bool))))),combb(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool))),real),hAPP(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(int,fun(int,fun(bool,bool))),fun(int,fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),int),combs(int,bool,bool))),hAPP(fun(real,fun(int,fun(int,bool))),fun(real,fun(int,fun(int,fun(bool,bool)))),hAPP(fun(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool)))),fun(fun(real,fun(int,fun(int,bool))),fun(real,fun(int,fun(int,fun(bool,bool))))),combb(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool))),real),hAPP(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),int),hAPP(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj))),hAPP(fun(int,fun(int,real)),fun(real,fun(int,fun(int,bool))),hAPP(fun(real,fun(fun(int,fun(int,real)),fun(int,fun(int,bool)))),fun(fun(int,fun(int,real)),fun(real,fun(int,fun(int,bool)))),combc(real,fun(int,fun(int,real)),fun(int,fun(int,bool))),hAPP(fun(real,fun(fun(int,real),fun(int,bool))),fun(real,fun(fun(int,fun(int,real)),fun(int,fun(int,bool)))),hAPP(fun(fun(fun(int,real),fun(int,bool)),fun(fun(int,fun(int,real)),fun(int,fun(int,bool)))),fun(fun(real,fun(fun(int,real),fun(int,bool))),fun(real,fun(fun(int,fun(int,real)),fun(int,fun(int,bool))))),combb(fun(fun(int,real),fun(int,bool)),fun(fun(int,fun(int,real)),fun(int,fun(int,bool))),real),combb(fun(int,real),fun(int,bool),int)),hAPP(fun(real,fun(real,bool)),fun(real,fun(fun(int,real),fun(int,bool))),hAPP(fun(fun(real,bool),fun(fun(int,real),fun(int,bool))),fun(fun(real,fun(real,bool)),fun(real,fun(fun(int,real),fun(int,bool)))),combb(fun(real,bool),fun(fun(int,real),fun(int,bool)),real),combb(real,bool,int)),fequal(real)))),hAPP(fun(int,real),fun(int,fun(int,real)),hAPP(fun(int,fun(fun(int,real),fun(int,real))),fun(fun(int,real),fun(int,fun(int,real))),combc(int,fun(int,real),fun(int,real)),hAPP(fun(int,fun(real,real)),fun(int,fun(fun(int,real),fun(int,real))),hAPP(fun(fun(real,real),fun(fun(int,real),fun(int,real))),fun(fun(int,fun(real,real)),fun(int,fun(fun(int,real),fun(int,real)))),combb(fun(real,real),fun(fun(int,real),fun(int,real)),int),combb(real,real,int)),hAPP(fun(int,real),fun(int,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(int,real),fun(int,fun(real,real))),combb(real,fun(real,real),int),inverse_divide(real)),real_1(int)))),real_1(int))))))),hAPP(fun(int,bool),fun(int,bool),hAPP(fun(bool,bool),fun(fun(int,bool),fun(int,bool)),combb(bool,bool,int),fNot),hAPP(int,fun(int,bool),hAPP(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),fequal(int)),zero_zero(int))))))) ).
fof(fact_4423_zEven__def,axiom,
zEven = hAPP(fun(int,bool),fun(int,bool),collect(int),hAPP(fun(int,fun(int,bool)),fun(int,bool),hAPP(fun(fun(int,bool),bool),fun(fun(int,fun(int,bool)),fun(int,bool)),combb(fun(int,bool),bool,int),fEx(int)),hAPP(fun(int,int),fun(int,fun(int,bool)),hAPP(fun(int,fun(fun(int,int),fun(int,bool))),fun(fun(int,int),fun(int,fun(int,bool))),combc(int,fun(int,int),fun(int,bool)),hAPP(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool))),hAPP(fun(fun(int,bool),fun(fun(int,int),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool)))),combb(fun(int,bool),fun(fun(int,int),fun(int,bool)),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_4424_summable__Leibniz_I5_J,axiom,
! [Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),zero_zero(real)),sequentially))
=> ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),Aa))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),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(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),sequentially)) ) ) ).
fof(fact_4425_setprod__dvd__setprod,axiom,
! [X_a,X_b] :
( ( comm_monoid_mult(X_b)
& dvd(X_b) )
=> ! [F,G,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(X_a,X_b,F,X_2)),hAPP(X_a,X_b,G,X_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),A_2)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2))) ) ) ).
fof(fact_4426_Pow__Compl,axiom,
! [X_a,A_2] : hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),uminus_uminus(fun(X_a,bool)),A_2)) = hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool),collect(fun(X_a,bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool),hAPP(fun(fun(fun(X_a,bool),bool),bool),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool)),combb(fun(fun(X_a,bool),bool),bool,fun(X_a,bool)),fEx(fun(X_a,bool))),hAPP(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(fun(X_a,bool),bool))),combc(fun(X_a,bool),fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),hAPP(fun(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)))),combb(fun(fun(X_a,bool),fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),bool)),fun(X_a,bool)),combs(fun(X_a,bool),bool,bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool))),hAPP(fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),fun(bool,bool)))),combb(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool)),fun(X_a,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(fun(X_a,bool),bool),fun(fun(X_a,bool),fun(bool,bool))),combb(bool,fun(bool,bool),fun(X_a,bool)),fconj)),hAPP(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),fun(fun(X_a,bool),bool))),combc(fun(X_a,bool),fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool)),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool))),hAPP(fun(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool))),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool)))),combb(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),fun(X_a,bool)),fun(fun(X_a,bool),bool)),fun(X_a,bool)),combb(fun(X_a,bool),bool,fun(X_a,bool))),fequal(fun(X_a,bool)))),uminus_uminus(fun(X_a,bool)))))),hAPP(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool),hAPP(fun(fun(fun(X_a,bool),bool),bool),fun(fun(fun(X_a,bool),fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),bool)),combb(fun(fun(X_a,bool),bool),bool,fun(X_a,bool)),hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),A_2)),pow(X_a))))) ).
fof(fact_4427_lim__def,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_3] : hAPP(fun(nat,X_a),X_a,lim(X_a),X_3) = hAPP(fun(X_a,bool),X_a,the(X_a),hAPP(filter(nat),fun(X_a,bool),hAPP(fun(X_a,fun(filter(nat),bool)),fun(filter(nat),fun(X_a,bool)),combc(X_a,filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3)),sequentially)) ) ).
fof(fact_4428_limI,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_3,L_2] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),L_2),sequentially))
=> hAPP(fun(nat,X_a),X_a,lim(X_a),X_3) = ti(X_a,L_2) ) ) ).
fof(fact_4429_LIMSEQ__neg__powr,axiom,
! [S_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_3))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(real,fun(nat,real)),combc(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),powr),real_1(nat))),hAPP(real,real,uminus_uminus(real),S_3))),zero_zero(real)),sequentially)) ) ).
fof(fact_4430_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R_1] : hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),plus_plus(real),R_1)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc))))),R_1),sequentially)) ).
fof(fact_4431_LIMSEQ__inverse__realpow__zero,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X_1))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(real,fun(nat,real),power_power(real),X_1))),zero_zero(real)),sequentially)) ) ).
fof(fact_4432_Bseq__inverse,axiom,
! [X_a] :
( real_n1866405975lgebra(X_a)
=> ! [X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> ( ti(X_a,Aa) != zero_zero(X_a)
=> hBOOL(hAPP(fun(nat,X_a),bool,bseq(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),inverse_inverse(X_a)),X_3))) ) ) ) ).
fof(fact_4433_LIMSEQ__imp__Cauchy,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3)) ) ) ).
fof(fact_4434_LIMSEQ__power__zero,axiom,
! [X_a] :
( real_n2089651433ebra_1(X_a)
=> ! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X_1)),one_one(real)))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1)),zero_zero(X_a)),sequentially)) ) ) ).
fof(fact_4435_LIMSEQ__divide__realpow__zero,axiom,
! [Aa,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),one_one(real)),X_1))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),Aa)),hAPP(real,fun(nat,real),power_power(real),X_1))),zero_zero(real)),sequentially)) ) ).
fof(fact_4436_LIMSEQ__unique,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [Ba,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Ba),sequentially))
=> ti(X_a,Aa) = ti(X_a,Ba) ) ) ) ).
fof(fact_4437_tendsto__inverse,axiom,
! [X_a,X_b] :
( real_n1866405975lgebra(X_b)
=> ! [F,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),Aa),A_2))
=> ( ti(X_b,Aa) != zero_zero(X_b)
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),inverse_inverse(X_b)),F)),hAPP(X_b,X_b,inverse_inverse(X_b),Aa)),A_2)) ) ) ) ).
fof(fact_4438_LIMSEQ__inverse__real__of__nat,axiom,
hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc))),zero_zero(real)),sequentially)) ).
fof(fact_4439_LIMSEQ__inverse,axiom,
! [X_a] :
( real_n1866405975lgebra(X_a)
=> ! [X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> ( ti(X_a,Aa) != zero_zero(X_a)
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),inverse_inverse(X_a)),X_3)),hAPP(X_a,X_a,inverse_inverse(X_a),Aa)),sequentially)) ) ) ) ).
fof(fact_4440_LIMSEQ__rabs__zero,axiom,
! [F] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)),zero_zero(real)),sequentially))
<=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),F),zero_zero(real)),sequentially)) ) ).
fof(fact_4441_LIMSEQ__linear,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [L,X_3,X_1] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),X_1),sequentially))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),L))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),X_3),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),times_times(nat)),L))),X_1),sequentially)) ) ) ) ).
fof(fact_4442_LIMSEQ__imp__rabs,axiom,
! [F,L] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),F),L),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),abs_abs(real)),F)),hAPP(real,real,abs_abs(real),L)),sequentially)) ) ).
fof(fact_4443_LIMSEQ__minus__cancel,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_3)),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially)) ) ) ).
fof(fact_4444_LIMSEQ__minus,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),X_3)),hAPP(X_a,X_a,uminus_uminus(X_a),Aa)),sequentially)) ) ) ).
fof(fact_4445_tendsto__minus__cancel,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [F,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),hAPP(X_b,X_b,uminus_uminus(X_b),Aa)),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),Aa),A_2)) ) ) ).
fof(fact_4446_tendsto__minus,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [F,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),hAPP(X_b,X_b,uminus_uminus(X_b),Aa)),A_2)) ) ) ).
fof(fact_4447_mult_Otendsto,axiom,
! [X_a,X_b] :
( real_normed_algebra(X_b)
=> ! [G,Ba,F,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),Aa),A_2))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ba),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Aa),Ba)),A_2)) ) ) ) ).
fof(fact_4448_mult__left_Otendsto,axiom,
! [X_a,X_b] :
( real_normed_algebra(X_b)
=> ! [Y_1,G,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),Y_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),Aa),Y_1)),A_2)) ) ) ).
fof(fact_4449_mult__right_Otendsto,axiom,
! [X_a,X_b] :
( real_normed_algebra(X_b)
=> ! [X_1,G,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1),Aa)),A_2)) ) ) ).
fof(fact_4450_LIMSEQ__mult,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y_3,Ba,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_3),Ba),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_3)),Y_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Ba)),sequentially)) ) ) ) ).
fof(fact_4451_mult__left_OLIMSEQ,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y_1,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),X_3)),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Y_1)),sequentially)) ) ) ).
fof(fact_4452_mult__right_OLIMSEQ,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X_1,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1)),X_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Aa)),sequentially)) ) ) ).
fof(fact_4453_tendsto__dist,axiom,
! [X_a,X_b] :
( metric_space(X_b)
=> ! [G,M,F,L,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_2))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_a,fun(X_b,real)),fun(fun(X_a,X_b),fun(X_a,real)),combs(X_a,X_b,real),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,real)),hAPP(fun(X_b,fun(X_b,real)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,real))),combb(X_b,fun(X_b,real),X_a),dist_dist(X_b)),F)),G)),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),L),M)),A_2)) ) ) ) ).
fof(fact_4454_of__real_Otendsto,axiom,
! [X_a,X_b] :
( ( real_algebra_1(X_b)
& real_normed_vector(X_b) )
=> ! [G,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),G)),hAPP(real,X_b,of_real(X_b),Aa)),A_2)) ) ) ).
fof(fact_4455_tendsto__const,axiom,
! [X_a,X_b] :
( topological_space(X_b)
=> ! [K,A_2] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),K),A_2)) ) ).
fof(fact_4456_bounded__bilinear_Otendsto,axiom,
! [X_d,X_b,X_c,X_a] :
( ( real_normed_vector(X_a)
& real_normed_vector(X_c)
& real_normed_vector(X_b) )
=> ! [G,Ba,F,Aa,A_2,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> ( hBOOL(hAPP(filter(X_d),bool,hAPP(X_a,fun(filter(X_d),bool),hAPP(fun(X_d,X_a),fun(X_a,fun(filter(X_d),bool)),tendsto(X_d,X_a),F),Aa),A_2))
=> ( hBOOL(hAPP(filter(X_d),bool,hAPP(X_b,fun(filter(X_d),bool),hAPP(fun(X_d,X_b),fun(X_b,fun(filter(X_d),bool)),tendsto(X_d,X_b),G),Ba),A_2))
=> hBOOL(hAPP(filter(X_d),bool,hAPP(X_c,fun(filter(X_d),bool),hAPP(fun(X_d,X_c),fun(X_c,fun(filter(X_d),bool)),tendsto(X_d,X_c),hAPP(fun(X_d,X_b),fun(X_d,X_c),hAPP(fun(X_d,fun(X_b,X_c)),fun(fun(X_d,X_b),fun(X_d,X_c)),combs(X_d,X_b,X_c),hAPP(fun(X_d,X_a),fun(X_d,fun(X_b,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_d,X_a),fun(X_d,fun(X_b,X_c))),combb(X_a,fun(X_b,X_c),X_d),Prod),F)),G)),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)),A_2)) ) ) ) ) ).
fof(fact_4457_tendsto__norm,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [F,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),hAPP(X_b,real,norm_norm(X_b),Aa)),A_2)) ) ) ).
fof(fact_4458_LIMSEQ__norm,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),X_3)),hAPP(X_a,real,norm_norm(X_a),Aa)),sequentially)) ) ) ).
fof(fact_4459_LIMSEQ__Suc__iff,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [F,L] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L),sequentially))
<=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L),sequentially)) ) ) ).
fof(fact_4460_LIMSEQ__Suc,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [F,L] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L),sequentially)) ) ) ).
fof(fact_4461_LIMSEQ__imp__Suc,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [F,L] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),L),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L),sequentially)) ) ) ).
fof(fact_4462_LIMSEQ__const,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [K] : hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),K)),K),sequentially)) ) ).
fof(fact_4463_LIMSEQ__const__iff,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [K,L] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),combk(X_a,nat),K)),L),sequentially))
<=> ti(X_a,K) = ti(X_a,L) ) ) ).
fof(fact_4464_bounded__bilinear_OLIMSEQ,axiom,
! [X_b,X_c,X_a] :
( ( real_normed_vector(X_a)
& real_normed_vector(X_c)
& real_normed_vector(X_b) )
=> ! [Y_3,Ba,X_3,Aa,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),Y_3),Ba),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_c,fun(filter(nat),bool),hAPP(fun(nat,X_c),fun(X_c,fun(filter(nat),bool)),tendsto(nat,X_c),hAPP(fun(nat,X_b),fun(nat,X_c),hAPP(fun(nat,fun(X_b,X_c)),fun(fun(nat,X_b),fun(nat,X_c)),combs(nat,X_b,X_c),hAPP(fun(nat,X_a),fun(nat,fun(X_b,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(nat,X_a),fun(nat,fun(X_b,X_c))),combb(X_a,fun(X_b,X_c),nat),Prod),X_3)),Y_3)),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)),sequentially)) ) ) ) ) ).
fof(fact_4465_LIMSEQ__pow,axiom,
! [X_a] :
( ( power(X_a)
& real_normed_algebra(X_a) )
=> ! [M,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(nat,fun(nat,X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a)),combc(nat,nat,X_a),hAPP(fun(nat,X_a),fun(nat,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(nat,X_a),fun(nat,fun(nat,X_a))),combb(X_a,fun(nat,X_a),nat),power_power(X_a)),X_3)),M)),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),Aa),M)),sequentially)) ) ) ).
fof(fact_4466_of__real_OLIMSEQ,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),of_real(X_a)),X_3)),hAPP(real,X_a,of_real(X_a),Aa)),sequentially)) ) ) ).
fof(fact_4467_divide_OLIMSEQ,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_1,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_3)),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Y_1)),sequentially)) ) ) ).
fof(fact_4468_LIMSEQ__norm__zero,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),X_3)),zero_zero(real)),sequentially))
<=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),zero_zero(X_a)),sequentially)) ) ) ).
fof(fact_4469_LIMSEQ__divide,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_3,Ba,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_3),Ba),sequentially))
=> ( ti(X_a,Ba) != zero_zero(X_a)
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a)),X_3)),Y_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Ba)),sequentially)) ) ) ) ) ).
fof(fact_4470_tendsto__divide,axiom,
! [X_a,X_b] :
( real_normed_field(X_b)
=> ! [G,Ba,F,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),Aa),A_2))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ba),A_2))
=> ( ti(X_b,Ba) != zero_zero(X_b)
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),Aa),Ba)),A_2)) ) ) ) ) ).
fof(fact_4471_tendsto__norm__zero__iff,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [F,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),A_2))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),A_2)) ) ) ).
fof(fact_4472_tendsto__norm__zero,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [F,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),A_2)) ) ) ).
fof(fact_4473_tendsto__norm__zero__cancel,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [F,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),A_2)) ) ) ).
fof(fact_4474_divide_Otendsto,axiom,
! [X_a,X_b] :
( real_normed_field(X_b)
=> ! [Y_1,G,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),G)),Y_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),Aa),Y_1)),A_2)) ) ) ).
fof(fact_4475_LIMSEQ__diff__approach__zero2,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [G,F,L_2] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L_2),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),G)),zero_zero(X_a)),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),G),L_2),sequentially)) ) ) ) ).
fof(fact_4476_LIMSEQ__diff__approach__zero,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [F,G,L_2] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),G),L_2),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),G)),zero_zero(X_a)),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),L_2),sequentially)) ) ) ) ).
fof(fact_4477_tendsto__add,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [G,Ba,F,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),Aa),A_2))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ba),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),Aa),Ba)),A_2)) ) ) ) ).
fof(fact_4478_tendsto__diff,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [G,Ba,F,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),Aa),A_2))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ba),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),Aa),Ba)),A_2)) ) ) ) ).
fof(fact_4479_LIMSEQ__diff__const,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [Ba,F,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),Ba)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba)),sequentially)) ) ) ).
fof(fact_4480_LIMSEQ__diff,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [Y_3,Ba,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_3),Ba),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),X_3)),Y_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),Ba)),sequentially)) ) ) ) ).
fof(fact_4481_LIMSEQ__ignore__initial__segment,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [K,F,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),Aa),sequentially)) ) ) ).
fof(fact_4482_LIMSEQ__add__const,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [Ba,F,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),F)),Ba)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Ba)),sequentially)) ) ) ).
fof(fact_4483_LIMSEQ__add,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [Y_3,Ba,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_3),Ba),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_3)),Y_3)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),Ba)),sequentially)) ) ) ) ).
fof(fact_4484_LIMSEQ__offset,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [F,K,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),Aa),sequentially)) ) ) ).
fof(fact_4485_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R_1] : hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),plus_plus(real),R_1)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))),R_1),sequentially)) ).
fof(fact_4486_LIMSEQ__add__minus,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [Y_3,Ba,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_3),Ba),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),plus_plus(X_a)),X_3)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),uminus_uminus(X_a)),Y_3))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa),hAPP(X_a,X_a,uminus_uminus(X_a),Ba))),sequentially)) ) ) ) ).
fof(fact_4487_LIMSEQ__rabs__realpow__zero2,axiom,
! [C_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),C_1)),one_one(real)))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),power_power(real),C_1)),zero_zero(real)),sequentially)) ) ).
fof(fact_4488_LIMSEQ__rabs__realpow__zero,axiom,
! [C_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),C_1)),one_one(real)))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),power_power(real),hAPP(real,real,abs_abs(real),C_1))),zero_zero(real)),sequentially)) ) ).
fof(fact_4489_summable__LIMSEQ__zero,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),F),zero_zero(X_a)),sequentially)) ) ) ).
fof(fact_4490_monoseq__le,axiom,
! [X_1,Aa] :
( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),Aa))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),X_1),sequentially))
=> ( ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,Aa,N_2)),X_1))
& ! [M_3,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,Aa,M_3)),hAPP(nat,real,Aa,N_2))) ) )
| ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),hAPP(nat,real,Aa,N_2)))
& ! [M_3,N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,Aa,N_2)),hAPP(nat,real,Aa,M_3))) ) ) ) ) ) ).
fof(fact_4491_tendsto__mono,axiom,
! [X_a,X_b] :
( topological_space(X_b)
=> ! [F,L,A_2,A_6] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less_eq(filter(X_a)),A_2),A_6))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_6))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_2)) ) ) ) ).
fof(fact_4492_LIMSEQ__le,axiom,
! [Y_3,Y_1,X_3,X_1] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_3),X_1),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Y_3),Y_1),sequentially))
=> ( ? [N_3] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,X_3,N_2)),hAPP(nat,real,Y_3,N_2))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Y_1)) ) ) ) ).
fof(fact_4493_LIMSEQ__le__const2,axiom,
! [Aa,X_3,X_1] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_3),X_1),sequentially))
=> ( ? [N_3] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,X_3,N_2)),Aa)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Aa)) ) ) ).
fof(fact_4494_LIMSEQ__le__const,axiom,
! [Aa,X_3,X_1] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_3),X_1),sequentially))
=> ( ? [N_3] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),hAPP(nat,real,X_3,N_2))) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_1)) ) ) ).
fof(fact_4495_LIMSEQ__realpow__zero,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),one_one(real)))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(real,fun(nat,real),power_power(real),X_1)),zero_zero(real)),sequentially)) ) ) ).
fof(fact_4496_summable__sumr__LIMSEQ__suminf,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& t2_space(X_a) )
=> ! [F] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),F))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))),hAPP(fun(nat,X_a),X_a,suminf(X_a),F)),sequentially)) ) ) ).
fof(fact_4497_sums__def,axiom,
! [X_a] :
( ( comm_monoid_add(X_a)
& topological_space(X_a) )
=> ! [F,S_3] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),F),S_3))
<=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(fun(nat,bool),X_a),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combb(fun(nat,bool),X_a,nat),hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F)),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)))),S_3),sequentially)) ) ) ).
fof(fact_4498_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R_1] : hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),times_times(real),R_1)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),plus_plus(real),one_one(real))),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),uminus_uminus(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),suc)))))),R_1),sequentially)) ).
fof(fact_4499_summable__Leibniz_I1_J,axiom,
! [Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),zero_zero(real)),sequentially))
=> ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),Aa))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))) ) ) ).
fof(fact_4500_zeroseq__arctan__series,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),X_1)),one_one(real)))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),hAPP(real,fun(real,real),inverse_divide(real),one_one(real))),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),hAPP(real,fun(nat,real),power_power(real),X_1)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),nat),plus_plus(nat)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,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_4501_Rats__eq__int__div__nat,axiom,
field_char_0_Rats(real) = hAPP(fun(real,bool),fun(real,bool),collect(real),hAPP(fun(real,fun(int,bool)),fun(real,bool),hAPP(fun(fun(int,bool),bool),fun(fun(real,fun(int,bool)),fun(real,bool)),combb(fun(int,bool),bool,real),fEx(int)),hAPP(fun(real,fun(int,fun(nat,bool))),fun(real,fun(int,bool)),hAPP(fun(fun(int,fun(nat,bool)),fun(int,bool)),fun(fun(real,fun(int,fun(nat,bool))),fun(real,fun(int,bool))),combb(fun(int,fun(nat,bool)),fun(int,bool),real),hAPP(fun(fun(nat,bool),bool),fun(fun(int,fun(nat,bool)),fun(int,bool)),combb(fun(nat,bool),bool,int),fEx(nat))),hAPP(fun(nat,bool),fun(real,fun(int,fun(nat,bool))),hAPP(fun(real,fun(fun(nat,bool),fun(int,fun(nat,bool)))),fun(fun(nat,bool),fun(real,fun(int,fun(nat,bool)))),combc(real,fun(nat,bool),fun(int,fun(nat,bool))),hAPP(fun(real,fun(int,fun(fun(nat,bool),fun(nat,bool)))),fun(real,fun(fun(nat,bool),fun(int,fun(nat,bool)))),hAPP(fun(fun(int,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,bool),fun(int,fun(nat,bool)))),fun(fun(real,fun(int,fun(fun(nat,bool),fun(nat,bool)))),fun(real,fun(fun(nat,bool),fun(int,fun(nat,bool))))),combb(fun(int,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,bool),fun(int,fun(nat,bool))),real),combc(int,fun(nat,bool),fun(nat,bool))),hAPP(fun(real,fun(int,fun(nat,fun(bool,bool)))),fun(real,fun(int,fun(fun(nat,bool),fun(nat,bool)))),hAPP(fun(fun(int,fun(nat,fun(bool,bool))),fun(int,fun(fun(nat,bool),fun(nat,bool)))),fun(fun(real,fun(int,fun(nat,fun(bool,bool)))),fun(real,fun(int,fun(fun(nat,bool),fun(nat,bool))))),combb(fun(int,fun(nat,fun(bool,bool))),fun(int,fun(fun(nat,bool),fun(nat,bool))),real),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(int,fun(nat,fun(bool,bool))),fun(int,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),int),combs(nat,bool,bool))),hAPP(fun(real,fun(int,fun(nat,bool))),fun(real,fun(int,fun(nat,fun(bool,bool)))),hAPP(fun(fun(int,fun(nat,bool)),fun(int,fun(nat,fun(bool,bool)))),fun(fun(real,fun(int,fun(nat,bool))),fun(real,fun(int,fun(nat,fun(bool,bool))))),combb(fun(int,fun(nat,bool)),fun(int,fun(nat,fun(bool,bool))),real),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(int,fun(nat,bool)),fun(int,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),int),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj))),hAPP(fun(int,fun(nat,real)),fun(real,fun(int,fun(nat,bool))),hAPP(fun(real,fun(fun(int,fun(nat,real)),fun(int,fun(nat,bool)))),fun(fun(int,fun(nat,real)),fun(real,fun(int,fun(nat,bool)))),combc(real,fun(int,fun(nat,real)),fun(int,fun(nat,bool))),hAPP(fun(real,fun(fun(nat,real),fun(nat,bool))),fun(real,fun(fun(int,fun(nat,real)),fun(int,fun(nat,bool)))),hAPP(fun(fun(fun(nat,real),fun(nat,bool)),fun(fun(int,fun(nat,real)),fun(int,fun(nat,bool)))),fun(fun(real,fun(fun(nat,real),fun(nat,bool))),fun(real,fun(fun(int,fun(nat,real)),fun(int,fun(nat,bool))))),combb(fun(fun(nat,real),fun(nat,bool)),fun(fun(int,fun(nat,real)),fun(int,fun(nat,bool))),real),combb(fun(nat,real),fun(nat,bool),int)),hAPP(fun(real,fun(real,bool)),fun(real,fun(fun(nat,real),fun(nat,bool))),hAPP(fun(fun(real,bool),fun(fun(nat,real),fun(nat,bool))),fun(fun(real,fun(real,bool)),fun(real,fun(fun(nat,real),fun(nat,bool)))),combb(fun(real,bool),fun(fun(nat,real),fun(nat,bool)),real),combb(real,bool,nat)),fequal(real)))),hAPP(fun(nat,real),fun(int,fun(nat,real)),hAPP(fun(int,fun(fun(nat,real),fun(nat,real))),fun(fun(nat,real),fun(int,fun(nat,real))),combc(int,fun(nat,real),fun(nat,real)),hAPP(fun(int,fun(real,real)),fun(int,fun(fun(nat,real),fun(nat,real))),hAPP(fun(fun(real,real),fun(fun(nat,real),fun(nat,real))),fun(fun(int,fun(real,real)),fun(int,fun(fun(nat,real),fun(nat,real)))),combb(fun(real,real),fun(fun(nat,real),fun(nat,real)),int),combb(real,real,nat)),hAPP(fun(int,real),fun(int,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(int,real),fun(int,fun(real,real))),combb(real,fun(real,real),int),inverse_divide(real)),real_1(int)))),real_1(nat))))))),hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot),hAPP(nat,fun(nat,bool),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),fequal(nat)),zero_zero(nat))))))) ).
fof(fact_4502_summable__Leibniz_I4_J,axiom,
! [Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),zero_zero(real)),sequentially))
=> ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),Aa))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(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(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),sequentially)) ) ) ).
fof(fact_4503_summable__Leibniz_H_I4_J,axiom,
! [N,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),zero_zero(real)),sequentially))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,Aa,N_2)))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,Aa,hAPP(nat,nat,suc,N_2))),hAPP(nat,real,Aa,N_2)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)),one_one(nat)))))) ) ) ) ).
fof(fact_4504_summable__Leibniz_H_I5_J,axiom,
! [Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),zero_zero(real)),sequentially))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,Aa,N_2)))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,Aa,hAPP(nat,nat,suc,N_2))),hAPP(nat,real,Aa,N_2)))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat))),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),hAPP(fun(nat,nat),fun(nat,fun(nat,nat)),hAPP(fun(nat,fun(nat,nat)),fun(fun(nat,nat),fun(nat,fun(nat,nat))),combb(nat,fun(nat,nat),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(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),sequentially)) ) ) ) ).
fof(fact_4505_Im_Otendsto,axiom,
! [X_a,G,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),hAPP(complex,real,im,Aa)),A_2)) ) ).
fof(fact_4506_Re_Otendsto,axiom,
! [X_a,G,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),hAPP(complex,real,re,Aa)),A_2)) ) ).
fof(fact_4507_cnj_OLIMSEQ,axiom,
! [X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(nat,complex),fun(nat,complex),hAPP(fun(complex,complex),fun(fun(nat,complex),fun(nat,complex)),combb(complex,complex,nat),cnj),X_3)),hAPP(complex,complex,cnj,Aa)),sequentially)) ) ).
fof(fact_4508_cnj_Otendsto,axiom,
! [X_a,G,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),hAPP(complex,complex,cnj,Aa)),A_2)) ) ).
fof(fact_4509_tendsto__Complex,axiom,
! [X_a,G,Ba,F,Aa,Net] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),Aa),Net))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),Ba),Net))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,real),fun(X_a,complex),hAPP(fun(X_a,fun(real,complex)),fun(fun(X_a,real),fun(X_a,complex)),combs(X_a,real,complex),hAPP(fun(X_a,real),fun(X_a,fun(real,complex)),hAPP(fun(real,fun(real,complex)),fun(fun(X_a,real),fun(X_a,fun(real,complex))),combb(real,fun(real,complex),X_a),complex_1),F)),G)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Aa),Ba)),Net)) ) ) ).
fof(fact_4510_less__filter__def,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less(filter(X_a)),A_2),B_1))
<=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less_eq(filter(X_a)),A_2),B_1))
& ~ hBOOL(hAPP(filter(X_a),bool,hAPP(filter(X_a),fun(filter(X_a),bool),ord_less_eq(filter(X_a)),B_1),A_2)) ) ) ).
fof(fact_4511_Re_OLIMSEQ,axiom,
! [X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),re),X_3)),hAPP(complex,real,re,Aa)),sequentially)) ) ).
fof(fact_4512_Im_OLIMSEQ,axiom,
! [X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,complex),fun(nat,real),hAPP(fun(complex,real),fun(fun(nat,complex),fun(nat,real)),combb(complex,real,nat),im),X_3)),hAPP(complex,real,im,Aa)),sequentially)) ) ).
fof(fact_4513_LIMSEQ__Complex,axiom,
! [Y_3,Ba,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_3),Aa),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Y_3),Ba),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(complex,fun(filter(nat),bool),hAPP(fun(nat,complex),fun(complex,fun(filter(nat),bool)),tendsto(nat,complex),hAPP(fun(nat,real),fun(nat,complex),hAPP(fun(nat,fun(real,complex)),fun(fun(nat,real),fun(nat,complex)),combs(nat,real,complex),hAPP(fun(nat,real),fun(nat,fun(real,complex)),hAPP(fun(real,fun(real,complex)),fun(fun(nat,real),fun(nat,fun(real,complex))),combb(real,fun(real,complex),nat),complex_1),X_3)),Y_3)),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,Aa),Ba)),sequentially)) ) ) ).
fof(fact_4514_Bolzano__nest__unique,axiom,
! [P_1,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),Ba))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),minus_minus(real)),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_fst(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba)))),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_snd(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba)))),zero_zero(real)),sequentially))
=> ? [L_3] :
( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(product_prod(real,real),real,product_fst(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N_2))),L_3))
& hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_fst(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba))),L_3),sequentially))
& ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),L_3),hAPP(product_prod(real,real),real,product_snd(real,real),hAPP(nat,product_prod(real,real),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba),N_2))))
& hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,product_prod(real,real)),fun(nat,real),hAPP(fun(product_prod(real,real),real),fun(fun(nat,product_prod(real,real)),fun(nat,real)),combb(product_prod(real,real),real,nat),product_snd(real,real)),hAPP(real,fun(nat,product_prod(real,real)),hAPP(real,fun(real,fun(nat,product_prod(real,real))),hAPP(fun(product_prod(real,real),bool),fun(real,fun(real,fun(nat,product_prod(real,real)))),bolzano_bisect,P_1),Aa),Ba))),L_3),sequentially)) ) ) ) ) ) ).
fof(fact_4515_summable__Leibniz_H_I3_J,axiom,
! [Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),zero_zero(real)),sequentially))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,Aa,N_2)))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,Aa,hAPP(nat,nat,suc,N_2))),hAPP(nat,real,Aa,N_2)))
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,fun(nat,bool)),fun(nat,real),hAPP(fun(fun(nat,bool),real),fun(fun(nat,fun(nat,bool)),fun(nat,real)),combb(fun(nat,bool),real,nat),hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(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(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),sequentially)) ) ) ) ).
fof(fact_4516_summable__Leibniz_H_I2_J,axiom,
! [N,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),zero_zero(real)),sequentially))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,Aa,N_2)))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,Aa,hAPP(nat,nat,suc,N_2))),hAPP(nat,real,Aa,N_2)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N)))),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa)))) ) ) ) ).
fof(fact_4517_LIMSEQ__iff__nz,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_3,L_2] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),L_2),sequentially))
<=> ! [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
=> ? [No] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),No))
& ! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_3,N_2)),L_2)),R_2)) ) ) ) ) ) ).
fof(fact_4518_LIMSEQ__D,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_1,X_3,L_2] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),L_2),sequentially))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
=> ? [No] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_3,N_2)),L_2))),R_1)) ) ) ) ) ).
fof(fact_4519_LIMSEQ__iff,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3,L_2] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),L_2),sequentially))
<=> ! [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
=> ? [No] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_3,N_2)),L_2))),R_2)) ) ) ) ) ).
fof(fact_4520_metric__LIMSEQ__D,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [R_1,X_3,L_2] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),L_2),sequentially))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
=> ? [No] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_3,N_2)),L_2)),R_1)) ) ) ) ) ).
fof(fact_4521_LIMSEQ__def,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_3,L_2] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),L_2),sequentially))
<=> ! [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
=> ? [No] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_3,N_2)),L_2)),R_2)) ) ) ) ) ).
fof(fact_4522_summable__Leibniz_I3_J,axiom,
! [Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),zero_zero(real)),sequentially))
=> ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),Aa))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(nat,real,Aa,zero_zero(nat))),zero_zero(real)))
=> ! [N_2] : hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_2)),one_one(nat))))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_2)))))) ) ) ) ).
fof(fact_4523_summable__Leibniz_I2_J,axiom,
! [Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),zero_zero(real)),sequentially))
=> ( hBOOL(hAPP(fun(nat,real),bool,monoseq(real),Aa))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(nat,real,Aa,zero_zero(nat))))
=> ! [N_2] : hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),hAPP(fun(nat,real),real,suminf(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_2)))),hAPP(fun(nat,bool),real,hAPP(fun(nat,real),fun(fun(nat,bool),real),big_co1399186613setsum(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),times_times(nat),hAPP(int,nat,number_number_of(nat),hAPP(int,int,bit0,hAPP(int,int,bit1,pls)))),N_2)),one_one(nat))))))) ) ) ) ).
fof(fact_4524_Sup__atLeastAtMost,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Y_1),X_1))
=> hAPP(fun(real,bool),real,complete_Sup_Sup(real),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_atLeastAtMost(real),Y_1),X_1)) = X_1 ) ).
fof(fact_4525_atLeastatMost__subset__iff,axiom,
! [X_a] :
( order(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),Aa),Ba)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C_1),D_1)))
<=> ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba))
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),D_1)) ) ) ) ) ).
fof(fact_4526_image__uminus__atLeastAtMost,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [X_1,Y_1] : hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,X_a),fun(fun(X_a,bool),fun(X_a,bool)),image(X_a,X_a),uminus_uminus(X_a)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),X_1),Y_1)) = hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),Y_1)),hAPP(X_a,X_a,uminus_uminus(X_a),X_1)) ) ).
fof(fact_4527_atLeastatMost__psubset__iff,axiom,
! [X_a] :
( order(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),Aa),Ba)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C_1),D_1)))
<=> ( ( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba))
| ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),D_1))
& ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),C_1),Aa))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),D_1)) ) ) )
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),D_1)) ) ) ) ).
fof(fact_4528_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),Aa),Ba)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),C_1),D_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Ba),D_1)) ) ) ) ) ).
fof(fact_4529_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastLessThan(X_a),Aa),Ba)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C_1),D_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),D_1)) ) ) ) ) ).
fof(fact_4530_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [X_a] :
( dense_linorder(X_a)
=> ! [Aa,Ba,C_1,D_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_gr676655987ssThan(X_a),Aa),Ba)),hAPP(X_a,fun(X_a,bool),hAPP(X_a,fun(X_a,fun(X_a,bool)),ord_atLeastAtMost(X_a),C_1),D_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Aa),Ba))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),C_1),Aa))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Ba),D_1)) ) ) ) ) ).
fof(fact_4531_summable,axiom,
! [Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),Aa),zero_zero(real)),sequentially))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(nat,real,Aa,N_2)))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,Aa,hAPP(nat,nat,suc,N_2))),hAPP(nat,real,Aa,N_2)))
=> hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(nat,fun(real,real)),fun(fun(nat,real),fun(nat,real)),combs(nat,real,real),hAPP(fun(nat,real),fun(nat,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(nat,real),fun(nat,fun(real,real))),combb(real,fun(real,real),nat),times_times(real)),hAPP(real,fun(nat,real),power_power(real),hAPP(int,real,number_number_of(real),min)))),Aa))) ) ) ) ).
fof(fact_4532_termdiffs__aux,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X_1,C_1,K_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),diffs(X_a),C_1)))),hAPP(X_a,fun(nat,X_a),power_power(X_a),K_3))))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),X_1)),hAPP(X_a,real,norm_norm(X_a),K_3)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,X_a),hAPP(fun(fun(nat,X_a),X_a),fun(fun(X_a,fun(nat,X_a)),fun(X_a,X_a)),combb(fun(nat,X_a),X_a,X_a),suminf(X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),C_1))),hAPP(fun(nat,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(nat,X_a),fun(X_a,fun(nat,X_a))),combc(X_a,fun(nat,X_a),fun(nat,X_a)),hAPP(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),hAPP(fun(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a)))),combb(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),X_a),combs(nat,X_a,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a))),hAPP(fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a)))),combb(fun(nat,X_a),fun(nat,fun(X_a,X_a)),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a))),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(X_a,fun(nat,X_a))),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combs(X_a,X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(X_a,fun(nat,X_a))),hAPP(fun(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a))),fun(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(X_a,fun(nat,X_a)))),combb(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),X_a),combc(nat,X_a,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a))),hAPP(fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a)))),combb(fun(nat,X_a),fun(nat,fun(X_a,X_a)),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),inverse_divide(X_a))),hAPP(fun(nat,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(nat,X_a),fun(X_a,fun(nat,X_a))),combc(X_a,fun(nat,X_a),fun(nat,X_a)),hAPP(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a))),hAPP(fun(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a))),fun(fun(X_a,fun(nat,fun(X_a,X_a))),fun(X_a,fun(fun(nat,X_a),fun(nat,X_a)))),combb(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),X_a),combs(nat,X_a,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a))),hAPP(fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),fun(fun(X_a,fun(nat,X_a)),fun(X_a,fun(nat,fun(X_a,X_a)))),combb(fun(nat,X_a),fun(nat,fun(X_a,X_a)),X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a))),hAPP(fun(X_a,X_a),fun(X_a,fun(nat,X_a)),hAPP(fun(X_a,fun(nat,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(nat,X_a))),combb(X_a,fun(nat,X_a),X_a),power_power(X_a)),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))))),combi(X_a))))),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),semiring_1_of_nat(X_a))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1)),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),minus_minus(nat)),hAPP(nat,nat,suc,zero_zero(nat))))))))),zero_zero(X_a)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ) ).
fof(fact_4533_finite__atLeastAtMost,axiom,
! [L,U_1] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),L),U_1))) ).
fof(fact_4534_finite__atLeastAtMost__int,axiom,
! [L,U_1] : hBOOL(hAPP(fun(int,bool),bool,finite_finite_1(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),L),U_1))) ).
fof(fact_4535_ex__nat__less,axiom,
! [P_1,N] :
( ? [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N))
& hBOOL(hAPP(nat,bool,P_1,M_3)) )
<=> ? [X_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),N)))
& hBOOL(hAPP(nat,bool,P_1,X_2)) ) ) ).
fof(fact_4536_all__nat__less,axiom,
! [P_1,N] :
( ! [M_3] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M_3),N))
=> hBOOL(hAPP(nat,bool,P_1,M_3)) )
<=> ! [X_2] :
( hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),X_2),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),N)))
=> hBOOL(hAPP(nat,bool,P_1,X_2)) ) ) ).
fof(fact_4537_SetInterval_Otransfer__nat__int__set__function__closures,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1))
=> hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),X_1),Y_1))) ) ).
fof(fact_4538_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L,U_1] : hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),L),hAPP(nat,nat,suc,U_1)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),L),U_1) ).
fof(fact_4539_fact__altdef__nat,axiom,
! [N] : hAPP(nat,nat,fact_fact(nat),N) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co302546470etprod(nat,nat),combi(nat)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),N)) ).
fof(fact_4540_image__add__atLeastAtMost,axiom,
! [K,I,J] : hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),I),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),J),K)) ).
fof(fact_4541_image__Suc__atLeastAtMost,axiom,
! [I,J] : hAPP(fun(nat,bool),fun(nat,bool),hAPP(fun(nat,nat),fun(fun(nat,bool),fun(nat,bool)),image(nat,nat),suc),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),I),J)) = hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,I)),hAPP(nat,nat,suc,J)) ).
fof(fact_4542_SetInterval_Otransfer__nat__int__set__functions_I2_J,axiom,
! [M,N] : hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),N) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),hAPP(nat,int,semiring_1_of_nat(int),M)),hAPP(nat,int,semiring_1_of_nat(int),N))) ).
fof(fact_4543_setsum__shift__bounds__cl__Suc__ivl,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,M,N] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,M)),hAPP(nat,nat,suc,N))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),N)) ) ).
fof(fact_4544_setsum__shift__bounds__cl__nat__ivl,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,M,K,N] : hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),M),K)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),K))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),K))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),N)) ) ).
fof(fact_4545_cnj_Ocont,axiom,
! [Aa] : hBOOL(hAPP(filter(complex),bool,hAPP(complex,fun(filter(complex),bool),hAPP(fun(complex,complex),fun(complex,fun(filter(complex),bool)),tendsto(complex,complex),cnj),hAPP(complex,complex,cnj,Aa)),hAPP(complex,filter(complex),at(complex),Aa))) ).
fof(fact_4546_Re_Ocont,axiom,
! [Aa] : hBOOL(hAPP(filter(complex),bool,hAPP(real,fun(filter(complex),bool),hAPP(fun(complex,real),fun(real,fun(filter(complex),bool)),tendsto(complex,real),re),hAPP(complex,real,re,Aa)),hAPP(complex,filter(complex),at(complex),Aa))) ).
fof(fact_4547_Im_Ocont,axiom,
! [Aa] : hBOOL(hAPP(filter(complex),bool,hAPP(real,fun(filter(complex),bool),hAPP(fun(complex,real),fun(real,fun(filter(complex),bool)),tendsto(complex,real),im),hAPP(complex,real,im,Aa)),hAPP(complex,filter(complex),at(complex),Aa))) ).
fof(fact_4548_cnj_OLIM,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [G,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),hAPP(complex,complex,cnj,L)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4549_LIM__add,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [G,M_2,F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),L_2),M_2)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4550_LIM__diff,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [G,M,F,L,X_1] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),X_1)))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M),hAPP(X_a,filter(X_a),at(X_a),X_1)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),L),M)),hAPP(X_a,filter(X_a),at(X_a),X_1))) ) ) ) ).
fof(fact_4551_LIM__not__zero,axiom,
! [X_a,X_b] :
( ( real_n2089651433ebra_1(X_b)
& zero(X_a)
& metric_space(X_a) )
=> ! [Aa,K] :
( ti(X_a,K) != zero_zero(X_a)
=> ~ hBOOL(hAPP(filter(X_b),bool,hAPP(X_a,fun(filter(X_b),bool),hAPP(fun(X_b,X_a),fun(X_a,fun(filter(X_b),bool)),tendsto(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),K)),zero_zero(X_a)),hAPP(X_b,filter(X_b),at(X_b),Aa))) ) ) ).
fof(fact_4552_divide_OLIM,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_field(X_b) )
=> ! [Y_1,G,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),G)),Y_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),inverse_divide(X_b),L),Y_1)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4553_divide_Ocont,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_1,Aa] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),inverse_divide(X_a)),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),Aa),Y_1)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ).
fof(fact_4554_of__real_Ocont,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [Aa] : hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),of_real(X_a)),hAPP(real,X_a,of_real(X_a),Aa)),hAPP(real,filter(real),at(real),Aa))) ) ).
fof(fact_4555_LIM__power,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& power(X_b)
& real_normed_algebra(X_b) )
=> ! [N,F,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(nat,fun(X_a,X_b),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(nat,X_b)),hAPP(fun(X_b,fun(nat,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(nat,X_b))),combb(X_b,fun(nat,X_b),X_a),power_power(X_b)),F)),N)),hAPP(nat,X_b,hAPP(X_b,fun(nat,X_b),power_power(X_b),L),N)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4556_LIM__const__eq,axiom,
! [X_a,X_b] :
( ( metric_space(X_b)
& real_n2089651433ebra_1(X_a) )
=> ! [K,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ti(X_b,K) = ti(X_b,L_2) ) ) ).
fof(fact_4557_LIM__compose,axiom,
! [X_a,X_b,X_c] :
( ( topological_space(X_c)
& topological_space(X_b)
& topological_space(X_a) )
=> ! [F,Aa,G,L] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),hAPP(X_a,X_b,G,L)),hAPP(X_a,filter(X_a),at(X_a),L)))
=> ( hBOOL(hAPP(filter(X_c),bool,hAPP(X_a,fun(filter(X_c),bool),hAPP(fun(X_c,X_a),fun(X_a,fun(filter(X_c),bool)),tendsto(X_c,X_a),F),L),hAPP(X_c,filter(X_c),at(X_c),Aa)))
=> hBOOL(hAPP(filter(X_c),bool,hAPP(X_b,fun(filter(X_c),bool),hAPP(fun(X_c,X_b),fun(X_b,fun(filter(X_c),bool)),tendsto(X_c,X_b),hAPP(fun(X_c,X_a),fun(X_c,X_b),hAPP(fun(X_a,X_b),fun(fun(X_c,X_a),fun(X_c,X_b)),combb(X_a,X_b,X_c),G),F)),hAPP(X_a,X_b,G,L)),hAPP(X_c,filter(X_c),at(X_c),Aa))) ) ) ) ).
fof(fact_4558_LIM__const__not__eq,axiom,
! [X_a,X_b] :
( ( real_n2089651433ebra_1(X_b)
& metric_space(X_a) )
=> ! [Aa,K,L_2] :
( ti(X_a,K) != ti(X_a,L_2)
=> ~ hBOOL(hAPP(filter(X_b),bool,hAPP(X_a,fun(filter(X_b),bool),hAPP(fun(X_b,X_a),fun(X_a,fun(filter(X_b),bool)),tendsto(X_b,X_a),hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),K)),L_2),hAPP(X_b,filter(X_b),at(X_b),Aa))) ) ) ).
fof(fact_4559_LIM__ident,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [Aa] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),combi(X_a)),Aa),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ).
fof(fact_4560_LIM__const,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& topological_space(X_b) )
=> ! [K,X_1] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),K),hAPP(X_a,filter(X_a),at(X_a),X_1))) ) ).
fof(fact_4561_bounded__bilinear_OLIM,axiom,
! [X_b,X_c,X_a,X_d] :
( ( topological_space(X_d)
& real_normed_vector(X_a)
& real_normed_vector(X_c)
& real_normed_vector(X_b) )
=> ! [G,M_2,F,L_2,Aa,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> ( hBOOL(hAPP(filter(X_d),bool,hAPP(X_a,fun(filter(X_d),bool),hAPP(fun(X_d,X_a),fun(X_a,fun(filter(X_d),bool)),tendsto(X_d,X_a),F),L_2),hAPP(X_d,filter(X_d),at(X_d),Aa)))
=> ( hBOOL(hAPP(filter(X_d),bool,hAPP(X_b,fun(filter(X_d),bool),hAPP(fun(X_d,X_b),fun(X_b,fun(filter(X_d),bool)),tendsto(X_d,X_b),G),M_2),hAPP(X_d,filter(X_d),at(X_d),Aa)))
=> hBOOL(hAPP(filter(X_d),bool,hAPP(X_c,fun(filter(X_d),bool),hAPP(fun(X_d,X_c),fun(X_c,fun(filter(X_d),bool)),tendsto(X_d,X_c),hAPP(fun(X_d,X_b),fun(X_d,X_c),hAPP(fun(X_d,fun(X_b,X_c)),fun(fun(X_d,X_b),fun(X_d,X_c)),combs(X_d,X_b,X_c),hAPP(fun(X_d,X_a),fun(X_d,fun(X_b,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_d,X_a),fun(X_d,fun(X_b,X_c))),combb(X_a,fun(X_b,X_c),X_d),Prod),F)),G)),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,L_2),M_2)),hAPP(X_d,filter(X_d),at(X_d),Aa))) ) ) ) ) ).
fof(fact_4562_LIM__mult,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_algebra(X_b) )
=> ! [G,M_2,F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),L_2),M_2)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4563_mult__left_OLIM,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_algebra(X_b) )
=> ! [Y_1,G,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),Y_1)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),L),Y_1)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4564_mult__right_OLIM,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_algebra(X_b) )
=> ! [X_1,G,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1)),G)),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1),L)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4565_mult__left_Ocont,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y_1,Aa] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),times_times(X_a)),Y_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Aa),Y_1)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ).
fof(fact_4566_mult__right_Ocont,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X_1,Aa] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1),Aa)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ).
fof(fact_4567_LIM__minus,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),hAPP(X_b,X_b,uminus_uminus(X_b),L_2)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4568_LIM__rabs,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [F,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),hAPP(real,real,abs_abs(real),L)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4569_LIM__inverse__fun,axiom,
! [X_a] :
( real_n1866405975lgebra(X_a)
=> ! [Aa] :
( ti(X_a,Aa) != zero_zero(X_a)
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),inverse_inverse(X_a)),hAPP(X_a,X_a,inverse_inverse(X_a),Aa)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4570_LIM__cong__limit,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& topological_space(X_b) )
=> ! [K_3,F,L_2,X_1] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),X_1)))
=> ( ti(X_b,K_3) = ti(X_b,L_2)
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),K_3),hAPP(X_a,filter(X_a),at(X_a),X_1))) ) ) ) ).
fof(fact_4571_LIM__unique,axiom,
! [X_a,X_b] :
( ( metric_space(X_b)
& real_n2089651433ebra_1(X_a) )
=> ! [M_2,F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),M_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ti(X_b,L_2) = ti(X_b,M_2) ) ) ) ).
fof(fact_4572_card__atLeastAtMost,axiom,
! [L,U_1] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),L),U_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,suc,U_1)),L) ).
fof(fact_4573_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L,U_1] : hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastLessThan(int),L),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),U_1),one_one(int))) = hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),L),U_1) ).
fof(fact_4574_LIM__mult__right__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_algebra(X_b) )
=> ! [C_1,F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),C_1)),F)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4575_mult__right_OLIM__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_algebra(X_b) )
=> ! [X_1,G,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(X_b,fun(X_b,X_b),times_times(X_b),X_1)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4576_LIM__mult__left__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_algebra(X_b) )
=> ! [C_1,F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),C_1)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4577_mult__left_OLIM__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_algebra(X_b) )
=> ! [Y_1,G,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),G)),Y_1)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4578_LIM__mult__zero,axiom,
! [X_b,X_a] :
( ( metric_space(X_a)
& real_normed_algebra(X_b) )
=> ! [G,F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4579_LIM__offset__zero__cancel,axiom,
! [X_b,X_a] :
( ( real_normed_vector(X_a)
& topological_space(X_b) )
=> ! [F,Aa,L_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa))),L_2),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a))))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4580_LIM__offset__zero,axiom,
! [X_b,X_a] :
( ( real_normed_vector(X_a)
& topological_space(X_b) )
=> ! [F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa))),L_2),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ).
fof(fact_4581_LIM__add__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [G,F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4582_LIM__isCont__iff,axiom,
! [X_b,X_a] :
( ( real_normed_vector(X_a)
& topological_space(X_b) )
=> ! [F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),hAPP(X_a,X_b,F,Aa)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa))),hAPP(X_a,X_b,F,Aa)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ).
fof(fact_4583_LIM__zero__cancel,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),L)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4584_LIM__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),L)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4585_LIM__zero__iff,axiom,
! [X_b,X_a] :
( ( metric_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),L)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4586_LIM__offset,axiom,
! [X_b,X_a] :
( ( real_normed_vector(X_a)
& topological_space(X_b) )
=> ! [K,F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),plus_plus(X_a)),K))),L_2),hAPP(X_a,filter(X_a),at(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Aa),K)))) ) ) ).
fof(fact_4587_divide_OLIM__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_field(X_b) )
=> ! [Y_1,G,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),inverse_divide(X_b)),G)),Y_1)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4588_LIM__add__minus,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [G,M,F,L,X_1] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),X_1)))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M),hAPP(X_a,filter(X_a),at(X_a),X_1)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),G))),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),plus_plus(X_b),L),hAPP(X_b,X_b,uminus_uminus(X_b),M))),hAPP(X_a,filter(X_a),at(X_a),X_1))) ) ) ) ).
fof(fact_4589_LIM__inverse,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_n1866405975lgebra(X_b) )
=> ! [F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( ti(X_b,L_2) != zero_zero(X_b)
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),inverse_inverse(X_b)),F)),hAPP(X_b,X_b,inverse_inverse(X_b),L_2)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4590_setsum__shift__lb__Suc0__0,axiom,
! [K,F] :
( hAPP(nat,nat,F,zero_zero(nat)) = zero_zero(nat)
=> hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,zero_zero(nat))),K)) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co1399186613setsum(nat,nat),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),K)) ) ).
fof(fact_4591_LIM__sgn,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( ti(X_b,L) != zero_zero(X_b)
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),sgn_sgn(X_b)),F)),hAPP(X_b,X_b,sgn_sgn(X_b),L)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4592_LIM__norm,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),hAPP(X_b,real,norm_norm(X_b),L)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4593_bounded__bilinear_OLIM__prod__zero,axiom,
! [X_c,X_a,X_b,X_d] :
( ( metric_space(X_d)
& real_normed_vector(X_b)
& real_normed_vector(X_a)
& real_normed_vector(X_c) )
=> ! [G,F,Aa,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> ( hBOOL(hAPP(filter(X_d),bool,hAPP(X_a,fun(filter(X_d),bool),hAPP(fun(X_d,X_a),fun(X_a,fun(filter(X_d),bool)),tendsto(X_d,X_a),F),zero_zero(X_a)),hAPP(X_d,filter(X_d),at(X_d),Aa)))
=> ( hBOOL(hAPP(filter(X_d),bool,hAPP(X_b,fun(filter(X_d),bool),hAPP(fun(X_d,X_b),fun(X_b,fun(filter(X_d),bool)),tendsto(X_d,X_b),G),zero_zero(X_b)),hAPP(X_d,filter(X_d),at(X_d),Aa)))
=> hBOOL(hAPP(filter(X_d),bool,hAPP(X_c,fun(filter(X_d),bool),hAPP(fun(X_d,X_c),fun(X_c,fun(filter(X_d),bool)),tendsto(X_d,X_c),hAPP(fun(X_d,X_b),fun(X_d,X_c),hAPP(fun(X_d,fun(X_b,X_c)),fun(fun(X_d,X_b),fun(X_d,X_c)),combs(X_d,X_b,X_c),hAPP(fun(X_d,X_a),fun(X_d,fun(X_b,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_d,X_a),fun(X_d,fun(X_b,X_c))),combb(X_a,fun(X_b,X_c),X_d),Prod),F)),G)),zero_zero(X_c)),hAPP(X_d,filter(X_d),at(X_d),Aa))) ) ) ) ) ).
fof(fact_4594_bounded__bilinear_OLIM__left__zero,axiom,
! [X_c,X_a,X_b,X_d] :
( ( topological_space(X_d)
& real_normed_vector(X_b)
& real_normed_vector(X_a)
& real_normed_vector(X_c) )
=> ! [C_1,F,Aa,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> ( hBOOL(hAPP(filter(X_d),bool,hAPP(X_a,fun(filter(X_d),bool),hAPP(fun(X_d,X_a),fun(X_a,fun(filter(X_d),bool)),tendsto(X_d,X_a),F),zero_zero(X_a)),hAPP(X_d,filter(X_d),at(X_d),Aa)))
=> hBOOL(hAPP(filter(X_d),bool,hAPP(X_c,fun(filter(X_d),bool),hAPP(fun(X_d,X_c),fun(X_c,fun(filter(X_d),bool)),tendsto(X_d,X_c),hAPP(X_b,fun(X_d,X_c),hAPP(fun(X_d,fun(X_b,X_c)),fun(X_b,fun(X_d,X_c)),combc(X_d,X_b,X_c),hAPP(fun(X_d,X_a),fun(X_d,fun(X_b,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_d,X_a),fun(X_d,fun(X_b,X_c))),combb(X_a,fun(X_b,X_c),X_d),Prod),F)),C_1)),zero_zero(X_c)),hAPP(X_d,filter(X_d),at(X_d),Aa))) ) ) ) ).
fof(fact_4595_bounded__bilinear_OLIM__right__zero,axiom,
! [X_c,X_a,X_b,X_d] :
( ( topological_space(X_d)
& real_normed_vector(X_b)
& real_normed_vector(X_a)
& real_normed_vector(X_c) )
=> ! [C_1,F,Aa,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> ( hBOOL(hAPP(filter(X_d),bool,hAPP(X_b,fun(filter(X_d),bool),hAPP(fun(X_d,X_b),fun(X_b,fun(filter(X_d),bool)),tendsto(X_d,X_b),F),zero_zero(X_b)),hAPP(X_d,filter(X_d),at(X_d),Aa)))
=> hBOOL(hAPP(filter(X_d),bool,hAPP(X_c,fun(filter(X_d),bool),hAPP(fun(X_d,X_c),fun(X_c,fun(filter(X_d),bool)),tendsto(X_d,X_c),hAPP(fun(X_d,X_b),fun(X_d,X_c),hAPP(fun(X_b,X_c),fun(fun(X_d,X_b),fun(X_d,X_c)),combb(X_b,X_c,X_d),hAPP(X_a,fun(X_b,X_c),Prod,C_1)),F)),zero_zero(X_c)),hAPP(X_d,filter(X_d),at(X_d),Aa))) ) ) ) ).
fof(fact_4596_LIM__rabs__zero__iff,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4597_LIM__rabs__zero,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4598_LIM__rabs__zero__cancel,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4599_LIM__of__real,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_algebra_1(X_b)
& real_normed_vector(X_b) )
=> ! [G,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),G)),hAPP(real,X_b,of_real(X_b),L)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4600_Re_OLIM,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [G,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),hAPP(complex,real,re,L)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4601_Im_OLIM,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [G,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),hAPP(complex,real,im,L)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4602_setsum__head__Suc,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,F,M)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,suc,M)),N))) ) ) ).
fof(fact_4603_cnj_OLIM__zero,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [G,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),hAPP(fun(X_a,complex),fun(X_a,complex),hAPP(fun(complex,complex),fun(fun(X_a,complex),fun(X_a,complex)),combb(complex,complex,X_a),cnj),G)),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4604_LIM__trans,axiom,
! [X_b,X_a] :
( ( metric_space(X_a)
& real_normed_vector(X_b) )
=> ! [L,F,G,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),G))),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4605_DERIV__iff2,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),F)),hAPP(X_a,X_a,F,X_1)))),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),minus_minus(X_a)),X_1))),D_2),hAPP(X_a,filter(X_a),at(X_a),X_1))) ) ) ).
fof(fact_4606_metric__LIM__compose2,axiom,
! [X_c,X_b,X_a] :
( ( metric_space(X_a)
& topological_space(X_b)
& topological_space(X_c) )
=> ! [G,C_1,F,Ba,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),Ba),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),C_1),hAPP(X_b,filter(X_b),at(X_b),Ba)))
=> ( ? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [X_2] :
( ( ti(X_a,X_2) != ti(X_a,Aa)
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_2),Aa)),D_3)) )
=> hAPP(X_a,X_b,F,X_2) != ti(X_b,Ba) ) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),C_1),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ) ).
fof(fact_4607_setsum__cl__ivl__Suc,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,N,M] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,N)),M))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),hAPP(nat,nat,suc,N))) = zero_zero(X_a) )
& ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,N)),M))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),hAPP(nat,nat,suc,N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),N))),hAPP(nat,X_a,F,hAPP(nat,nat,suc,N))) ) ) ) ).
fof(fact_4608_setsum__ub__add__nat,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,Pa,M,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),Pa))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),N))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),Pa)))) ) ) ).
fof(fact_4609_card__atLeastAtMost__int,axiom,
! [L,U_1] : hAPP(fun(int,bool),nat,finite_card(int),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),L),U_1)) = hAPP(int,nat,nat_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),U_1),L)),one_one(int))) ).
fof(fact_4610_setsum__cl__ivl__Suc2,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,N,M] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,N)),M))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),hAPP(nat,nat,suc,N))) = zero_zero(X_a) )
& ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),hAPP(nat,nat,suc,N)),M))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),F),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),hAPP(nat,nat,suc,N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,F,M)),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),suc)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),N))) ) ) ) ).
fof(fact_4611_DERIV__LIM__iff,axiom,
! [X_a] :
( ( inverse(X_a)
& real_normed_vector(X_a) )
=> ! [F,Aa,D_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Aa)))),hAPP(X_a,X_a,F,Aa)))),combi(X_a))),D_2),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a))))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),F)),hAPP(X_a,X_a,F,Aa)))),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),minus_minus(X_a)),Aa))),D_2),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4612_LIM__compose2,axiom,
! [X_c,X_b,X_a] :
( ( real_normed_vector(X_a)
& topological_space(X_b)
& topological_space(X_c) )
=> ! [G,C_1,F,Ba,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),Ba),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),C_1),hAPP(X_b,filter(X_b),at(X_b),Ba)))
=> ( ? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [X_2] :
( ( ti(X_a,X_2) != ti(X_a,Aa)
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_2),Aa))),D_3)) )
=> hAPP(X_a,X_b,F,X_2) != ti(X_b,Ba) ) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),C_1),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ) ).
fof(fact_4613_LIM__norm__zero__cancel,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4614_LIM__norm__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4615_LIM__norm__zero__iff,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4616_fact__div__fact,axiom,
! [N,M] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> hAPP(nat,nat,hAPP(nat,fun(nat,nat),div_div(nat),hAPP(nat,nat,fact_fact(nat),M)),hAPP(nat,nat,fact_fact(nat),N)) = hAPP(fun(nat,bool),nat,hAPP(fun(nat,nat),fun(fun(nat,bool),nat),big_co302546470etprod(nat,nat),combi(nat)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),M)) ) ).
fof(fact_4617_of__real_OLIM__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_algebra_1(X_b)
& real_normed_vector(X_b) )
=> ! [G,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4618_fact__altdef__int,axiom,
! [N] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> hAPP(int,int,fact_fact(int),N) = hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),N)) ) ).
fof(fact_4619_Re_OLIM__zero,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [G,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),re),G)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4620_Im_OLIM__zero,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [G,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(complex,fun(filter(X_a),bool),hAPP(fun(X_a,complex),fun(complex,fun(filter(X_a),bool)),tendsto(X_a,complex),G),zero_zero(complex)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),hAPP(fun(X_a,complex),fun(X_a,real),hAPP(fun(complex,real),fun(fun(X_a,complex),fun(X_a,real)),combb(complex,real,X_a),im),G)),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4621_gauss__sum,axiom,
! [X_a] :
( comm_semiring_1(X_a)
=> ! [N] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),one_one(X_a)),one_one(X_a))),hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),semiring_1_of_nat(X_a)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),one_one(nat)),N))) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(nat,X_a,semiring_1_of_nat(X_a),N)),one_one(X_a))) ) ).
fof(fact_4622_setsum__natinterval__difff,axiom,
! [X_a] :
( ab_group_add(X_a)
=> ! [F,M,N] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),N)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,F,M)),hAPP(nat,X_a,F,hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) )
& ( ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),minus_minus(X_a)),F)),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),F),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),plus_plus(nat)),one_one(nat))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),M),N)) = zero_zero(X_a) ) ) ) ).
fof(fact_4623_DERIV__D,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1)))),hAPP(X_a,X_a,F,X_1)))),combi(X_a))),D_2),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ).
fof(fact_4624_DERIV__iff,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,X_a)),combs(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),inverse_divide(X_a)),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(X_a,X_a),fun(X_a,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),X_a),minus_minus(X_a)),hAPP(fun(X_a,X_a),fun(X_a,X_a),hAPP(fun(X_a,X_a),fun(fun(X_a,X_a),fun(X_a,X_a)),combb(X_a,X_a,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1)))),hAPP(X_a,X_a,F,X_1)))),combi(X_a))),D_2),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ).
fof(fact_4625_Cauchy__product,axiom,
! [X_a] :
( ( real_normed_algebra(X_a)
& banach(X_a) )
=> ! [Ba,Aa] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),Aa)))
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),Ba)))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),Aa)),hAPP(fun(nat,X_a),X_a,suminf(X_a),Ba)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(nat,fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combs(nat,fun(nat,bool),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a)),hAPP(fun(fun(nat,X_a),fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a))),combb(fun(nat,X_a),fun(fun(nat,bool),X_a),nat),big_co1399186613setsum(nat,X_a)),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),nat),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),Aa))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,nat),fun(nat,X_a)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a))),combb(fun(nat,nat),fun(nat,X_a),nat),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),Ba)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))) ) ) ) ).
fof(fact_4626_setsum__triangle__reindex,axiom,
! [X_a] :
( comm_monoid_add(X_a)
=> ! [F,N] : hAPP(fun(product_prod(nat,nat),bool),X_a,hAPP(fun(product_prod(nat,nat),X_a),fun(fun(product_prod(nat,nat),bool),X_a),big_co1399186613setsum(product_prod(nat,nat),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(product_prod(nat,nat),X_a),product_prod_case(nat,nat,X_a),F)),hAPP(fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),collect(product_prod(nat,nat)),hAPP(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_prod_case(nat,nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(nat,fun(nat,fun(nat,bool))),combc(nat,nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(nat,bool))),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),fun(fun(nat,fun(nat,fun(nat,bool))),fun(nat,fun(nat,fun(nat,bool)))),combb(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),nat),combc(nat,nat,bool)),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,fun(nat,bool))),hAPP(fun(fun(nat,nat),fun(nat,fun(nat,bool))),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,fun(nat,bool)))),combb(fun(nat,nat),fun(nat,fun(nat,bool)),nat),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),ord_less(nat))),plus_plus(nat)))),N)))) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(nat,fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combs(nat,fun(nat,bool),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a)),hAPP(fun(fun(nat,X_a),fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a))),combb(fun(nat,X_a),fun(fun(nat,bool),X_a),nat),big_co1399186613setsum(nat,X_a)),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,nat),fun(nat,X_a)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a))),combb(fun(nat,nat),fun(nat,X_a),nat),hAPP(fun(nat,fun(nat,X_a)),fun(fun(nat,nat),fun(nat,X_a)),combs(nat,nat,X_a),F)),minus_minus(nat)))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastLessThan(nat),zero_zero(nat)),N)) ) ).
fof(fact_4627_Cauchy__product__sums,axiom,
! [X_a] :
( ( real_normed_algebra(X_a)
& banach(X_a) )
=> ! [Ba,Aa] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),Aa)))
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),Ba)))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,X_a),hAPP(fun(nat,fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,bool)),fun(nat,X_a)),combs(nat,fun(nat,bool),X_a),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a)),hAPP(fun(fun(nat,X_a),fun(fun(nat,bool),X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(fun(nat,bool),X_a))),combb(fun(nat,X_a),fun(fun(nat,bool),X_a),nat),big_co1399186613setsum(nat,X_a)),hAPP(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,X_a),fun(nat,X_a)),fun(fun(nat,fun(nat,X_a)),fun(nat,fun(nat,X_a))),combb(fun(nat,X_a),fun(nat,X_a),nat),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),Aa))),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a)),hAPP(fun(fun(nat,nat),fun(nat,X_a)),fun(fun(nat,fun(nat,nat)),fun(nat,fun(nat,X_a))),combb(fun(nat,nat),fun(nat,X_a),nat),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),Ba)),minus_minus(nat))))),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(fun(nat,X_a),X_a,suminf(X_a),Aa)),hAPP(fun(nat,X_a),X_a,suminf(X_a),Ba)))) ) ) ) ).
fof(fact_4628_LIM__cos__div__sin,axiom,
hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),hAPP(fun(real,real),fun(real,real),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,real)),combs(real,real,real),hAPP(fun(real,real),fun(real,fun(real,real)),hAPP(fun(real,fun(real,real)),fun(fun(real,real),fun(real,fun(real,real))),combb(real,fun(real,real),real),inverse_divide(real)),cos)),sin)),zero_zero(real)),hAPP(real,filter(real),at(real),hAPP(real,real,hAPP(real,fun(real,real),inverse_divide(real),pi),hAPP(int,real,number_number_of(real),hAPP(int,int,bit0,hAPP(int,int,bit1,pls))))))) ).
fof(fact_4629_lemma__termdiff5,axiom,
! [X_b,X_a] :
( ( real_normed_field(X_a)
& banach(X_b) )
=> ! [G,F,K] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
=> ( hBOOL(hAPP(fun(nat,real),bool,summable(real),F))
=> ( ! [H_1,N_2] :
( ti(X_a,H_1) != zero_zero(X_a)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),H_1)),K))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(nat,X_b,hAPP(X_a,fun(nat,X_b),G,H_1),N_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(nat,real,F,N_2)),hAPP(X_a,real,norm_norm(X_a),H_1)))) ) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,fun(nat,X_b)),fun(X_a,X_b),hAPP(fun(fun(nat,X_b),X_b),fun(fun(X_a,fun(nat,X_b)),fun(X_a,X_b)),combb(fun(nat,X_b),X_b,X_a),suminf(X_b)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ) ) ).
fof(fact_4630_lemma__termdiff4,axiom,
! [X_b,X_a] :
( ( real_normed_field(X_a)
& real_normed_vector(X_b) )
=> ! [F,K_3,K] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
=> ( ! [H_1] :
( ti(X_a,H_1) != zero_zero(X_a)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),H_1)),K))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_a,X_b,F,H_1))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),K_3),hAPP(X_a,real,norm_norm(X_a),H_1)))) ) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ) ).
fof(fact_4631_LIM__fun__not__zero,axiom,
! [F,L,C_1] :
( hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),F),L),hAPP(real,filter(real),at(real),C_1)))
=> ( L != zero_zero(real)
=> ? [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
& ! [X_2] :
( ( X_2 != C_1
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C_1),X_2))),R_2)) )
=> hAPP(real,real,F,X_2) != zero_zero(real) ) ) ) ) ).
fof(fact_4632_LIM__fun__gt__zero,axiom,
! [F,L,C_1] :
( hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),F),L),hAPP(real,filter(real),at(real),C_1)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
=> ? [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
& ! [X_2] :
( ( X_2 != C_1
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C_1),X_2))),R_2)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,F,X_2))) ) ) ) ) ).
fof(fact_4633_LIM__fun__less__zero,axiom,
! [F,L,C_1] :
( hBOOL(hAPP(filter(real),bool,hAPP(real,fun(filter(real),bool),hAPP(fun(real,real),fun(real,fun(filter(real),bool)),tendsto(real,real),F),L),hAPP(real,filter(real),at(real),C_1)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
=> ? [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
& ! [X_2] :
( ( X_2 != C_1
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),C_1),X_2))),R_2)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,X_2)),zero_zero(real))) ) ) ) ) ).
fof(fact_4634_aset_I8_J,axiom,
! [T_3,A_2,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),T_3),X_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),T_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),D_2))) ) ) ) ).
fof(fact_4635_aset_I6_J,axiom,
! [T_3,A_2,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),T_3),one_one(int))),A_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_2),T_3))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),D_2)),T_3)) ) ) ) ) ).
fof(fact_4636_bset_I8_J,axiom,
! [T_3,B_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),T_3),one_one(int))),B_1))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),T_3),X_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),T_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),D_2))) ) ) ) ) ).
fof(fact_4637_bset_I6_J,axiom,
! [T_3,B_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),X_2),T_3))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),D_2)),T_3)) ) ) ) ).
fof(fact_4638_LIMSEQ__SEQ__conv,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_3,L_2,Aa] :
( ! [S_2] :
( ( ! [N_2] : hAPP(nat,real,S_2,N_2) != Aa
& hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),S_2),Aa),sequentially)) )
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),X_3),S_2)),L_2),sequentially)) )
<=> hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),X_3),L_2),hAPP(real,filter(real),at(real),Aa))) ) ) ).
fof(fact_4639_bset_I3_J,axiom,
! [T_3,B_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),T_3),one_one(int))),B_1))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa) ) )
=> ( X_2 = T_3
=> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),D_2) = T_3 ) ) ) ) ).
fof(fact_4640_aset_I3_J,axiom,
! [T_3,A_2,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),T_3),one_one(int))),A_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa) ) )
=> ( X_2 = T_3
=> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),D_2) = T_3 ) ) ) ) ).
fof(fact_4641_aset_I7_J,axiom,
! [T_3,A_2,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),T_3),X_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),T_3),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),D_2))) ) ) ) ).
fof(fact_4642_bset_I5_J,axiom,
! [T_3,B_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),T_3))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),D_2)),T_3)) ) ) ) ).
fof(fact_4643_aset_I5_J,axiom,
! [T_3,A_2,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),T_3),A_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),T_3))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),D_2)),T_3)) ) ) ) ) ).
fof(fact_4644_bset_I7_J,axiom,
! [T_3,B_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),T_3),B_1))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),T_3),X_2))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),T_3),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),D_2))) ) ) ) ) ).
fof(fact_4645_aset_I4_J,axiom,
! [T_3,A_2,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),T_3),A_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa) ) )
=> ( X_2 != T_3
=> hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),D_2) != T_3 ) ) ) ) ).
fof(fact_4646_bset_I4_J,axiom,
! [T_3,B_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),T_3),B_1))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa) ) )
=> ( X_2 != T_3
=> hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),D_2) != T_3 ) ) ) ) ).
fof(fact_4647_periodic__finite__ex,axiom,
! [P_1,D_1] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_1))
=> ( ! [X_2,K_1] :
( hBOOL(hAPP(int,bool,P_1,X_2))
<=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_1)))) )
=> ( ? [X1] : hBOOL(hAPP(int,bool,P_1,X1))
<=> ? [X_2] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_1)))
& hBOOL(hAPP(int,bool,P_1,X_2)) ) ) ) ) ).
fof(fact_4648_LIM__eq,axiom,
! [X_b,X_a] :
( ( real_normed_vector(X_a)
& real_normed_vector(X_b) )
=> ! [F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
<=> ! [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
=> ? [S_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
& ! [X_2] :
( ( ti(X_a,X_2) != ti(X_a,Aa)
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_2),Aa))),S_1)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_2)),L_2))),R_2)) ) ) ) ) ) ).
fof(fact_4649_LIM__D,axiom,
! [X_b,X_a] :
( ( real_normed_vector(X_a)
& real_normed_vector(X_b) )
=> ! [R_1,F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
=> ? [S_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
& ! [X_2] :
( ( ti(X_a,X_2) != ti(X_a,Aa)
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_2),Aa))),S_1)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_2)),L_2))),R_1)) ) ) ) ) ) ).
fof(fact_4650_LIM__equal2,axiom,
! [X_b,X_a] :
( ( real_normed_vector(X_a)
& topological_space(X_b) )
=> ! [L,F,G,Aa,R_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_3))
=> ( ! [X_2] :
( ti(X_a,X_2) != ti(X_a,Aa)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_2),Aa))),R_3))
=> hAPP(X_a,X_b,F,X_2) = hAPP(X_a,X_b,G,X_2) ) )
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ) ).
fof(fact_4651_aset_I10_J,axiom,
! [T_3,A_2,D_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa) ) )
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),T_3)))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_2),D_2)),T_3))) ) ) ) ).
fof(fact_4652_bset_I10_J,axiom,
! [T_3,B_1,D_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa) ) )
=> ( ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),T_3)))
=> ~ hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_2),D_2)),T_3))) ) ) ) ).
fof(fact_4653_bset_I9_J,axiom,
! [T_3,B_1,D_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),T_3)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_2),D_2)),T_3))) ) ) ) ).
fof(fact_4654_aset_I9_J,axiom,
! [T_3,A_2,D_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),D_2))
=> ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),dvd_dvd(int),D_1),hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),T_3)))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,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_2),D_2)),T_3))) ) ) ) ).
fof(fact_4655_metric__LIM__D,axiom,
! [X_b,X_a] :
( ( metric_space(X_a)
& metric_space(X_b) )
=> ! [R_1,F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_1))
=> ? [S_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
& ! [X_2] :
( ( ti(X_a,X_2) != ti(X_a,Aa)
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_2),Aa)),S_1)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_2)),L_2)),R_1)) ) ) ) ) ) ).
fof(fact_4656_metric__LIM__equal2,axiom,
! [X_b,X_a] :
( ( metric_space(X_a)
& topological_space(X_b) )
=> ! [L,F,G,Aa,R_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_3))
=> ( ! [X_2] :
( ti(X_a,X_2) != ti(X_a,Aa)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_2),Aa)),R_3))
=> hAPP(X_a,X_b,F,X_2) = hAPP(X_a,X_b,G,X_2) ) )
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ) ).
fof(fact_4657_LIM__def,axiom,
! [X_b,X_a] :
( ( metric_space(X_a)
& metric_space(X_b) )
=> ! [F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
<=> ! [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
=> ? [S_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
& ! [X_2] :
( ( ti(X_a,X_2) != ti(X_a,Aa)
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_2),Aa)),S_1)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_2)),L_2)),R_2)) ) ) ) ) ) ).
fof(fact_4658_LIM__imp__LIM,axiom,
! [X_b,X_c,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_c)
& real_normed_vector(X_b) )
=> ! [G,M,F,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( ! [X_2] :
( ti(X_a,X_2) != ti(X_a,Aa)
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_c,real,norm_norm(X_c),hAPP(X_c,X_c,hAPP(X_c,fun(X_c,X_c),minus_minus(X_c),hAPP(X_a,X_c,G,X_2)),M))),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_2)),L)))) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),G),M),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4659_LIMSEQ__SEQ__conv1,axiom,
! [X_b,X_a] :
( ( metric_space(X_a)
& metric_space(X_b) )
=> ! [X_3,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),X_3),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ! [S_2] :
( ( ! [N_2] : hAPP(nat,X_a,S_2,N_2) != ti(X_a,Aa)
& hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),S_2),Aa),sequentially)) )
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(fun(nat,X_a),fun(nat,X_b),hAPP(fun(X_a,X_b),fun(fun(nat,X_a),fun(nat,X_b)),combb(X_a,X_b,nat),X_3),S_2)),L_2),sequentially)) ) ) ) ).
fof(fact_4660_metric__LIM__imp__LIM,axiom,
! [X_b,X_c,X_a] :
( ( topological_space(X_a)
& metric_space(X_c)
& metric_space(X_b) )
=> ! [G,M,F,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( ! [X_2] :
( ti(X_a,X_2) != ti(X_a,Aa)
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_c,real,hAPP(X_c,fun(X_c,real),dist_dist(X_c),hAPP(X_a,X_c,G,X_2)),M)),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_2)),L))) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),G),M),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4661_LIMSEQ__setprod,axiom,
! [X_a,X_b] :
( ( real_normed_algebra(X_b)
& comm_ring_1(X_b) )
=> ! [X_3,L_2,S] :
( ! [N_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),N_2),S))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(X_a,fun(nat,X_b),X_3,N_2)),hAPP(X_a,X_b,L_2,N_2)),sequentially)) )
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(fun(X_a,bool),fun(nat,X_b),hAPP(fun(nat,fun(fun(X_a,bool),X_b)),fun(fun(X_a,bool),fun(nat,X_b)),combc(nat,fun(X_a,bool),X_b),hAPP(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b)),hAPP(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),fun(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b))),combb(fun(X_a,X_b),fun(fun(X_a,bool),X_b),nat),big_co302546470etprod(X_a,X_b)),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),X_3))),S)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),L_2),S)),sequentially)) ) ) ).
fof(fact_4662_LIMSEQ__setsum,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [X_3,L_2,S] :
( ! [N_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),N_2),S))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(X_a,fun(nat,X_b),X_3,N_2)),hAPP(X_a,X_b,L_2,N_2)),sequentially)) )
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_b,fun(filter(nat),bool),hAPP(fun(nat,X_b),fun(X_b,fun(filter(nat),bool)),tendsto(nat,X_b),hAPP(fun(X_a,bool),fun(nat,X_b),hAPP(fun(nat,fun(fun(X_a,bool),X_b)),fun(fun(X_a,bool),fun(nat,X_b)),combc(nat,fun(X_a,bool),X_b),hAPP(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b)),hAPP(fun(fun(X_a,X_b),fun(fun(X_a,bool),X_b)),fun(fun(nat,fun(X_a,X_b)),fun(nat,fun(fun(X_a,bool),X_b))),combb(fun(X_a,X_b),fun(fun(X_a,bool),X_b),nat),big_co1399186613setsum(X_a,X_b)),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),X_3))),S)),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),L_2),S)),sequentially)) ) ) ).
fof(fact_4663_LIMSEQ__I,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_3,L_2] :
( ! [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
=> ? [No] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(nat,X_a,X_3,N_2)),L_2))),R_2)) ) )
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),L_2),sequentially)) ) ) ).
fof(fact_4664_LIMSEQ__inverse__zero,axiom,
! [X_3] :
( ! [R_2] :
? [N_3] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),N_3),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),R_2),hAPP(nat,real,X_3,N_2))) )
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),X_3)),zero_zero(real)),sequentially)) ) ).
fof(fact_4665_LIMSEQ__SEQ__conv2,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_3,L_2,Aa] :
( ! [S_2] :
( ( ! [N_2] : hAPP(nat,real,S_2,N_2) != Aa
& hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),S_2),Aa),sequentially)) )
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,real),fun(nat,X_a),hAPP(fun(real,X_a),fun(fun(nat,real),fun(nat,X_a)),combb(real,X_a,nat),X_3),S_2)),L_2),sequentially)) )
=> hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),X_3),L_2),hAPP(real,filter(real),at(real),Aa))) ) ) ).
fof(fact_4666_metric__LIMSEQ__I,axiom,
! [X_a] :
( metric_space(X_a)
=> ! [X_3,L_2] :
( ! [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
=> ? [No] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),No),N_2))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),hAPP(nat,X_a,X_3,N_2)),L_2)),R_2)) ) )
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),L_2),sequentially)) ) ) ).
fof(fact_4667_LIM__I,axiom,
! [X_b,X_a] :
( ( real_normed_vector(X_a)
& real_normed_vector(X_b) )
=> ! [F,L_2,Aa] :
( ! [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
=> ? [S_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
& ! [X_2] :
( ( ti(X_a,X_2) != ti(X_a,Aa)
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_2),Aa))),S_1)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,norm_norm(X_b),hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),minus_minus(X_b),hAPP(X_a,X_b,F,X_2)),L_2))),R_2)) ) ) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4668_exp__series__add,axiom,
! [X_a] :
( real_field(X_a)
=> ! [N,X_1,Y_1] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),hAPP(nat,real,real_1(nat),hAPP(nat,nat,fact_fact(nat),N)))),hAPP(nat,X_a,hAPP(X_a,fun(nat,X_a),power_power(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1),Y_1)),N)) = hAPP(fun(nat,bool),X_a,hAPP(fun(nat,X_a),fun(fun(nat,bool),X_a),big_co1399186613setsum(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,X_a),fun(nat,fun(X_a,X_a)),hAPP(fun(X_a,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,fun(X_a,X_a))),combb(X_a,fun(X_a,X_a),nat),times_times(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1)))),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),fact_fact(nat)),hAPP(nat,fun(nat,nat),minus_minus(nat),N)))))),hAPP(fun(nat,nat),fun(nat,X_a),hAPP(fun(nat,X_a),fun(fun(nat,nat),fun(nat,X_a)),combb(nat,X_a,nat),hAPP(X_a,fun(nat,X_a),power_power(X_a),Y_1)),hAPP(nat,fun(nat,nat),minus_minus(nat),N))))),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),N)) ) ).
fof(fact_4669_norm__scaleR,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,X] : hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,abs_abs(real),A_1)),hAPP(X_a,real,norm_norm(X_a),X)) ) ).
fof(fact_4670_scaleR__minus1__left,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(int,real,number_number_of(real),min)),X) = hAPP(X_a,X_a,uminus_uminus(X_a),X) ) ).
fof(fact_4671_scaleR__left_OCauchy,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1,X_3] :
( hBOOL(hAPP(fun(nat,real),bool,cauchy(real),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_3)),X_1))) ) ) ).
fof(fact_4672_scaleR_Oprod__diff__prod,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X,Y,A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Y)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),A_1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),A_1)),B))),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),B))) ) ).
fof(fact_4673_sgn__scaleR,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_4,X] : hAPP(X_a,X_a,sgn_sgn(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,sgn_sgn(real),R_4)),hAPP(X_a,X_a,sgn_sgn(X_a),X)) ) ).
fof(fact_4674_scaleR__left_Odiff,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [X,Y,Xa_1] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),X),Y)),Xa_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_1)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Y),Xa_1)) ) ).
fof(fact_4675_scaleR_Odiff__left,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,A_5,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),A_5)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_5),B)) ) ).
fof(fact_4676_scaleR__left__diff__distrib,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [A_1,B,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),A_1),B)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)) ) ).
fof(fact_4677_scaleR__left_Oadd,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [X,Y,Xa_1] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),X),Y)),Xa_1) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_1)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Y),Xa_1)) ) ).
fof(fact_4678_scaleR_Oadd__left,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,A_5,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),A_5)),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_5),B)) ) ).
fof(fact_4679_real__vector_Oscale__left__distrib,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [A_1,B,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),A_1),B)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)) ) ).
fof(fact_4680_scaleR_Oadd__right,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,B,B_4] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),B),B_4)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B_4)) ) ).
fof(fact_4681_scaleR__right_Oadd,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [A_1,X,Y] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y)) ) ).
fof(fact_4682_scaleR_Odiff__right,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,B,B_4] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),B),B_4)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B_4)) ) ).
fof(fact_4683_scaleR__right_Odiff,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [A_1,X,Y] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y)) ) ).
fof(fact_4684_real__vector_Oscale__one,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),one_one(real)),X) = ti(X_a,X) ) ).
fof(fact_4685_scaleR__left__imp__eq,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [X,Y,A_1] :
( A_1 != zero_zero(real)
=> ( hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y)
=> ti(X_a,X) = ti(X_a,Y) ) ) ) ).
fof(fact_4686_scaleR__cancel__left,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [Aa,X_1,Y_1] :
( hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Aa),X_1) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Aa),Y_1)
<=> ( ti(X_a,X_1) = ti(X_a,Y_1)
| Aa = zero_zero(real) ) ) ) ).
fof(fact_4687_scaleR__right__imp__eq,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [A_1,B,X] :
( ti(X_a,X) != zero_zero(X_a)
=> ( hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)
=> A_1 = B ) ) ) ).
fof(fact_4688_scaleR__cancel__right,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [Aa,X_1,Ba] :
( hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Aa),X_1) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ba),X_1)
<=> ( Aa = Ba
| ti(X_a,X_1) = zero_zero(X_a) ) ) ) ).
fof(fact_4689_scaleR_Ozero__right,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_4690_scaleR__right_Ozero,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),zero_zero(X_a)) = zero_zero(X_a) ) ).
fof(fact_4691_scaleR__eq__0__iff,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [Aa,X_1] :
( hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Aa),X_1) = zero_zero(X_a)
<=> ( Aa = zero_zero(real)
| ti(X_a,X_1) = zero_zero(X_a) ) ) ) ).
fof(fact_4692_scaleR__left_Ozero,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),zero_zero(real)),X) = zero_zero(X_a) ) ).
fof(fact_4693_scaleR_Ozero__left,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),zero_zero(real)),B) = zero_zero(X_a) ) ).
fof(fact_4694_divide_OscaleR,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [R_4,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),X)),Y) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),inverse_divide(X_a),X),Y)) ) ).
fof(fact_4695_of__real__def,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [R_4] : hAPP(real,X_a,of_real(X_a),R_4) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),one_one(X_a)) ) ).
fof(fact_4696_of__real_OscaleR,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [R_4,X] : hAPP(real,X_a,of_real(X_a),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_4),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),hAPP(real,X_a,of_real(X_a),X)) ) ).
fof(fact_4697_scaleR__left__commute,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [A_1,B,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) ) ).
fof(fact_4698_scaleR_OscaleR__right,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,R_4,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),B)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) ) ).
fof(fact_4699_scaleR__right_OscaleR,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [Ra,R_4,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Ra),X)) ) ).
fof(fact_4700_scaleR_OscaleR__left,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_4,A_1,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_4),A_1)),B) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) ) ).
fof(fact_4701_scaleR__left_OscaleR,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_4,X,Xa_1] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_4),X)),Xa_1) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_1)) ) ).
fof(fact_4702_bounded__bilinear_OscaleR__left,axiom,
! [X_a,X_c,X_b] :
( ( real_normed_vector(X_b)
& real_normed_vector(X_c)
& real_normed_vector(X_a) )
=> ! [R_1,Aa,Ba,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1),Aa)),Ba) = hAPP(X_c,X_c,hAPP(real,fun(X_c,X_c),scaleR_scaleR(X_c),R_1),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)) ) ) ).
fof(fact_4703_bounded__bilinear_OscaleR__right,axiom,
! [X_a,X_c,X_b] :
( ( real_normed_vector(X_b)
& real_normed_vector(X_c)
& real_normed_vector(X_a) )
=> ! [Aa,R_1,Ba,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),R_1),Ba)) = hAPP(X_c,X_c,hAPP(real,fun(X_c,X_c),scaleR_scaleR(X_c),R_1),hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),Prod,Aa),Ba)) ) ) ).
fof(fact_4704_cnj_OscaleR,axiom,
! [R_4,X] : hAPP(complex,complex,cnj,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_4),X)) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_4),hAPP(complex,complex,cnj,X)) ).
fof(fact_4705_Re_OscaleR,axiom,
! [R_4,X] : hAPP(complex,real,re,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_4),X)) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_4),hAPP(complex,real,re,X)) ).
fof(fact_4706_Im_OscaleR,axiom,
! [R_4,X] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_4),X)) = hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),R_4),hAPP(complex,real,im,X)) ).
fof(fact_4707_mult_OscaleR__right,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [A_1,R_4,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),B)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) ) ).
fof(fact_4708_mult__scaleR__right,axiom,
! [X_a] :
( real_algebra(X_a)
=> ! [X,A_1,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),Y)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) ) ).
fof(fact_4709_mult__right_OscaleR,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Xa_1,R_4,X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),Xa_1),X)) ) ).
fof(fact_4710_mult__scaleR__left,axiom,
! [X_a] :
( real_algebra(X_a)
=> ! [A_1,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)),Y) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) ) ).
fof(fact_4711_mult_OscaleR__left,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [R_4,A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),A_1)),B) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),A_1),B)) ) ).
fof(fact_4712_mult__left_OscaleR,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [R_4,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),X)),Y) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X),Y)) ) ).
fof(fact_4713_real__scaleR__def,axiom,
! [A_1,X] : hAPP(real,real,hAPP(real,fun(real,real),scaleR_scaleR(real),A_1),X) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),X) ).
fof(fact_4714_real__vector_Oscale__scale,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [A_1,B,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),B),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),A_1),B)),X) ) ).
fof(fact_4715_scaleR__conv__of__real,axiom,
! [X_a] :
( real_algebra_1(X_a)
=> ! [R_4,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(real,X_a,of_real(X_a),R_4)),X) ) ).
fof(fact_4716_complex__scaleR,axiom,
! [R_4,A_1,B] : hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_4),hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,A_1),B)) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),A_1)),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),B)) ).
fof(fact_4717_complex__Im__scaleR,axiom,
! [R_4,X] : hAPP(complex,real,im,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_4),X)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),hAPP(complex,real,im,X)) ).
fof(fact_4718_complex__Re__scaleR,axiom,
! [R_4,X] : hAPP(complex,real,re,hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_4),X)) = hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),hAPP(complex,real,re,X)) ).
fof(fact_4719_scaleR_Ominus__right,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),B)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) ) ).
fof(fact_4720_scaleR__minus__right,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [A_1,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),hAPP(X_a,X_a,uminus_uminus(X_a),X)) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) ) ).
fof(fact_4721_scaleR_Ominus__left,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,uminus_uminus(real),A_1)),B) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),B)) ) ).
fof(fact_4722_scaleR__minus__left,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [A_1,X] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,uminus_uminus(real),A_1)),X) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) ) ).
fof(fact_4723_scaleR__left_Ominus,axiom,
! [X_a] :
( real_vector(X_a)
=> ! [X,Xa_1] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,uminus_uminus(real),X)),Xa_1) = hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X),Xa_1)) ) ).
fof(fact_4724_inverse__scaleR__distrib,axiom,
! [X_a] :
( ( divisi14063676e_zero(X_a)
& real_div_algebra(X_a) )
=> ! [A_1,X] : hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),X)) ) ).
fof(fact_4725_scaleR_Oflip,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> hBOOL(hAPP(fun(X_a,fun(real,X_a)),bool,bounded_bilinear(X_a,real,X_a),hAPP(fun(real,fun(X_a,X_a)),fun(X_a,fun(real,X_a)),combc(real,X_a,X_a),scaleR_scaleR(X_a)))) ) ).
fof(fact_4726_scaleR__right_Osummable,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1)),X_3))) ) ) ).
fof(fact_4727_scaleR__left_Osummable,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1,X_3] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_3)),X_1))) ) ) ).
fof(fact_4728_scaleR__right_Osuminf,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),X_3))
=> hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1),hAPP(fun(nat,X_a),X_a,suminf(X_a),X_3)) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1)),X_3)) ) ) ).
fof(fact_4729_scaleR__right_OCauchy,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_1,X_3] :
( hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),X_3))
=> hBOOL(hAPP(fun(nat,X_a),bool,cauchy(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1)),X_3))) ) ) ).
fof(fact_4730_scaleR__right_Osums,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_1,X_3,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),X_3),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1)),X_3)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1),Aa))) ) ) ).
fof(fact_4731_scaleR__left_Osums,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1,X_3,Aa] :
( hBOOL(hAPP(real,bool,hAPP(fun(nat,real),fun(real,bool),sums(real),X_3),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_3)),X_1)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Aa),X_1))) ) ) ).
fof(fact_4732_sgn__div__norm,axiom,
! [X_a] :
( sgn_div_norm(X_a)
=> ! [X] : hAPP(X_a,X_a,sgn_sgn(X_a),X) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),hAPP(X_a,real,norm_norm(X_a),X))),X) ) ).
fof(fact_4733_scaleR_Otendsto,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [G,Ba,F,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),Aa),A_2))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Ba),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Aa),Ba)),A_2)) ) ) ) ).
fof(fact_4734_scaleR__left_Otendsto,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [X_1,G,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),G)),X_1)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),Aa),X_1)),A_2)) ) ) ).
fof(fact_4735_scaleR__right_Otendsto,axiom,
! [X_a,X_b] :
( real_normed_vector(X_b)
=> ! [R_1,G,Aa,A_2] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),Aa),A_2))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),R_1)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),R_1),Aa)),A_2)) ) ) ).
fof(fact_4736_scaleR__right_OLIMSEQ,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_1,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(X_a,X_a),fun(fun(nat,X_a),fun(nat,X_a)),combb(X_a,X_a,nat),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1)),X_3)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1),Aa)),sequentially)) ) ) ).
fof(fact_4737_scaleR__right_Osetsum,axiom,
! [X_b,X_a] :
( real_vector(X_a)
=> ! [Aa,G,A_2] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Aa),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),A_2)) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(fun(X_b,X_a),fun(X_b,X_a),hAPP(fun(X_a,X_a),fun(fun(X_b,X_a),fun(X_b,X_a)),combb(X_a,X_a,X_b),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Aa)),G)),A_2) ) ).
fof(fact_4738_scaleR__left_Osetsum,axiom,
! [X_b,X_a] :
( real_vector(X_a)
=> ! [G,A_2,X_1] : hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(fun(X_b,bool),real,hAPP(fun(X_b,real),fun(fun(X_b,bool),real),big_co1399186613setsum(X_b,real),G),A_2)),X_1) = hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),hAPP(X_a,fun(X_b,X_a),hAPP(fun(X_b,fun(X_a,X_a)),fun(X_a,fun(X_b,X_a)),combc(X_b,X_a,X_a),hAPP(fun(X_b,real),fun(X_b,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(X_b,real),fun(X_b,fun(X_a,X_a))),combb(real,fun(X_a,X_a),X_b),scaleR_scaleR(X_a)),G)),X_1)),A_2) ) ).
fof(fact_4739_scaleR__right_Ocont,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_1,Aa] : hBOOL(hAPP(filter(X_a),bool,hAPP(X_a,fun(filter(X_a),bool),hAPP(fun(X_a,X_a),fun(X_a,fun(filter(X_a),bool)),tendsto(X_a,X_a),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1),Aa)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ).
fof(fact_4740_scaleR__left_Ocont,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1,Aa] : hBOOL(hAPP(filter(real),bool,hAPP(X_a,fun(filter(real),bool),hAPP(fun(real,X_a),fun(X_a,fun(filter(real),bool)),tendsto(real,X_a),hAPP(X_a,fun(real,X_a),hAPP(fun(real,fun(X_a,X_a)),fun(X_a,fun(real,X_a)),combc(real,X_a,X_a),scaleR_scaleR(X_a)),X_1)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Aa),X_1)),hAPP(real,filter(real),at(real),Aa))) ) ).
fof(fact_4741_scaleR__right_OLIM,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [R_1,G,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),R_1)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),R_1),L)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4742_scaleR_OLIM__right__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [C_1,F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),C_1)),F)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4743_scaleR__right_OLIM__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [R_1,G,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),R_1)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4744_scaleR__left_OLIM,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [X_1,G,L,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),G)),X_1)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),L),X_1)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4745_LIM__scaleR,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [G,M_2,F,L_2,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M_2),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),hAPP(X_b,X_b,hAPP(real,fun(X_b,X_b),scaleR_scaleR(X_b),L_2),M_2)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4746_scaleR_OLIMSEQ,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [Y_3,Ba,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_3),Aa),sequentially))
=> ( hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),Y_3),Ba),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_3)),Y_3)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Aa),Ba)),sequentially)) ) ) ) ).
fof(fact_4747_scaleR__left_OLIMSEQ,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1,X_3,Aa] :
( hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),X_3),Aa),sequentially))
=> hBOOL(hAPP(filter(nat),bool,hAPP(X_a,fun(filter(nat),bool),hAPP(fun(nat,X_a),fun(X_a,fun(filter(nat),bool)),tendsto(nat,X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_3)),X_1)),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),Aa),X_1)),sequentially)) ) ) ).
fof(fact_4748_scaleR__left_Osuminf,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1,X_3] :
( hBOOL(hAPP(fun(nat,real),bool,summable(real),X_3))
=> hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(fun(nat,real),real,suminf(real),X_3)),X_1) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(X_a,fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(X_a,fun(nat,X_a)),combc(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),X_3)),X_1)) ) ) ).
fof(fact_4749_complex__sgn__def,axiom,
! [X] : hAPP(complex,complex,sgn_sgn(complex),X) = hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),hAPP(real,real,inverse_inverse(real),hAPP(complex,real,norm_norm(complex),X))),X) ).
fof(fact_4750_nonzero__inverse__scaleR__distrib,axiom,
! [X_a] :
( real_div_algebra(X_a)
=> ! [X,A_1] :
( A_1 != zero_zero(real)
=> ( ti(X_a,X) != zero_zero(X_a)
=> hAPP(X_a,X_a,inverse_inverse(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_1),X)) = hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),hAPP(real,real,inverse_inverse(real),A_1)),hAPP(X_a,X_a,inverse_inverse(X_a),X)) ) ) ) ).
fof(fact_4751_complex__scaleR__def,axiom,
! [R_4,X] : hAPP(complex,complex,hAPP(real,fun(complex,complex),scaleR_scaleR(complex),R_4),X) = hAPP(real,complex,hAPP(real,fun(real,complex),complex_1,hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),hAPP(complex,real,re,X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),R_4),hAPP(complex,real,im,X))) ).
fof(fact_4752_scaleR_OLIM__left__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [C_1,F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),C_1)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4753_scaleR__left_OLIM__zero,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [X_1,G,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(X_b,fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(X_b,fun(X_a,X_b)),combc(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),G)),X_1)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4754_scaleR_OLIM__prod__zero,axiom,
! [X_b,X_a] :
( ( metric_space(X_a)
& real_normed_vector(X_b) )
=> ! [G,F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),zero_zero(X_b)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4755_summable__exp__generic,axiom,
! [X_a] :
( ( real_n2089651433ebra_1(X_a)
& banach(X_a) )
=> ! [X_1] : hBOOL(hAPP(fun(nat,X_a),bool,summable(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1)))) ) ).
fof(fact_4756_summable__norm__exp,axiom,
! [X_a] :
( ( real_n2089651433ebra_1(X_a)
& banach(X_a) )
=> ! [X_1] : hBOOL(hAPP(fun(nat,real),bool,summable(real),hAPP(fun(nat,X_a),fun(nat,real),hAPP(fun(X_a,real),fun(fun(nat,X_a),fun(nat,real)),combb(X_a,real,nat),norm_norm(X_a)),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))))) ) ).
fof(fact_4757_lemma__exp__ext,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X_2] : hAPP(X_a,X_a,exp(X_a),X_2) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_2))) ) ).
fof(fact_4758_exp__def,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X_1] : hAPP(X_a,X_a,exp(X_a),X_1) = hAPP(fun(nat,X_a),X_a,suminf(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))) ) ).
fof(fact_4759_exp__converges,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(fun(nat,X_a),fun(X_a,bool),sums(X_a),hAPP(fun(nat,X_a),fun(nat,X_a),hAPP(fun(nat,fun(X_a,X_a)),fun(fun(nat,X_a),fun(nat,X_a)),combs(nat,X_a,X_a),hAPP(fun(nat,real),fun(nat,fun(X_a,X_a)),hAPP(fun(real,fun(X_a,X_a)),fun(fun(nat,real),fun(nat,fun(X_a,X_a))),combb(real,fun(X_a,X_a),nat),scaleR_scaleR(X_a)),hAPP(fun(nat,real),fun(nat,real),hAPP(fun(real,real),fun(fun(nat,real),fun(nat,real)),combb(real,real,nat),inverse_inverse(real)),hAPP(fun(nat,nat),fun(nat,real),hAPP(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),combb(nat,real,nat),real_1(nat)),fact_fact(nat))))),hAPP(X_a,fun(nat,X_a),power_power(X_a),X_1))),hAPP(X_a,X_a,exp(X_a),X_1))) ) ).
fof(fact_4760_scaleR__left_Opos__bounded,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X] :
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X_2),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_2)),K_2))) ) ) ).
fof(fact_4761_scaleR_Opos__bounded,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [A_3,B_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),B_3))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A_3)),hAPP(X_a,real,norm_norm(X_a),B_3))),K_2))) ) ) ).
fof(fact_4762_scaleR_Ononneg__bounded,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [A_3,B_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),B_3))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A_3)),hAPP(X_a,real,norm_norm(X_a),B_3))),K_2))) ) ) ).
fof(fact_4763_scaleR__left_Ononneg__bounded,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X] :
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X_2),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_2)),K_2))) ) ) ).
fof(fact_4764_scaleR__right_Opos__bounded,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_4] :
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ) ).
fof(fact_4765_scaleR__left_Obounded,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X] :
? [K_2] :
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),X_2),X))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),X_2)),K_2))) ) ).
fof(fact_4766_scaleR_Obounded,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ? [K_2] :
! [A_3,B_3] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),A_3),B_3))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(real,real,norm_norm(real),A_3)),hAPP(X_a,real,norm_norm(X_a),B_3))),K_2))) ) ).
fof(fact_4767_scaleR__right_Ononneg__bounded,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_4] :
? [K_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K_2))
& ! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ) ).
fof(fact_4768_scaleR__right_Obounded,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_4] :
? [K_2] :
! [X_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_4),X_2))),hAPP(real,real,hAPP(real,fun(real,real),times_times(real),hAPP(X_a,real,norm_norm(X_a),X_2)),K_2))) ) ).
fof(fact_4769_LIM__cong,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& topological_space(X_b) )
=> ! [L,M,F,G,Aa,Ba] :
( ti(X_a,Aa) = ti(X_a,Ba)
=> ( ! [X_2] :
( ti(X_a,X_2) != ti(X_a,Ba)
=> hAPP(X_a,X_b,F,X_2) = hAPP(X_a,X_b,G,X_2) )
=> ( ti(X_b,L) = ti(X_b,M)
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),M),hAPP(X_a,filter(X_a),at(X_a),Ba))) ) ) ) ) ) ).
fof(fact_4770_tendsto__setsum,axiom,
! [X_b,X_a,X_c] :
( real_normed_vector(X_c)
=> ! [F,Aa,A_2,S] :
( ! [I_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),S))
=> hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),hAPP(X_a,fun(X_b,X_c),F,I_2)),hAPP(X_a,X_c,Aa,I_2)),A_2)) )
=> hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),hAPP(fun(X_a,bool),fun(X_b,X_c),hAPP(fun(X_b,fun(fun(X_a,bool),X_c)),fun(fun(X_a,bool),fun(X_b,X_c)),combc(X_b,fun(X_a,bool),X_c),hAPP(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c)),hAPP(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c))),combb(fun(X_a,X_c),fun(fun(X_a,bool),X_c),X_b),big_co1399186613setsum(X_a,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),F))),S)),hAPP(fun(X_a,bool),X_c,hAPP(fun(X_a,X_c),fun(fun(X_a,bool),X_c),big_co1399186613setsum(X_a,X_c),Aa),S)),A_2)) ) ) ).
fof(fact_4771_metric__LIM__I,axiom,
! [X_b,X_a] :
( ( metric_space(X_a)
& metric_space(X_b) )
=> ! [F,L_2,Aa] :
( ! [R_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),R_2))
=> ? [S_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),S_1))
& ! [X_2] :
( ( ti(X_a,X_2) != ti(X_a,Aa)
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_2),Aa)),S_1)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_b,real,hAPP(X_b,fun(X_b,real),dist_dist(X_b),hAPP(X_a,X_b,F,X_2)),L_2)),R_2)) ) ) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_2),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4772_trivial__limit__sequentially,axiom,
~ hBOOL(hAPP(filter(nat),bool,trivial_limit(nat),sequentially)) ).
fof(fact_4773_LIM__equal,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& topological_space(X_b) )
=> ! [L,F,G,Aa] :
( ! [X_2] :
( ti(X_a,X_2) != ti(X_a,Aa)
=> hAPP(X_a,X_b,F,X_2) = hAPP(X_a,X_b,G,X_2) )
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),hAPP(X_a,filter(X_a),at(X_a),Aa)))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),G),L),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ).
fof(fact_4774_tendsto__unique,axiom,
! [X_a,X_b] :
( t2_space(X_b)
=> ! [L_1,F,L,A_2] :
( ~ hBOOL(hAPP(filter(X_a),bool,trivial_limit(X_a),A_2))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L),A_2))
=> ( hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),L_1),A_2))
=> ti(X_b,L) = ti(X_b,L_1) ) ) ) ) ).
fof(fact_4775_pred__nat__def,axiom,
pred_nat = hAPP(fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),collect(product_prod(nat,nat)),hAPP(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_prod_case(nat,nat,bool),hAPP(fun(nat,nat),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,nat),fun(nat,fun(nat,bool))),combb(nat,fun(nat,bool),nat),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),fequal(nat))),suc))) ).
fof(fact_4776_setprod_Ocong,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [G,H,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(X_a,X_b,G,X_2) = hAPP(X_a,X_b,H,X_2) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),G),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),H),A_2) ) ) ) ).
fof(fact_4777_splitE2,axiom,
! [X_a,X_b,X_c,Q_1,P_1,Z_1] :
( hBOOL(hAPP(X_a,bool,Q_1,hAPP(product_prod(X_b,X_c),X_a,hAPP(fun(X_b,fun(X_c,X_a)),fun(product_prod(X_b,X_c),X_a),product_prod_case(X_b,X_c,X_a),P_1),Z_1)))
=> ~ ! [X_2,Y_2] :
( ti(product_prod(X_b,X_c),Z_1) = hAPP(X_c,product_prod(X_b,X_c),hAPP(X_b,fun(X_c,product_prod(X_b,X_c)),product_Pair(X_b,X_c),X_2),Y_2)
=> ~ hBOOL(hAPP(X_a,bool,Q_1,hAPP(X_c,X_a,hAPP(X_b,fun(X_c,X_a),P_1,X_2),Y_2))) ) ) ).
fof(fact_4778_split__cong,axiom,
! [X_c,X_b,X_a,Pa,F,G,Qa] :
( ! [X_2,Y_2] :
( hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_2),Y_2) = ti(product_prod(X_a,X_b),Qa)
=> hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),F,X_2),Y_2) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),G,X_2),Y_2) )
=> ( ti(product_prod(X_a,X_b),Pa) = ti(product_prod(X_a,X_b),Qa)
=> hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),F),Pa) = hAPP(product_prod(X_a,X_b),X_c,hAPP(fun(X_a,fun(X_b,X_c)),fun(product_prod(X_a,X_b),X_c),product_prod_case(X_a,X_b,X_c),G),Qa) ) ) ).
fof(fact_4779_setprod__1_H,axiom,
! [X_a,X_b] :
( comm_monoid_mult(X_b)
=> ! [F,F_1] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),F_1))
=> hAPP(X_a,X_b,F,X_2) = one_one(X_b) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co302546470etprod(X_a,X_b),F),F_1) = one_one(X_b) ) ) ).
fof(fact_4780_dvd__setsum,axiom,
! [X_a,X_b] :
( comm_semiring_1(X_b)
=> ! [D_1,F,A_2] :
( ! [I_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I_2),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),D_1),hAPP(X_a,X_b,F,I_2))) )
=> hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),dvd_dvd(X_b),D_1),hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2))) ) ) ).
fof(fact_4781_setsum_Ocong,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [G,H,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(X_a,X_b,G,X_2) = hAPP(X_a,X_b,H,X_2) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),G),A_2) = hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),H),A_2) ) ) ) ).
fof(fact_4782_setsum__0_H,axiom,
! [X_a,X_b] :
( comm_monoid_add(X_b)
=> ! [F,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hAPP(X_a,X_b,F,X_2) = zero_zero(X_b) )
=> hAPP(fun(X_a,bool),X_b,hAPP(fun(X_a,X_b),fun(fun(X_a,bool),X_b),big_co1399186613setsum(X_a,X_b),F),A_2) = zero_zero(X_b) ) ) ).
fof(fact_4783_real__LIM__sandwich__zero,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [G,F,Aa] :
( hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),F),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa)))
=> ( ! [X_2] :
( ti(X_a,X_2) != ti(X_a,Aa)
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(X_a,real,G,X_2))) )
=> ( ! [X_2] :
( ti(X_a,X_2) != ti(X_a,Aa)
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(X_a,real,G,X_2)),hAPP(X_a,real,F,X_2))) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(real,fun(filter(X_a),bool),hAPP(fun(X_a,real),fun(real,fun(filter(X_a),bool)),tendsto(X_a,real),G),zero_zero(real)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ) ).
fof(fact_4784_gcd__coprime__exists__int,axiom,
! [A_1,B] :
( hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B) != zero_zero(int)
=> ? [A_4,B_2] :
( A_1 = 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_1),B))
& B = hAPP(int,int,hAPP(int,fun(int,int),times_times(int),B_2),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_1),B))
& hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),A_4),B_2) = one_one(int) ) ) ).
fof(fact_4785_exE__some,axiom,
! [X_a,C_1,P_1] :
( ? [X1] : hBOOL(hAPP(X_a,bool,P_1,X1))
=> ( ti(X_a,C_1) = hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1)
=> hBOOL(hAPP(X_a,bool,P_1,C_1)) ) ) ).
fof(fact_4786_setprod_Orelated,axiom,
! [X_b,X_a] :
( comm_monoid_mult(X_a)
=> ! [H,G,S,R_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,one_one(X_a)),one_one(X_a)))
=> ( ! [X1,Y1,X2,Y2] :
( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,X1),X2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,Y1),Y2)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X1),Y1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),X2),Y2))) )
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),S))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,hAPP(X_b,X_a,H,X_2)),hAPP(X_b,X_a,G,X_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),H),S)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co302546470etprod(X_b,X_a),G),S))) ) ) ) ) ) ).
fof(fact_4787_split__paired__Eps,axiom,
! [X_a,X_b,P_1] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),P_1) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool)),hAPP(fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),fun(fun(X_a,fun(X_b,product_prod(X_a,X_b))),fun(X_a,fun(X_b,bool))),combb(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool),X_a),hAPP(fun(product_prod(X_a,X_b),bool),fun(fun(X_b,product_prod(X_a,X_b)),fun(X_b,bool)),combb(product_prod(X_a,X_b),bool,X_b),P_1)),product_Pair(X_a,X_b)))) ).
fof(fact_4788_Eps__split__eq,axiom,
! [X_a,X_b,X_1,Y_1] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),hAPP(fun(X_b,bool),fun(X_a,fun(X_b,bool)),hAPP(fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_b,bool),fun(X_a,fun(X_b,bool))),combc(X_a,fun(X_b,bool),fun(X_b,bool)),hAPP(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool))),hAPP(fun(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool))),fun(fun(X_a,fun(bool,bool)),fun(X_a,fun(fun(X_b,bool),fun(X_b,bool)))),combb(fun(bool,bool),fun(fun(X_b,bool),fun(X_b,bool)),X_a),combb(bool,bool,X_b)),hAPP(fun(X_a,bool),fun(X_a,fun(bool,bool)),hAPP(fun(bool,fun(bool,bool)),fun(fun(X_a,bool),fun(X_a,fun(bool,bool))),combb(bool,fun(bool,bool),X_a),fconj),hAPP(X_a,fun(X_a,bool),fequal(X_a),X_1)))),hAPP(X_b,fun(X_b,bool),fequal(X_b),Y_1)))) = hAPP(X_b,product_prod(X_a,X_b),hAPP(X_a,fun(X_b,product_prod(X_a,X_b)),product_Pair(X_a,X_b),X_1),Y_1) ).
fof(fact_4789_Eps__split,axiom,
! [X_a,X_b,P_1] : hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(X_a,X_b),bool),product_prod_case(X_a,X_b,bool),P_1)) = hAPP(fun(product_prod(X_a,X_b),bool),product_prod(X_a,X_b),hilbert_Eps(product_prod(X_a,X_b)),hAPP(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool),hAPP(fun(product_prod(X_a,X_b),fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_b),fun(product_prod(X_a,X_b),bool)),combs(product_prod(X_a,X_b),X_b,bool),hAPP(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool)),hAPP(fun(X_a,fun(X_b,bool)),fun(fun(product_prod(X_a,X_b),X_a),fun(product_prod(X_a,X_b),fun(X_b,bool))),combb(X_a,fun(X_b,bool),product_prod(X_a,X_b)),P_1),product_fst(X_a,X_b))),product_snd(X_a,X_b))) ).
fof(fact_4790_tfl__some,axiom,
! [X_a,P_3,X_2] :
( hBOOL(hAPP(X_a,bool,P_3,X_2))
=> hBOOL(hAPP(X_a,bool,P_3,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_3))) ) ).
fof(fact_4791_someI,axiom,
! [X_a,P_1,X_1] :
( hBOOL(hAPP(X_a,bool,P_1,X_1))
=> hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))) ) ).
fof(fact_4792_someI__ex,axiom,
! [X_a,P_1] :
( ? [X1] : hBOOL(hAPP(X_a,bool,P_1,X1))
=> hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))) ) ).
fof(fact_4793_some__eq__ex,axiom,
! [X_a,P_1] :
( hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1)))
<=> ? [X1] : hBOOL(hAPP(X_a,bool,P_1,X1)) ) ).
fof(fact_4794_some__eq__trivial,axiom,
! [X_a,X_1] : hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),fequal(X_a)),X_1)) = ti(X_a,X_1) ).
fof(fact_4795_some__sym__eq__trivial,axiom,
! [X_a,X_1] : hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),hAPP(X_a,fun(X_a,bool),fequal(X_a),X_1)) = ti(X_a,X_1) ).
fof(fact_4796_some__equality,axiom,
! [X_a,P_1,Aa] :
( hBOOL(hAPP(X_a,bool,P_1,Aa))
=> ( ! [X_2] :
( hBOOL(hAPP(X_a,bool,P_1,X_2))
=> ti(X_a,X_2) = ti(X_a,Aa) )
=> hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1) = ti(X_a,Aa) ) ) ).
fof(fact_4797_dvd_Ostrict__monoD,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y_1,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),strict_mono(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combs(nat,fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool))),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),nat),combs(nat,bool,bool)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool))),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),nat),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj)),dvd_dvd(nat)))),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(fun(nat,bool),fun(nat,bool)),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combb(fun(nat,bool),fun(nat,bool),nat),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat))))),F))
=> ( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_1),X_1)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,X_1)),hAPP(nat,X_a,F,Y_1))) ) ) ) ).
fof(fact_4798_dvd_Ostrict__monoI,axiom,
! [X_a] :
( order(X_a)
=> ! [F] :
( ! [X_2,Y_2] :
( ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2))
& ~ hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),Y_2),X_2)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(nat,X_a,F,X_2)),hAPP(nat,X_a,F,Y_2))) )
=> hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),strict_mono(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combs(nat,fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool))),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),nat),combs(nat,bool,bool)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool))),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),nat),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj)),dvd_dvd(nat)))),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(fun(nat,bool),fun(nat,bool)),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combb(fun(nat,bool),fun(nat,bool),nat),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat))))),F)) ) ) ).
fof(fact_4799_dvd_Ostrict__mono__mono,axiom,
! [X_a] :
( order(X_a)
=> ! [F] :
( hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),strict_mono(nat,X_a),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(nat,fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combs(nat,fun(nat,bool),fun(nat,bool)),hAPP(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool))),hAPP(fun(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool))),fun(fun(nat,fun(nat,fun(bool,bool))),fun(nat,fun(fun(nat,bool),fun(nat,bool)))),combb(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),nat),combs(nat,bool,bool)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool))),hAPP(fun(fun(nat,bool),fun(nat,fun(bool,bool))),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,fun(bool,bool)))),combb(fun(nat,bool),fun(nat,fun(bool,bool)),nat),hAPP(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj)),dvd_dvd(nat)))),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),hAPP(fun(fun(nat,bool),fun(nat,bool)),fun(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool))),combb(fun(nat,bool),fun(nat,bool),nat),hAPP(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),combb(bool,bool,nat),fNot)),hAPP(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),dvd_dvd(nat))))),F))
=> hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),mono(nat,X_a),dvd_dvd(nat)),F)) ) ) ).
fof(fact_4800_dvd_OmonoD,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y_1,F] :
( hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),mono(nat,X_a),dvd_dvd(nat)),F))
=> ( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_1),Y_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,X_1)),hAPP(nat,X_a,F,Y_1))) ) ) ) ).
fof(fact_4801_dvd_OmonoI,axiom,
! [X_a] :
( order(X_a)
=> ! [F] :
( ! [X_2,Y_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),dvd_dvd(nat),X_2),Y_2))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(nat,X_a,F,X_2)),hAPP(nat,X_a,F,Y_2))) )
=> hBOOL(hAPP(fun(nat,X_a),bool,hAPP(fun(nat,fun(nat,bool)),fun(fun(nat,X_a),bool),mono(nat,X_a),dvd_dvd(nat)),F)) ) ) ).
fof(fact_4802_setsum_Orelated,axiom,
! [X_b,X_a] :
( comm_monoid_add(X_a)
=> ! [H,G,S,R_3] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,zero_zero(X_a)),zero_zero(X_a)))
=> ( ! [X1,Y1,X2,Y2] :
( ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,X1),X2))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,Y1),Y2)) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X1),Y1)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X2),Y2))) )
=> ( hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),S))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_b,bool),bool,hAPP(X_b,fun(fun(X_b,bool),bool),member(X_b),X_2),S))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,hAPP(X_b,X_a,H,X_2)),hAPP(X_b,X_a,G,X_2))) )
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),R_3,hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),H),S)),hAPP(fun(X_b,bool),X_a,hAPP(fun(X_b,X_a),fun(fun(X_b,bool),X_a),big_co1399186613setsum(X_b,X_a),G),S))) ) ) ) ) ) ).
fof(fact_4803_min__Suc__number__of,axiom,
! [V_1,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(int,nat,number_number_of(nat),V_1)),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),zero_zero(nat)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),N))) ).
fof(fact_4804_min__number__of__Suc,axiom,
! [N,V_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,suc,N)),hAPP(int,nat,number_number_of(nat),V_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),zero_zero(nat)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),N),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))))) ).
fof(fact_4805_min__less__iff__conj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z_1,X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y_1)) ) ) ) ).
fof(fact_4806_min__less__iff__disj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1)),Z_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),Z_1)) ) ) ) ).
fof(fact_4807_min__max_Oless__infI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),B),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)) ) ) ).
fof(fact_4808_min__max_Oless__infI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),A_1),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)) ) ) ).
fof(fact_4809_min__max_Oinf__assoc,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z)) ) ).
fof(fact_4810_min__max_Oinf_Oassoc,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),C)) ) ).
fof(fact_4811_min__max_Oinf__left__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z)) ) ).
fof(fact_4812_min__max_Oinf_Oleft__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),C)) ) ).
fof(fact_4813_min__max_Oinf__left__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) ) ).
fof(fact_4814_min__max_Oinf_Oleft__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B) ) ).
fof(fact_4815_min__max_Oinf__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),X) ) ).
fof(fact_4816_min__max_Oinf_Ocommute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),B),A_1) ) ).
fof(fact_4817_min__max_Oinf_Oidem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),A_1) = ti(X_a,A_1) ) ).
fof(fact_4818_min__Suc__Suc,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,suc,M_1)),hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),M_1),N_1)) ).
fof(fact_4819_min__0R,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),N_1),zero_zero(nat)) = zero_zero(nat) ).
fof(fact_4820_min__0L,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N_1) = zero_zero(nat) ).
fof(fact_4821_min__diff,axiom,
! [M_1,I_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),M_1),I_1)),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),I_1)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),M_1),N_1)),I_1) ).
fof(fact_4822_min__diff__distrib__left,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),Z)) ) ).
fof(fact_4823_min__add__distrib__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),Z)) ) ).
fof(fact_4824_Min_Oidem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),X) = ti(X_a,X) ) ).
fof(fact_4825_min__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ord(X_b) )
=> ! [U,V] :
( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(int,X_b,number_number_of(X_b),U) )
& ( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_min(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(int,X_b,number_number_of(X_b),V) ) ) ) ).
fof(fact_4826_min__le__iff__disj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1)),Z_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y_1),Z_1)) ) ) ) ).
fof(fact_4827_min__max_Ole__infE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,A_1,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)))
=> ~ ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B)) ) ) ) ).
fof(fact_4828_min__max_Oinf__mono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,D,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),C),D))) ) ) ) ).
fof(fact_4829_min__max_Oinf__greatest,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z,X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Z))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z))) ) ) ) ).
fof(fact_4830_min__max_Ole__infI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,X,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),A_1))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B))) ) ) ) ).
fof(fact_4831_min__max_Oinf__absorb2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = ti(X_a,Y) ) ) ).
fof(fact_4832_min__max_Oinf__absorb1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X),Y))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y) = ti(X_a,X) ) ) ).
fof(fact_4833_min__max_Ole__infI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)) ) ) ).
fof(fact_4834_min__max_Ole__infI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),A_1),B)),X)) ) ) ).
fof(fact_4835_min__max_Ole__inf__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y_1),Z_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Z_1)) ) ) ) ).
fof(fact_4836_min__max_Ole__iff__inf,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1))
<=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X_1),Y_1) = ti(X_a,X_1) ) ) ).
fof(fact_4837_min__max_Oinf__le2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),Y)) ) ).
fof(fact_4838_min__max_Oinf__le1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),X)) ) ).
fof(fact_4839_min__Suc2,axiom,
! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),M),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),zero_zero(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),ord_min(nat)),N))),M) ).
fof(fact_4840_min__Suc1,axiom,
! [N,M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_min(nat),hAPP(nat,nat,suc,N)),M) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),zero_zero(nat)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),ord_min(nat),N))),M) ).
fof(fact_4841_dual__max,axiom,
! [X_a] :
( linorder(X_a)
=> hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),max(X_a),hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,bool)),combc(X_a,X_a,bool),ord_less_eq(X_a))) = ord_min(X_a) ) ).
fof(fact_4842_min__leastL,axiom,
! [X_a] :
( ord(X_a)
=> ! [X,Least] :
( ! [X_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Least),X) = ti(X_a,Least) ) ) ).
fof(fact_4843_min__leastR,axiom,
! [X_a] :
( order(X_a)
=> ! [X,Least] :
( ! [X_2] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Least),X_2))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Least) = ti(X_a,Least) ) ) ).
fof(fact_4844_min__ord__min,axiom,
! [X_a] :
( ord(X_a)
=> ord_min(X_a) = hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),min_1(X_a),ord_less_eq(X_a)) ) ).
fof(fact_4845_vanishes__mult__bounded,axiom,
! [Y_3,X_3] :
( ? [A_3] :
( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A_3))
& ! [N_2] : hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_3,N_2))),A_3)) )
=> ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,Y_3))
=> hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),times_times(rat)),X_3)),Y_3))) ) ) ).
fof(fact_4846_SetInterval_Otransfer__int__nat__set__functions,axiom,
! [N,M] :
( hBOOL(hAPP(int,bool,nat_is_nat,M))
=> ( hBOOL(hAPP(int,bool,nat_is_nat,N))
=> hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),M),N) = hAPP(fun(nat,bool),fun(int,bool),hAPP(fun(nat,int),fun(fun(nat,bool),fun(int,bool)),image(nat,int),semiring_1_of_nat(int)),hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),hAPP(int,nat,nat_1,M)),hAPP(int,nat,nat_1,N))) ) ) ).
fof(fact_4847_is__nat__def,axiom,
! [X_1] :
( hBOOL(hAPP(int,bool,nat_is_nat,X_1))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X_1)) ) ).
fof(fact_4848_Nat__Transfer_Otransfer__int__nat__function__closures_I9_J,axiom,
! [Z] : hBOOL(hAPP(int,bool,nat_is_nat,hAPP(nat,int,semiring_1_of_nat(int),Z))) ).
fof(fact_4849_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_4850_Nat__Transfer_Otransfer__int__nat__function__closures_I6_J,axiom,
hBOOL(hAPP(int,bool,nat_is_nat,one_one(int))) ).
fof(fact_4851_Nat__Transfer_Otransfer__int__nat__function__closures_I5_J,axiom,
hBOOL(hAPP(int,bool,nat_is_nat,zero_zero(int))) ).
fof(fact_4852_Nat__Transfer_Otransfer__int__nat__set__function__closures_I6_J,axiom,
! [X_1,A_2] :
( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),A_2))
=> hBOOL(hAPP(int,bool,nat_is_nat,X_1)) ) ) ).
fof(fact_4853_Nat__Transfer_Otransfer__int__nat__function__closures_I4_J,axiom,
! [N_1,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_1))) ) ).
fof(fact_4854_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,hAPP(int,fun(int,int),div_mod(int),X),Y))) ) ) ).
fof(fact_4855_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_4856_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,hAPP(int,fun(int,int),nat_tsub,X),Y))) ) ) ).
fof(fact_4857_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_4858_vanishes__minus,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_3))
=> hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(rat,rat),fun(fun(nat,rat),fun(nat,rat)),combb(rat,rat,nat),uminus_uminus(rat)),X_3))) ) ).
fof(fact_4859_vanishes__const,axiom,
! [C_1] :
( hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(rat,fun(nat,rat),combk(rat,nat),C_1)))
<=> C_1 = zero_zero(rat) ) ).
fof(fact_4860_vanishes__diff,axiom,
! [Y_3,X_3] :
( hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_3))
=> ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,Y_3))
=> hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),minus_minus(rat)),X_3)),Y_3))) ) ) ).
fof(fact_4861_vanishes__add,axiom,
! [Y_3,X_3] :
( hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_3))
=> ( hBOOL(hAPP(fun(nat,rat),bool,vanishes,Y_3))
=> hBOOL(hAPP(fun(nat,rat),bool,vanishes,hAPP(fun(nat,rat),fun(nat,rat),hAPP(fun(nat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,rat)),combs(nat,rat,rat),hAPP(fun(nat,rat),fun(nat,fun(rat,rat)),hAPP(fun(rat,fun(rat,rat)),fun(fun(nat,rat),fun(nat,fun(rat,rat))),combb(rat,fun(rat,rat),nat),plus_plus(rat)),X_3)),Y_3))) ) ) ).
fof(fact_4862_SetInterval_Otransfer__int__nat__set__function__closures,axiom,
! [Y_1,X_1] :
( hBOOL(hAPP(int,bool,nat_is_nat,X_1))
=> hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),X_1),Y_1))) ) ).
fof(fact_4863_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_4864_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(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,hAPP(int,fun(int,int),gcd_gcd(int),X),Y))) ) ) ).
fof(fact_4865_transfer__int__nat__factorial__closure,axiom,
! [X] :
( hBOOL(hAPP(int,bool,nat_is_nat,X))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),hAPP(int,int,fact_fact(int),X))) ) ).
fof(fact_4866_transfer__int__nat__set__relations_I2_J,axiom,
! [A_2,X_1] :
( hBOOL(hAPP(int,bool,nat_is_nat,X_1))
=> ( hBOOL(hAPP(fun(int,bool),bool,nat_nat_set,A_2))
=> ( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_1),A_2))
<=> hBOOL(hAPP(fun(nat,bool),bool,hAPP(nat,fun(fun(nat,bool),bool),member(nat),hAPP(int,nat,nat_1,X_1)),hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),A_2))) ) ) ) ).
fof(fact_4867_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_4868_transfer__int__nat__sum__prod2_I2_J,axiom,
! [X_a,F,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(int,bool,nat_is_nat,hAPP(X_a,int,F,X_2))) )
=> hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co302546470etprod(X_a,int),F),A_2) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co302546470etprod(X_a,nat),hAPP(fun(X_a,int),fun(X_a,nat),hAPP(fun(int,nat),fun(fun(X_a,int),fun(X_a,nat)),combb(int,nat,X_a),nat_1),F)),A_2)) ) ).
fof(fact_4869_transfer__int__nat__sum__prod2_I1_J,axiom,
! [X_a,F,A_2] :
( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(int,bool,nat_is_nat,hAPP(X_a,int,F,X_2))) )
=> hAPP(fun(X_a,bool),int,hAPP(fun(X_a,int),fun(fun(X_a,bool),int),big_co1399186613setsum(X_a,int),F),A_2) = hAPP(nat,int,semiring_1_of_nat(int),hAPP(fun(X_a,bool),nat,hAPP(fun(X_a,nat),fun(fun(X_a,bool),nat),big_co1399186613setsum(X_a,nat),hAPP(fun(X_a,int),fun(X_a,nat),hAPP(fun(int,nat),fun(fun(X_a,int),fun(X_a,nat)),combb(int,nat,X_a),nat_1),F)),A_2)) ) ).
fof(fact_4870_vanishesD,axiom,
! [R_1,X_3] :
( hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_3))
=> ( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R_1))
=> ? [K_1] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_2))
=> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_3,N_2))),R_1)) ) ) ) ).
fof(fact_4871_vanishes__def,axiom,
! [X_3] :
( hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_3))
<=> ! [R_2] :
( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R_2))
=> ? [K_1] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_2))
=> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_3,N_2))),R_2)) ) ) ) ).
fof(fact_4872_vanishesI,axiom,
! [X_3] :
( ! [R_2] :
( hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R_2))
=> ? [K_1] :
! [N_2] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less_eq(nat),K_1),N_2))
=> hBOOL(hAPP(rat,bool,hAPP(rat,fun(rat,bool),ord_less(rat),hAPP(rat,rat,abs_abs(rat),hAPP(nat,rat,X_3,N_2))),R_2)) ) )
=> hBOOL(hAPP(fun(nat,rat),bool,vanishes,X_3)) ) ).
fof(fact_4873_EulerFermat_Oaux,axiom,
! [X_a,Q_1,P_1] :
( hBOOL(hAPP(X_a,bool,P_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1)))
=> ( hBOOL(hAPP(X_a,bool,Q_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),Q_1)))
=> ( hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1) = hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),Q_1)
=> ? [A_3] :
( hBOOL(hAPP(X_a,bool,P_1,A_3))
& hBOOL(hAPP(X_a,bool,Q_1,A_3)) ) ) ) ) ).
fof(fact_4874_someI2,axiom,
! [X_a,Q_1,P_1,Aa] :
( hBOOL(hAPP(X_a,bool,P_1,Aa))
=> ( ! [X_2] :
( hBOOL(hAPP(X_a,bool,P_1,X_2))
=> hBOOL(hAPP(X_a,bool,Q_1,X_2)) )
=> hBOOL(hAPP(X_a,bool,Q_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))) ) ) ).
fof(fact_4875_some1__equality,axiom,
! [X_a,Aa,P_1] :
( ? [X_2] :
( hBOOL(hAPP(X_a,bool,P_1,X_2))
& ! [Y_2] :
( hBOOL(hAPP(X_a,bool,P_1,Y_2))
=> ti(X_a,Y_2) = ti(X_a,X_2) ) )
=> ( hBOOL(hAPP(X_a,bool,P_1,Aa))
=> hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1) = ti(X_a,Aa) ) ) ).
fof(fact_4876_someI2__ex,axiom,
! [X_a,Q_1,P_1] :
( ? [X1] : hBOOL(hAPP(X_a,bool,P_1,X1))
=> ( ! [X_2] :
( hBOOL(hAPP(X_a,bool,P_1,X_2))
=> hBOOL(hAPP(X_a,bool,Q_1,X_2)) )
=> hBOOL(hAPP(X_a,bool,Q_1,hAPP(fun(X_a,bool),X_a,hilbert_Eps(X_a),P_1))) ) ) ).
fof(fact_4877_Powp__mono,axiom,
! [X_a,A_2,B_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),A_2),B_1))
=> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(fun(X_a,bool),bool),fun(fun(fun(X_a,bool),bool),bool),ord_less_eq(fun(fun(X_a,bool),bool)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),A_2)),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),B_1))) ) ).
fof(fact_4878_Powp__Pow__eq,axiom,
! [X_a,A_2,X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),powp(X_a),hAPP(fun(X_a,bool),fun(X_a,bool),hAPP(fun(X_a,fun(fun(X_a,bool),bool)),fun(fun(X_a,bool),fun(X_a,bool)),combc(X_a,fun(X_a,bool),bool),member(X_a)),A_2)),X_2))
<=> hBOOL(hAPP(fun(fun(X_a,bool),bool),bool,hAPP(fun(X_a,bool),fun(fun(fun(X_a,bool),bool),bool),member(fun(X_a,bool)),X_2),hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),pow(X_a),A_2))) ) ).
fof(fact_4879_zcongm__def,axiom,
! [M,X_2,Xa] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcongm,M),X_2),Xa))
<=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,X_2),Xa),M)) ) ).
fof(fact_4880_cppi,axiom,
! [A_2,P_2,P_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ( ? [Z_2] :
! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),Z_2),X_2))
=> ( hBOOL(hAPP(int,bool,P_1,X_2))
<=> hBOOL(hAPP(int,bool,P_2,X_2)) ) )
=> ( ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),A_2))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,P_1,X_2))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),X_2),D_2))) ) )
=> ( ! [X_2,K_1] :
( hBOOL(hAPP(int,bool,P_2,X_2))
<=> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_2)))) )
=> ( ? [X1] : hBOOL(hAPP(int,bool,P_1,X1))
<=> ( ? [X_2] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
& hBOOL(hAPP(int,bool,P_2,X_2)) )
| ? [X_2] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
& ? [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),A_2))
& hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),Xa),X_2))) ) ) ) ) ) ) ) ) ).
fof(fact_4881_bijzcong__zcong__prod,axiom,
! [A_2,B_1,M] :
( hBOOL(hAPP(fun(product_prod(fun(int,bool),fun(int,bool)),bool),bool,hAPP(product_prod(fun(int,bool),fun(int,bool)),fun(fun(product_prod(fun(int,bool),fun(int,bool)),bool),bool),member(product_prod(fun(int,bool),fun(int,bool))),hAPP(fun(int,bool),product_prod(fun(int,bool),fun(int,bool)),hAPP(fun(int,bool),fun(fun(int,bool),product_prod(fun(int,bool),fun(int,bool))),product_Pair(fun(int,bool),fun(int,bool)),A_2),B_1)),hAPP(fun(int,fun(int,bool)),fun(product_prod(fun(int,bool),fun(int,bool)),bool),bijR(int,int),hAPP(int,fun(int,fun(int,bool)),zcongm,M))))
=> hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),zcong,hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),A_2)),hAPP(fun(int,bool),int,hAPP(fun(int,int),fun(fun(int,bool),int),big_co302546470etprod(int,int),combi(int)),B_1)),M)) ) ).
fof(fact_4882_cpmi,axiom,
! [B_1,P_2,P_1,D_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),zero_zero(int)),D_2))
=> ( ? [Z_2] :
! [X_2] :
( hBOOL(hAPP(int,bool,hAPP(int,fun(int,bool),ord_less(int),X_2),Z_2))
=> ( hBOOL(hAPP(int,bool,P_1,X_2))
<=> hBOOL(hAPP(int,bool,P_2,X_2)) ) )
=> ( ! [X_2] :
( ! [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
=> ! [Xb] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xb),B_1))
=> X_2 != hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xb),Xa) ) )
=> ( hBOOL(hAPP(int,bool,P_1,X_2))
=> hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),D_2))) ) )
=> ( ! [X_2,K_1] :
( hBOOL(hAPP(int,bool,P_2,X_2))
<=> hBOOL(hAPP(int,bool,P_2,hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),X_2),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),K_1),D_2)))) )
=> ( ? [X1] : hBOOL(hAPP(int,bool,P_1,X1))
<=> ( ? [X_2] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
& hBOOL(hAPP(int,bool,P_2,X_2)) )
| ? [X_2] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),X_2),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),one_one(int)),D_2)))
& ? [Xa] :
( hBOOL(hAPP(fun(int,bool),bool,hAPP(int,fun(fun(int,bool),bool),member(int),Xa),B_1))
& hBOOL(hAPP(int,bool,P_1,hAPP(int,int,hAPP(int,fun(int,int),plus_plus(int),Xa),X_2))) ) ) ) ) ) ) ) ) ).
fof(fact_4883_fin__bijRl,axiom,
! [X_a,X_b,A_2,B_1,P_1] :
( hBOOL(hAPP(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool,hAPP(product_prod(fun(X_a,bool),fun(X_b,bool)),fun(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool),member(product_prod(fun(X_a,bool),fun(X_b,bool))),hAPP(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,bool),fun(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool))),product_Pair(fun(X_a,bool),fun(X_b,bool)),A_2),B_1)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bijR(X_a,X_b),P_1)))
=> hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2)) ) ).
fof(fact_4884_fin__bijRr,axiom,
! [X_a,X_b,A_2,B_1,P_1] :
( hBOOL(hAPP(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool,hAPP(product_prod(fun(X_a,bool),fun(X_b,bool)),fun(fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bool),member(product_prod(fun(X_a,bool),fun(X_b,bool))),hAPP(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool)),hAPP(fun(X_a,bool),fun(fun(X_b,bool),product_prod(fun(X_a,bool),fun(X_b,bool))),product_Pair(fun(X_a,bool),fun(X_b,bool)),A_2),B_1)),hAPP(fun(X_a,fun(X_b,bool)),fun(product_prod(fun(X_a,bool),fun(X_b,bool)),bool),bijR(X_a,X_b),P_1)))
=> hBOOL(hAPP(fun(X_b,bool),bool,finite_finite_1(X_b),B_1)) ) ).
fof(fact_4885_increasing__LIMSEQ,axiom,
! [L,F] :
( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_2)),hAPP(nat,real,F,hAPP(nat,nat,suc,N_2))))
=> ( ! [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(nat,real,F,N_2)),L))
=> ( ! [E] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
=> ? [N_2] : hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),L),hAPP(real,real,hAPP(real,fun(real,real),plus_plus(real),hAPP(nat,real,F,N_2)),E))) )
=> hBOOL(hAPP(filter(nat),bool,hAPP(real,fun(filter(nat),bool),hAPP(fun(nat,real),fun(real,fun(filter(nat),bool)),tendsto(nat,real),F),L),sequentially)) ) ) ) ).
fof(fact_4886_isCont__LIM__compose2,axiom,
! [X_c,X_b,X_a] :
( ( real_normed_vector(X_a)
& topological_space(X_b)
& topological_space(X_c) )
=> ! [G,L,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> ( hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),L),hAPP(X_b,filter(X_b),at(X_b),hAPP(X_a,X_b,F,Aa))))
=> ( ? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [X_2] :
( ( ti(X_a,X_2) != ti(X_a,Aa)
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,norm_norm(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X_2),Aa))),D_3)) )
=> hAPP(X_a,X_b,F,X_2) != hAPP(X_a,X_b,F,Aa) ) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),L),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ) ).
fof(fact_4887_isCont__LIM__compose,axiom,
! [X_a,X_b,X_c] :
( ( topological_space(X_c)
& topological_space(X_b)
& topological_space(X_a) )
=> ! [F,Aa,G,L] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),L))
=> ( hBOOL(hAPP(filter(X_c),bool,hAPP(X_a,fun(filter(X_c),bool),hAPP(fun(X_c,X_a),fun(X_a,fun(filter(X_c),bool)),tendsto(X_c,X_a),F),L),hAPP(X_c,filter(X_c),at(X_c),Aa)))
=> hBOOL(hAPP(filter(X_c),bool,hAPP(X_b,fun(filter(X_c),bool),hAPP(fun(X_c,X_b),fun(X_b,fun(filter(X_c),bool)),tendsto(X_c,X_b),hAPP(fun(X_c,X_a),fun(X_c,X_b),hAPP(fun(X_a,X_b),fun(fun(X_c,X_a),fun(X_c,X_b)),combb(X_a,X_b,X_c),G),F)),hAPP(X_a,X_b,G,L)),hAPP(X_c,filter(X_c),at(X_c),Aa))) ) ) ) ).
fof(fact_4888_isCont__def,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& topological_space(X_b) )
=> ! [F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),F),hAPP(X_a,X_b,F,Aa)),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ).
fof(fact_4889_CARAT__DERIV,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [F,X_1,L] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),L))
<=> ? [G_1] :
( ! [Z_2] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,F,Z_2)),hAPP(X_a,X_a,F,X_1)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),times_times(X_a),hAPP(X_a,X_a,G_1,Z_2)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Z_2),X_1))
& hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),G_1),X_1))
& hAPP(X_a,X_a,G_1,X_1) = ti(X_a,L) ) ) ) ).
fof(fact_4890_isCont__real__root,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,N)),X_1)) ) ).
fof(fact_4891_isCont__ln,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),ln),X_1)) ) ).
fof(fact_4892_isCont__abs,axiom,
! [Aa] : hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),abs_abs(real)),Aa)) ).
fof(fact_4893_isCont__sgn,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> ( hAPP(X_a,X_b,F,Aa) != zero_zero(X_b)
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),sgn_sgn(X_b)),F)),Aa)) ) ) ) ).
fof(fact_4894_isCont__inverse,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_n1866405975lgebra(X_b) )
=> ! [F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> ( hAPP(X_a,X_b,F,Aa) != zero_zero(X_b)
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),inverse_inverse(X_b)),F)),Aa)) ) ) ) ).
fof(fact_4895_isCont__rabs,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),F),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),hAPP(fun(X_a,real),fun(X_a,real),hAPP(fun(real,real),fun(fun(X_a,real),fun(X_a,real)),combb(real,real,X_a),abs_abs(real)),F)),Aa)) ) ) ).
fof(fact_4896_isCont__minus,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_b,X_b),fun(fun(X_a,X_b),fun(X_a,X_b)),combb(X_b,X_b,X_a),uminus_uminus(X_b)),F)),Aa)) ) ) ).
fof(fact_4897_mult__right_OisCont,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [X_1,Aa] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(X_a,fun(X_a,X_a),times_times(X_a),X_1)),Aa)) ) ).
fof(fact_4898_mult__left_OisCont,axiom,
! [X_a] :
( real_normed_algebra(X_a)
=> ! [Y_1,Aa] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),times_times(X_a)),Y_1)),Aa)) ) ).
fof(fact_4899_isCont__mult,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_algebra(X_b) )
=> ! [G,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),Aa)) ) ) ) ).
fof(fact_4900_mult_OisCont,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_algebra(X_b) )
=> ! [G,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),times_times(X_b)),F)),G)),Aa)) ) ) ) ).
fof(fact_4901_isCont__norm,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),hAPP(fun(X_a,X_b),fun(X_a,real),hAPP(fun(X_b,real),fun(fun(X_a,X_b),fun(X_a,real)),combb(X_b,real,X_a),norm_norm(X_b)),F)),Aa)) ) ) ).
fof(fact_4902_isCont__const,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& topological_space(X_b) )
=> ! [K,Aa] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(X_b,fun(X_a,X_b),combk(X_b,X_a),K)),Aa)) ) ).
fof(fact_4903_isCont__ident,axiom,
! [X_a] :
( topological_space(X_a)
=> ! [Aa] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),combi(X_a)),Aa)) ) ).
fof(fact_4904_isCont__o2,axiom,
! [X_b,X_c,X_a] :
( ( topological_space(X_a)
& topological_space(X_c)
& topological_space(X_b) )
=> ! [G,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> ( hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,X_c),fun(X_b,bool),isCont(X_b,X_c),G),hAPP(X_a,X_b,F,Aa)))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_c),fun(X_a,bool),isCont(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),Aa)) ) ) ) ).
fof(fact_4905_isCont__power,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& power(X_b)
& real_normed_algebra(X_b) )
=> ! [N,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(nat,fun(X_a,X_b),hAPP(fun(X_a,fun(nat,X_b)),fun(nat,fun(X_a,X_b)),combc(X_a,nat,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(nat,X_b)),hAPP(fun(X_b,fun(nat,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(nat,X_b))),combb(X_b,fun(nat,X_b),X_a),power_power(X_b)),F)),N)),Aa)) ) ) ).
fof(fact_4906_of__real_OisCont,axiom,
! [X_a] :
( ( real_algebra_1(X_a)
& real_normed_vector(X_a) )
=> ! [Aa] : hBOOL(hAPP(real,bool,hAPP(fun(real,X_a),fun(real,bool),isCont(real,X_a),of_real(X_a)),Aa)) ) ).
fof(fact_4907_isCont__of__real,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_n2089651433ebra_1(X_b) )
=> ! [F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),F),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,real),fun(X_a,X_b),hAPP(fun(real,X_b),fun(fun(X_a,real),fun(X_a,X_b)),combb(real,X_b,X_a),of_real(X_b)),F)),Aa)) ) ) ).
fof(fact_4908_divide_OisCont,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [Y_1,Aa] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(X_a,fun(X_a,X_a),hAPP(fun(X_a,fun(X_a,X_a)),fun(X_a,fun(X_a,X_a)),combc(X_a,X_a,X_a),inverse_divide(X_a)),Y_1)),Aa)) ) ).
fof(fact_4909_isCont__diff,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [G,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),minus_minus(X_b)),F)),G)),Aa)) ) ) ) ).
fof(fact_4910_isCont__add,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [G,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,X_b),fun(X_a,fun(X_b,X_b)),hAPP(fun(X_b,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,fun(X_b,X_b))),combb(X_b,fun(X_b,X_b),X_a),plus_plus(X_b)),F)),G)),Aa)) ) ) ) ).
fof(fact_4911_Im_OisCont,axiom,
! [Aa] : hBOOL(hAPP(complex,bool,hAPP(fun(complex,real),fun(complex,bool),isCont(complex,real),im),Aa)) ).
fof(fact_4912_Re_OisCont,axiom,
! [Aa] : hBOOL(hAPP(complex,bool,hAPP(fun(complex,real),fun(complex,bool),isCont(complex,real),re),Aa)) ).
fof(fact_4913_cnj_OisCont,axiom,
! [Aa] : hBOOL(hAPP(complex,bool,hAPP(fun(complex,complex),fun(complex,bool),isCont(complex,complex),cnj),Aa)) ).
fof(fact_4914_DERIV__isCont,axiom,
! [X_a] :
( real_normed_field(X_a)
=> ! [F,X_1,D_2] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),hAPP(fun(X_a,X_a),fun(X_a,fun(X_a,bool)),deriv(X_a),F),X_1),D_2))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),F),X_1)) ) ) ).
fof(fact_4915_isCont__real__sqrt,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),sqrt),X_1)) ).
fof(fact_4916_isCont__cos,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),cos),X_1)) ).
fof(fact_4917_isCont__arctan,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),arctan),X_1)) ).
fof(fact_4918_isCont__sin,axiom,
! [X_1] : hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),sin),X_1)) ).
fof(fact_4919_isCont__exp,axiom,
! [X_a] :
( ( real_normed_field(X_a)
& banach(X_a) )
=> ! [X_1] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),exp(X_a)),X_1)) ) ).
fof(fact_4920_isCont__tan,axiom,
! [X_1] :
( hAPP(real,real,cos,X_1) != zero_zero(real)
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),tan),X_1)) ) ).
fof(fact_4921_bounded__bilinear_OisCont,axiom,
! [X_b,X_c,X_a,X_d] :
( ( topological_space(X_d)
& real_normed_vector(X_a)
& real_normed_vector(X_c)
& real_normed_vector(X_b) )
=> ! [G,F,Aa,Prod] :
( hBOOL(hAPP(fun(X_a,fun(X_b,X_c)),bool,bounded_bilinear(X_a,X_b,X_c),Prod))
=> ( hBOOL(hAPP(X_d,bool,hAPP(fun(X_d,X_a),fun(X_d,bool),isCont(X_d,X_a),F),Aa))
=> ( hBOOL(hAPP(X_d,bool,hAPP(fun(X_d,X_b),fun(X_d,bool),isCont(X_d,X_b),G),Aa))
=> hBOOL(hAPP(X_d,bool,hAPP(fun(X_d,X_c),fun(X_d,bool),isCont(X_d,X_c),hAPP(fun(X_d,X_b),fun(X_d,X_c),hAPP(fun(X_d,fun(X_b,X_c)),fun(fun(X_d,X_b),fun(X_d,X_c)),combs(X_d,X_b,X_c),hAPP(fun(X_d,X_a),fun(X_d,fun(X_b,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_d,X_a),fun(X_d,fun(X_b,X_c))),combb(X_a,fun(X_b,X_c),X_d),Prod),F)),G)),Aa)) ) ) ) ) ).
fof(fact_4922_scaleR__right_OisCont,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [R_1,Aa] : hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_a),fun(X_a,bool),isCont(X_a,X_a),hAPP(real,fun(X_a,X_a),scaleR_scaleR(X_a),R_1)),Aa)) ) ).
fof(fact_4923_scaleR__left_OisCont,axiom,
! [X_a] :
( real_normed_vector(X_a)
=> ! [X_1,Aa] : hBOOL(hAPP(real,bool,hAPP(fun(real,X_a),fun(real,bool),isCont(real,X_a),hAPP(X_a,fun(real,X_a),hAPP(fun(real,fun(X_a,X_a)),fun(X_a,fun(real,X_a)),combc(real,X_a,X_a),scaleR_scaleR(X_a)),X_1)),Aa)) ) ).
fof(fact_4924_isCont__scaleR,axiom,
! [X_b,X_a] :
( ( topological_space(X_a)
& real_normed_vector(X_b) )
=> ! [G,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,real),fun(X_a,bool),isCont(X_a,real),F),Aa))
=> ( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),G),Aa))
=> hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),hAPP(fun(X_a,X_b),fun(X_a,X_b),hAPP(fun(X_a,fun(X_b,X_b)),fun(fun(X_a,X_b),fun(X_a,X_b)),combs(X_a,X_b,X_b),hAPP(fun(X_a,real),fun(X_a,fun(X_b,X_b)),hAPP(fun(real,fun(X_b,X_b)),fun(fun(X_a,real),fun(X_a,fun(X_b,X_b))),combb(real,fun(X_b,X_b),X_a),scaleR_scaleR(X_b)),F)),G)),Aa)) ) ) ) ).
fof(fact_4925_isCont__root__zero,axiom,
! [N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,N)),zero_zero(real))) ) ).
fof(fact_4926_isCont__root__neg,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),zero_zero(real)))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,N)),X_1)) ) ) ).
fof(fact_4927_isCont__root__pos,axiom,
! [X_1,N] :
( hBOOL(hAPP(nat,bool,hAPP(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),X_1))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),hAPP(nat,fun(real,real),root,N)),X_1)) ) ) ).
fof(fact_4928_isCont__iff,axiom,
! [X_b,X_a] :
( ( real_normed_vector(X_a)
& topological_space(X_b) )
=> ! [F,X_1] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),X_1))
<=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_b,fun(filter(X_a),bool),hAPP(fun(X_a,X_b),fun(X_b,fun(filter(X_a),bool)),tendsto(X_a,X_b),hAPP(fun(X_a,X_a),fun(X_a,X_b),hAPP(fun(X_a,X_b),fun(fun(X_a,X_a),fun(X_a,X_b)),combb(X_a,X_b,X_a),F),hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X_1))),hAPP(X_a,X_b,F,X_1)),hAPP(X_a,filter(X_a),at(X_a),zero_zero(X_a)))) ) ) ).
fof(fact_4929_isCont__arccos,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),arccos),X_1)) ) ) ).
fof(fact_4930_isCont__arcsin,axiom,
! [X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(int,real,number_number_of(real),min)),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),one_one(real)))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),arcsin),X_1)) ) ) ).
fof(fact_4931_metric__isCont__LIM__compose2,axiom,
! [X_c,X_b,X_a] :
( ( metric_space(X_a)
& topological_space(X_b)
& topological_space(X_c) )
=> ! [G,L,F,Aa] :
( hBOOL(hAPP(X_a,bool,hAPP(fun(X_a,X_b),fun(X_a,bool),isCont(X_a,X_b),F),Aa))
=> ( hBOOL(hAPP(filter(X_b),bool,hAPP(X_c,fun(filter(X_b),bool),hAPP(fun(X_b,X_c),fun(X_c,fun(filter(X_b),bool)),tendsto(X_b,X_c),G),L),hAPP(X_b,filter(X_b),at(X_b),hAPP(X_a,X_b,F,Aa))))
=> ( ? [D_3] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_3))
& ! [X_2] :
( ( ti(X_a,X_2) != ti(X_a,Aa)
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(X_a,real,hAPP(X_a,fun(X_a,real),dist_dist(X_a),X_2),Aa)),D_3)) )
=> hAPP(X_a,X_b,F,X_2) != hAPP(X_a,X_b,F,Aa) ) )
=> hBOOL(hAPP(filter(X_a),bool,hAPP(X_c,fun(filter(X_a),bool),hAPP(fun(X_a,X_c),fun(X_c,fun(filter(X_a),bool)),tendsto(X_a,X_c),hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),G),F)),L),hAPP(X_a,filter(X_a),at(X_a),Aa))) ) ) ) ) ).
fof(fact_4932_LIM__less__bound,axiom,
! [F,Ba,X_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Ba),X_1))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(real,bool),bool,hAPP(real,fun(fun(real,bool),bool),member(real),X_2),hAPP(real,fun(real,bool),hAPP(real,fun(real,fun(real,bool)),ord_gr676655987ssThan(real),Ba),X_1)))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,F,X_2))) )
=> ( hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),X_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),hAPP(real,real,F,X_1))) ) ) ) ).
fof(fact_4933_DERIV__inverse__function,axiom,
! [Ba,Aa,F,G,X_1,D_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),hAPP(real,real,G,X_1)),D_2))
=> ( D_2 != zero_zero(real)
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_1),Ba))
=> ( ! [Y_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),Y_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Y_2),Ba)) )
=> hAPP(real,real,F,hAPP(real,real,G,Y_2)) = Y_2 )
=> ( hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),G),X_1))
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),G),X_1),hAPP(real,real,inverse_inverse(real),D_2))) ) ) ) ) ) ) ).
fof(fact_4934_isCont__setsum,axiom,
! [X_a,X_c,X_b] :
( ( topological_space(X_b)
& real_normed_vector(X_c) )
=> ! [F,X_1,A_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,finite_finite_1(X_a),A_2))
=> ( ! [X_2] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),X_2),A_2))
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,X_c),fun(X_b,bool),isCont(X_b,X_c),hAPP(X_a,fun(X_b,X_c),F,X_2)),X_1)) )
=> hBOOL(hAPP(X_b,bool,hAPP(fun(X_b,X_c),fun(X_b,bool),isCont(X_b,X_c),hAPP(fun(X_a,bool),fun(X_b,X_c),hAPP(fun(X_b,fun(fun(X_a,bool),X_c)),fun(fun(X_a,bool),fun(X_b,X_c)),combc(X_b,fun(X_a,bool),X_c),hAPP(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c)),hAPP(fun(fun(X_a,X_c),fun(fun(X_a,bool),X_c)),fun(fun(X_b,fun(X_a,X_c)),fun(X_b,fun(fun(X_a,bool),X_c))),combb(fun(X_a,X_c),fun(fun(X_a,bool),X_c),X_b),big_co1399186613setsum(X_a,X_c)),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),F))),A_2)),X_1)) ) ) ) ).
fof(fact_4935_isCont__inv__fun,axiom,
! [G,F,X_1,D_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
=> ( ! [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),X_1))),D_1))
=> hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2 )
=> ( ! [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),X_1))),D_1))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)) )
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),G),hAPP(real,real,F,X_1))) ) ) ) ).
fof(fact_4936_DERIV__isconst2,axiom,
! [X_1,F,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),Ba))
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Ba)) )
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),X_2)) )
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Ba)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),zero_zero(real))) )
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_1))
=> ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_1),Ba))
=> hAPP(real,real,F,X_1) = hAPP(real,real,F,Aa) ) ) ) ) ) ).
fof(fact_4937_DERIV__isconst__end,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),Ba))
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Ba)) )
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),X_2)) )
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Ba)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),zero_zero(real))) )
=> hAPP(real,real,F,Ba) = hAPP(real,real,F,Aa) ) ) ) ).
fof(fact_4938_DERIV__isconst1,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),Ba))
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Ba)) )
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),X_2)) )
=> ( ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),X_2),Ba)) )
=> hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),hAPP(fun(real,real),fun(real,fun(real,bool)),deriv(real),F),X_2),zero_zero(real))) )
=> ! [X_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),Aa),X_2))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),X_2),Ba)) )
=> hAPP(real,real,F,X_2) = hAPP(real,real,F,Aa) ) ) ) ) ).
fof(fact_4939_lemma__isCont__inj2,axiom,
! [G,F,X_1,D_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
=> ( ! [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),X_1))),D_1))
=> hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2 )
=> ( ! [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),X_1))),D_1))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)) )
=> ? [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),X_1))),D_1))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,Z_2)),hAPP(real,real,F,X_1))) ) ) ) ) ).
fof(fact_4940_isCont__inv__fun__inv,axiom,
! [G,F,X_1,D_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
=> ( ! [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),X_1))),D_1))
=> hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2 )
=> ( ! [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),X_1))),D_1))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)) )
=> ? [E] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
& ! [Y_2] :
( ( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Y_2),hAPP(real,real,F,X_1)))))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Y_2),hAPP(real,real,F,X_1)))),E)) )
=> hAPP(real,real,F,hAPP(real,real,G,Y_2)) = Y_2 ) ) ) ) ) ).
fof(fact_4941_lemma__isCont__inj,axiom,
! [G,F,X_1,D_1] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),zero_zero(real)),D_1))
=> ( ! [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),X_1))),D_1))
=> hAPP(real,real,G,hAPP(real,real,F,Z_2)) = Z_2 )
=> ( ! [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),X_1))),D_1))
=> hBOOL(hAPP(real,bool,hAPP(fun(real,real),fun(real,bool),isCont(real,real),F),Z_2)) )
=> ? [Z_2] :
( hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less_eq(real),hAPP(real,real,abs_abs(real),hAPP(real,real,hAPP(real,fun(real,real),minus_minus(real),Z_2),X_1))),D_1))
& hBOOL(hAPP(real,bool,hAPP(real,fun(real,bool),ord_less(real),hAPP(real,real,F,X_1)),hAPP(real,real,F,Z_2))) ) ) ) ) ).
fof(fact_4942_SetInterval_Otransfer__nat__int__set__functions_I1_J,axiom,
! [N] : hAPP(nat,fun(nat,bool),ord_atMost(nat),N) = hAPP(fun(int,bool),fun(nat,bool),hAPP(fun(int,nat),fun(fun(int,bool),fun(nat,bool)),image(int,nat),nat_1),hAPP(int,fun(int,bool),hAPP(int,fun(int,fun(int,bool)),ord_atLeastAtMost(int),zero_zero(int)),hAPP(nat,int,semiring_1_of_nat(int),N))) ).
fof(fact_4943_max__Suc__number__of,axiom,
! [V_1,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(int,nat,number_number_of(nat),V_1)),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),hAPP(bool,fun(nat,fun(nat,nat)),if(nat),hAPP(int,bool,nat_neg,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),hAPP(nat,nat,suc,N)),hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(int,nat,nat_1,hAPP(int,int,number_number_of(int),hAPP(int,int,pred,V_1)))),N))) ).
fof(fact_4944_finite__atMost,axiom,
! [K] : hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),hAPP(nat,fun(nat,bool),ord_atMost(nat),K))) ).
fof(fact_4945_minus__min__eq__max,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) ) ).
fof(fact_4946_minus__max__eq__min,axiom,
! [X_a] :
( linord219039673up_add(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,uminus_uminus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,uminus_uminus(X_a),X)),hAPP(X_a,X_a,uminus_uminus(X_a),Y)) ) ).
fof(fact_4947_min__max_Oinf__sup__distrib2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,Z,X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Z),X)) ) ).
fof(fact_4948_min__max_Osup__inf__distrib2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,Z,X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z)),X) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),X)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Z),X)) ) ).
fof(fact_4949_min__max_Oinf__sup__distrib1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z)) ) ).
fof(fact_4950_min__max_Osup__inf__distrib1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z)) ) ).
fof(fact_4951_min__max_Oinf__sup__absorb,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)) = ti(X_a,X) ) ).
fof(fact_4952_min__max_Osup__inf__absorb,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)) = ti(X_a,X) ) ).
fof(fact_4953_min__max_Odistrib__inf__le,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),Z))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)))) ) ).
fof(fact_4954_min__max_Odistrib__sup__le,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),Y),Z))),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_min(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z)))) ) ).
fof(fact_4955_max__ord__max,axiom,
! [X_a] :
( ord(X_a)
=> ord_max(X_a) = hAPP(fun(X_a,fun(X_a,bool)),fun(X_a,fun(X_a,X_a)),max(X_a),ord_less_eq(X_a)) ) ).
fof(fact_4956_atLeast0AtMost,axiom,
! [N] : hAPP(nat,fun(nat,bool),hAPP(nat,fun(nat,fun(nat,bool)),ord_atLeastAtMost(nat),zero_zero(nat)),N) = hAPP(nat,fun(nat,bool),ord_atMost(nat),N) ).
fof(fact_4957_less__max__iff__disj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z_1,X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_1),Y_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),X_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Z_1),Y_1)) ) ) ) ).
fof(fact_4958_max__less__iff__conj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1,Z_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_1),Y_1)),Z_1))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X_1),Z_1))
& hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),Y_1),Z_1)) ) ) ) ).
fof(fact_4959_min__max_Oless__supI2,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,X,B] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),B))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B))) ) ) ).
fof(fact_4960_min__max_Oless__supI1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,X,A_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),A_1))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B))) ) ) ).
fof(fact_4961_card__atMost,axiom,
! [U_1] : hAPP(fun(nat,bool),nat,finite_card(nat),hAPP(nat,fun(nat,bool),ord_atMost(nat),U_1)) = hAPP(nat,nat,suc,U_1) ).
fof(fact_4962_min__max_Osup__assoc,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)) ) ).
fof(fact_4963_min__max_Osup_Oassoc,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B)),C) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B),C)) ) ).
fof(fact_4964_min__max_Osup__left__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Z)) ) ).
fof(fact_4965_min__max_Osup_Oleft__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,C] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),C)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B),C)) ) ).
fof(fact_4966_min__max_Osup__left__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y) ) ).
fof(fact_4967_min__max_Osup_Oleft__idem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B)) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B) ) ).
fof(fact_4968_min__max_Osup__commute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),X) ) ).
fof(fact_4969_min__max_Osup_Ocommute,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),B),A_1) ) ).
fof(fact_4970_min__max_Osup_Oidem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),A_1) = ti(X_a,A_1) ) ).
fof(fact_4971_max__Suc__Suc,axiom,
! [M_1,N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,suc,M_1)),hAPP(nat,nat,suc,N_1)) = hAPP(nat,nat,suc,hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),M_1),N_1)) ).
fof(fact_4972_atMost__eq__iff,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X_1,Y_1] :
( hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),X_1) = hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),Y_1)
<=> ti(X_a,X_1) = ti(X_a,Y_1) ) ) ).
fof(fact_4973_max__0R,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),N_1),zero_zero(nat)) = N_1 ).
fof(fact_4974_max__0L,axiom,
! [N_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N_1) = N_1 ).
fof(fact_4975_nat__minus__add__max,axiom,
! [N_1,M_1] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),plus_plus(nat),hAPP(nat,nat,hAPP(nat,fun(nat,nat),minus_minus(nat),N_1),M_1)),M_1) = hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),N_1),M_1) ).
fof(fact_4976_max__diff__distrib__left,axiom,
! [X_a] :
( ordered_ab_group_add(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),minus_minus(X_a),Y),Z)) ) ).
fof(fact_4977_max__add__distrib__left,axiom,
! [X_a] :
( ordere236663937imp_le(X_a)
=> ! [X,Y,Z] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y)),Z) = hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),X),Z)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),plus_plus(X_a),Y),Z)) ) ).
fof(fact_4978_Max_Oidem,axiom,
! [X_a] :
( linorder(X_a)
=> ! [X] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),X) = ti(X_a,X) ) ).
fof(fact_4979_max__Suc1,axiom,
! [N,M] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),hAPP(nat,nat,suc,N)),M) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),hAPP(nat,nat,suc,N)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),ord_max(nat),N))),M) ).
fof(fact_4980_max__Suc2,axiom,
! [M,N] : hAPP(nat,nat,hAPP(nat,fun(nat,nat),ord_max(nat),M),hAPP(nat,nat,suc,N)) = hAPP(nat,nat,hAPP(fun(nat,nat),fun(nat,nat),hAPP(nat,fun(fun(nat,nat),fun(nat,nat)),nat_case(nat),hAPP(nat,nat,suc,N)),hAPP(fun(nat,nat),fun(nat,nat),hAPP(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),combb(nat,nat,nat),suc),hAPP(nat,fun(nat,nat),hAPP(fun(nat,fun(nat,nat)),fun(nat,fun(nat,nat)),combc(nat,nat,nat),ord_max(nat)),N))),M) ).
fof(fact_4981_lessThan__Suc__atMost,axiom,
! [K] : hAPP(nat,fun(nat,bool),ord_lessThan(nat),hAPP(nat,nat,suc,K)) = hAPP(nat,fun(nat,bool),ord_atMost(nat),K) ).
fof(fact_4982_finite__nat__iff__bounded__le,axiom,
! [S] :
( hBOOL(hAPP(fun(nat,bool),bool,finite_finite_1(nat),S))
<=> ? [K_1] : hBOOL(hAPP(fun(nat,bool),bool,hAPP(fun(nat,bool),fun(fun(nat,bool),bool),ord_less_eq(fun(nat,bool)),S),hAPP(nat,fun(nat,bool),ord_atMost(nat),K_1))) ) ).
fof(fact_4983_atMost__subset__iff,axiom,
! [X_a] :
( order(X_a)
=> ! [X_1,Y_1] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(fun(X_a,bool),fun(fun(X_a,bool),bool),ord_less_eq(fun(X_a,bool)),hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),X_1)),hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),Y_1)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),X_1),Y_1)) ) ) ).
fof(fact_4984_max__number__of,axiom,
! [X_b] :
( ( number(X_b)
& ord(X_b) )
=> ! [U,V] :
( ( hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(int,X_b,number_number_of(X_b),V) )
& ( ~ hBOOL(hAPP(X_b,bool,hAPP(X_b,fun(X_b,bool),ord_less_eq(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)))
=> hAPP(X_b,X_b,hAPP(X_b,fun(X_b,X_b),ord_max(X_b),hAPP(int,X_b,number_number_of(X_b),U)),hAPP(int,X_b,number_number_of(X_b),V)) = hAPP(int,X_b,number_number_of(X_b),U) ) ) ) ).
fof(fact_4985_atMost__iff,axiom,
! [X_a] :
( ord(X_a)
=> ! [I,K] :
( hBOOL(hAPP(fun(X_a,bool),bool,hAPP(X_a,fun(fun(X_a,bool),bool),member(X_a),I),hAPP(X_a,fun(X_a,bool),ord_atMost(X_a),K)))
<=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),I),K)) ) ) ).
fof(fact_4986_le__max__iff__disj,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z_1,X_1,Y_1] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X_1),Y_1)))
<=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),X_1))
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z_1),Y_1)) ) ) ) ).
fof(fact_4987_min__max_Ole__supE,axiom,
! [X_a] :
( linorder(X_a)
=> ! [A_1,B,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B)),X))
=> ~ ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),X))
=> ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),X)) ) ) ) ).
fof(fact_4988_min__max_Osup__mono,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,D,A_1,C] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),C))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),D))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B)),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),C),D))) ) ) ) ).
fof(fact_4989_min__max_Osup__least,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Z,Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Z),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),Y),Z)),X)) ) ) ) ).
fof(fact_4990_min__max_Ole__supI,axiom,
! [X_a] :
( linorder(X_a)
=> ! [B,A_1,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),A_1),X))
=> ( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),B),X))
=> hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),A_1),B)),X)) ) ) ) ).
fof(fact_4991_min__max_Osup__absorb1,axiom,
! [X_a] :
( linorder(X_a)
=> ! [Y,X] :
( hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),ord_less_eq(X_a),Y),X))
=> hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),ord_max(X_a),X),Y) = ti(X_a,X) ) ) ).
%----Arities (446)
fof(arity_fun___Complete__Lattice_Ocomplete__lattice,axiom,
! [T_2,T_1] :
( comple44059174attice(T_1)
=> comple44059174attice(fun(T_2,T_1)) ) ).
fof(arity_fun___Lattices_Oboolean__algebra,axiom,
! [T_2,T_1] :
( boolean_algebra(T_1)
=> boolean_algebra(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Opreorder,axiom,
! [T_2,T_1] :
( preorder(T_1)
=> preorder(fun(T_2,T_1)) ) ).
fof(arity_fun___Finite__Set_Ofinite,axiom,
! [T_2,T_1] :
( ( finite_finite(T_1)
& finite_finite(T_2) )
=> finite_finite(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Oorder,axiom,
! [T_2,T_1] :
( order(T_1)
=> order(fun(T_2,T_1)) ) ).
fof(arity_fun___Orderings_Oord,axiom,
! [T_2,T_1] :
( ord(T_1)
=> ord(fun(T_2,T_1)) ) ).
fof(arity_fun___Groups_Ouminus,axiom,
! [T_2,T_1] :
( uminus(T_1)
=> uminus(fun(T_2,T_1)) ) ).
fof(arity_fun___Groups_Ominus,axiom,
! [T_2,T_1] :
( minus(T_1)
=> minus(fun(T_2,T_1)) ) ).
fof(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(int) ).
fof(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
fof(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord626643107strict(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs(int) ).
fof(arity_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(int) ).
fof(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
fof(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
fof(arity_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(int) ).
fof(arity_Int_Oint___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(int) ).
fof(arity_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(int) ).
fof(arity_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord1278240602ring_1(int) ).
fof(arity_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
fof(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
fof(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
fof(arity_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
fof(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
fof(arity_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring(int) ).
fof(arity_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs(int) ).
fof(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(int) ).
fof(arity_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(int) ).
fof(arity_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring(int) ).
fof(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
fof(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
fof(arity_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(int) ).
fof(arity_Int_Oint___Divides_Osemiring__div,axiom,
semiring_div(int) ).
fof(arity_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring(int) ).
fof(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
fof(arity_Int_Oint___Int_Onumber__semiring,axiom,
number_semiring(int) ).
fof(arity_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add(int) ).
fof(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
fof(arity_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring(int) ).
fof(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
fof(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
fof(arity_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
fof(arity_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1(int) ).
fof(arity_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(int) ).
fof(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
fof(arity_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(int) ).
fof(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
fof(arity_Int_Oint___Divides_Oring__div,axiom,
ring_div(int) ).
fof(arity_Int_Oint___Rings_Omult__zero,axiom,
mult_zero(int) ).
fof(arity_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring(int) ).
fof(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
fof(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
fof(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
fof(arity_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
fof(arity_Int_Oint___Orderings_Oord,axiom,
ord(int) ).
fof(arity_Int_Oint___Groups_Ouminus,axiom,
uminus(int) ).
fof(arity_Int_Oint___Groups_Osgn__if,axiom,
sgn_if(int) ).
fof(arity_Int_Oint___Groups_Oabs__if,axiom,
abs_if(int) ).
fof(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
fof(arity_Int_Oint___Groups_Ominus,axiom,
minus(int) ).
fof(arity_Int_Oint___Power_Opower,axiom,
power(int) ).
fof(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
fof(arity_Int_Oint___Groups_Oplus,axiom,
plus(int) ).
fof(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
fof(arity_Int_Oint___Rings_Oidom,axiom,
idom(int) ).
fof(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
fof(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
fof(arity_Int_Oint___Rings_Odvd,axiom,
dvd(int) ).
fof(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(nat) ).
fof(arity_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(nat) ).
fof(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
fof(arity_Nat_Onat___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(nat) ).
fof(arity_Nat_Onat___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(nat) ).
fof(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semiring,axiom,
linordered_semiring(nat) ).
fof(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
fof(arity_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(nat) ).
fof(arity_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(nat) ).
fof(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
fof(arity_Nat_Onat___Rings_Oordered__semiring,axiom,
ordered_semiring(nat) ).
fof(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(nat) ).
fof(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
fof(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
fof(arity_Nat_Onat___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(nat) ).
fof(arity_Nat_Onat___Divides_Osemiring__div,axiom,
semiring_div(nat) ).
fof(arity_Nat_Onat___Rings_Ocomm__semiring,axiom,
comm_semiring(nat) ).
fof(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
fof(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
fof(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
fof(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
fof(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
fof(arity_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult(nat) ).
fof(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring__0,axiom,
semiring_0(nat) ).
fof(arity_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero(nat) ).
fof(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
fof(arity_Nat_Onat___Rings_Osemiring,axiom,
semiring(nat) ).
fof(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
fof(arity_Nat_Onat___Groups_Ominus,axiom,
minus(nat) ).
fof(arity_Nat_Onat___Power_Opower,axiom,
power(nat) ).
fof(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
fof(arity_Nat_Onat___Groups_Oplus,axiom,
plus(nat) ).
fof(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
fof(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
fof(arity_Nat_Onat___Rings_Odvd,axiom,
dvd(nat) ).
fof(arity_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduc,axiom,
semiri456707255roduct(rat) ).
fof(arity_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(rat) ).
fof(arity_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(rat) ).
fof(arity_Rat_Orat___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(rat) ).
fof(arity_Rat_Orat___Fields_Olinordered__field__inverse__zero,axiom,
linord1117847801e_zero(rat) ).
fof(arity_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
archim1804426504_field(rat) ).
fof(arity_Rat_Orat___Rings_Olinordered__semiring__1__strict,axiom,
linord626643107strict(rat) ).
fof(arity_Rat_Orat___Fields_Odivision__ring__inverse__zero,axiom,
divisi14063676e_zero(rat) ).
fof(arity_Rat_Orat___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(rat) ).
fof(arity_Rat_Orat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(rat) ).
fof(arity_Rat_Orat___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs(rat) ).
fof(arity_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim1727834104eiling(rat) ).
fof(arity_Rat_Orat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(rat) ).
fof(arity_Rat_Orat___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(rat) ).
fof(arity_Rat_Orat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(rat) ).
fof(arity_Rat_Orat___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(rat) ).
fof(arity_Rat_Orat___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(rat) ).
fof(arity_Rat_Orat___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(rat) ).
fof(arity_Rat_Orat___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(rat) ).
fof(arity_Rat_Orat___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(rat) ).
fof(arity_Rat_Orat___Rings_Olinordered__semiring__1,axiom,
linord1278240602ring_1(rat) ).
fof(arity_Rat_Orat___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(rat) ).
fof(arity_Rat_Orat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(rat) ).
fof(arity_Rat_Orat___Rings_Olinordered__semiring,axiom,
linordered_semiring(rat) ).
fof(arity_Rat_Orat___Fields_Ofield__inverse__zero,axiom,
field_inverse_zero(rat) ).
fof(arity_Rat_Orat___Rings_Olinordered__semidom,axiom,
linordered_semidom(rat) ).
fof(arity_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder(rat) ).
fof(arity_Rat_Orat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(rat) ).
fof(arity_Rat_Orat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(rat) ).
fof(arity_Rat_Orat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(rat) ).
fof(arity_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field(rat) ).
fof(arity_Rat_Orat___Rings_Oordered__semiring,axiom,
ordered_semiring(rat) ).
fof(arity_Rat_Orat___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs(rat) ).
fof(arity_Rat_Orat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(rat) ).
fof(arity_Rat_Orat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(rat) ).
fof(arity_Rat_Orat___Rings_Olinordered__ring,axiom,
linordered_ring(rat) ).
fof(arity_Rat_Orat___Rings_Olinordered__idom,axiom,
linordered_idom(rat) ).
fof(arity_Rat_Orat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(rat) ).
fof(arity_Rat_Orat___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(rat) ).
fof(arity_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring(rat) ).
fof(arity_Rat_Orat___Rings_Ocomm__semiring,axiom,
comm_semiring(rat) ).
fof(arity_Rat_Orat___Nat_Osemiring__char__0,axiom,
semiring_char_0(rat) ).
fof(arity_Rat_Orat___Int_Onumber__semiring,axiom,
number_semiring(rat) ).
fof(arity_Rat_Orat___Groups_Oab__group__add,axiom,
ab_group_add(rat) ).
fof(arity_Rat_Orat___Rings_Ozero__neq__one,axiom,
zero_neq_one(rat) ).
fof(arity_Rat_Orat___Rings_Oordered__ring,axiom,
ordered_ring(rat) ).
fof(arity_Rat_Orat___Orderings_Opreorder,axiom,
preorder(rat) ).
fof(arity_Rat_Orat___Orderings_Olinorder,axiom,
linorder(rat) ).
fof(arity_Rat_Orat___Groups_Omonoid__mult,axiom,
monoid_mult(rat) ).
fof(arity_Rat_Orat___Rings_Ocomm__ring__1,axiom,
comm_ring_1(rat) ).
fof(arity_Rat_Orat___Groups_Omonoid__add,axiom,
monoid_add(rat) ).
fof(arity_Rat_Orat___Rings_Osemiring__1,axiom,
semiring_1(rat) ).
fof(arity_Rat_Orat___Rings_Osemiring__0,axiom,
semiring_0(rat) ).
fof(arity_Rat_Orat___Rat_Ofield__char__0,axiom,
field_char_0(rat) ).
fof(arity_Rat_Orat___Groups_Ogroup__add,axiom,
group_add(rat) ).
fof(arity_Rat_Orat___Rings_Omult__zero,axiom,
mult_zero(rat) ).
fof(arity_Rat_Orat___Rings_Ocomm__ring,axiom,
comm_ring(rat) ).
fof(arity_Rat_Orat___Orderings_Oorder,axiom,
order(rat) ).
fof(arity_Rat_Orat___Int_Oring__char__0,axiom,
ring_char_0(rat) ).
fof(arity_Rat_Orat___Int_Onumber__ring,axiom,
number_ring(rat) ).
fof(arity_Rat_Orat___Rings_Osemiring,axiom,
semiring(rat) ).
fof(arity_Rat_Orat___Fields_Oinverse,axiom,
inverse(rat) ).
fof(arity_Rat_Orat___Orderings_Oord,axiom,
ord(rat) ).
fof(arity_Rat_Orat___Groups_Ouminus,axiom,
uminus(rat) ).
fof(arity_Rat_Orat___Groups_Osgn__if,axiom,
sgn_if(rat) ).
fof(arity_Rat_Orat___Groups_Oabs__if,axiom,
abs_if(rat) ).
fof(arity_Rat_Orat___Rings_Oring__1,axiom,
ring_1(rat) ).
fof(arity_Rat_Orat___Groups_Ominus,axiom,
minus(rat) ).
fof(arity_Rat_Orat___Fields_Ofield,axiom,
field(rat) ).
fof(arity_Rat_Orat___Power_Opower,axiom,
power(rat) ).
fof(arity_Rat_Orat___Groups_Ozero,axiom,
zero(rat) ).
fof(arity_Rat_Orat___Groups_Oplus,axiom,
plus(rat) ).
fof(arity_Rat_Orat___Rings_Oring,axiom,
ring(rat) ).
fof(arity_Rat_Orat___Rings_Oidom,axiom,
idom(rat) ).
fof(arity_Rat_Orat___Int_Onumber,axiom,
number(rat) ).
fof(arity_Rat_Orat___Groups_Oone,axiom,
one(rat) ).
fof(arity_Rat_Orat___Rings_Odvd,axiom,
dvd(rat) ).
fof(arity_HOL_Obool___Complete__Lattice_Ocomplete__lattice,axiom,
comple44059174attice(bool) ).
fof(arity_HOL_Obool___Lattices_Oboolean__algebra,axiom,
boolean_algebra(bool) ).
fof(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
fof(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite(bool) ).
fof(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
fof(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
fof(arity_HOL_Obool___Groups_Ouminus,axiom,
uminus(bool) ).
fof(arity_HOL_Obool___Groups_Ominus,axiom,
minus(bool) ).
fof(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossp,axiom,
semiri456707255roduct(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(real) ).
fof(arity_RealDef_Oreal___Fields_Olinordered__field__inverse__zero,axiom,
linord1117847801e_zero(real) ).
fof(arity_RealDef_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
archim1804426504_field(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semiring__1__strict,axiom,
linord626643107strict(real) ).
fof(arity_RealDef_Oreal___RealVector_Oreal__normed__div__algebra,axiom,
real_n1866405975lgebra(real) ).
fof(arity_RealDef_Oreal___Fields_Odivision__ring__inverse__zero,axiom,
divisi14063676e_zero(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(real) ).
fof(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra__1,axiom,
real_n2089651433ebra_1(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs(real) ).
fof(arity_RealDef_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
archim1727834104eiling(real) ).
fof(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra,axiom,
real_normed_algebra(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(real) ).
fof(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(real) ).
fof(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(real) ).
fof(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(real) ).
fof(arity_RealDef_Oreal___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(real) ).
fof(arity_RealDef_Oreal___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(real) ).
fof(arity_RealDef_Oreal___RealVector_Otopological__space,axiom,
topological_space(real) ).
fof(arity_RealDef_Oreal___RealVector_Oreal__normed__field,axiom,
real_normed_field(real) ).
fof(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(real) ).
fof(arity_RealDef_Oreal___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semiring__1,axiom,
linord1278240602ring_1(real) ).
fof(arity_RealDef_Oreal___RealVector_Oreal__div__algebra,axiom,
real_div_algebra(real) ).
fof(arity_RealDef_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(real) ).
fof(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semiring,axiom,
linordered_semiring(real) ).
fof(arity_RealDef_Oreal___RealVector_Oreal__algebra__1,axiom,
real_algebra_1(real) ).
fof(arity_RealDef_Oreal___Fields_Ofield__inverse__zero,axiom,
field_inverse_zero(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom(real) ).
fof(arity_RealDef_Oreal___Orderings_Odense__linorder,axiom,
dense_linorder(real) ).
fof(arity_RealDef_Oreal___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(real) ).
fof(arity_RealDef_Oreal___RealVector_Osgn__div__norm,axiom,
sgn_div_norm(real) ).
fof(arity_RealDef_Oreal___RealVector_Oreal__algebra,axiom,
real_algebra(real) ).
fof(arity_RealDef_Oreal___RealVector_Ometric__space,axiom,
metric_space(real) ).
fof(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(real) ).
fof(arity_RealDef_Oreal___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(real) ).
fof(arity_RealDef_Oreal___Fields_Olinordered__field,axiom,
linordered_field(real) ).
fof(arity_RealDef_Oreal___Rings_Oordered__semiring,axiom,
ordered_semiring(real) ).
fof(arity_RealDef_Oreal___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs(real) ).
fof(arity_RealDef_Oreal___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(real) ).
fof(arity_RealDef_Oreal___RealVector_Oreal__vector,axiom,
real_vector(real) ).
fof(arity_RealDef_Oreal___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__ring,axiom,
linordered_ring(real) ).
fof(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
fof(arity_RealDef_Oreal___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(real) ).
fof(arity_RealDef_Oreal___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(real) ).
fof(arity_RealDef_Oreal___RealVector_Oreal__field,axiom,
real_field(real) ).
fof(arity_RealDef_Oreal___RealVector_Odist__norm,axiom,
dist_norm(real) ).
fof(arity_RealDef_Oreal___Fields_Odivision__ring,axiom,
division_ring(real) ).
fof(arity_RealDef_Oreal___Rings_Ocomm__semiring,axiom,
comm_semiring(real) ).
fof(arity_RealDef_Oreal___RealVector_Ot2__space,axiom,
t2_space(real) ).
fof(arity_RealDef_Oreal___Nat_Osemiring__char__0,axiom,
semiring_char_0(real) ).
fof(arity_RealDef_Oreal___Int_Onumber__semiring,axiom,
number_semiring(real) ).
fof(arity_RealDef_Oreal___Groups_Oab__group__add,axiom,
ab_group_add(real) ).
fof(arity_RealDef_Oreal___Rings_Ozero__neq__one,axiom,
zero_neq_one(real) ).
fof(arity_RealDef_Oreal___Rings_Oordered__ring,axiom,
ordered_ring(real) ).
fof(arity_RealDef_Oreal___Orderings_Opreorder,axiom,
preorder(real) ).
fof(arity_RealDef_Oreal___Orderings_Olinorder,axiom,
linorder(real) ).
fof(arity_RealDef_Oreal___Groups_Omonoid__mult,axiom,
monoid_mult(real) ).
fof(arity_RealDef_Oreal___Rings_Ocomm__ring__1,axiom,
comm_ring_1(real) ).
fof(arity_RealDef_Oreal___Groups_Omonoid__add,axiom,
monoid_add(real) ).
fof(arity_RealDef_Oreal___Rings_Osemiring__1,axiom,
semiring_1(real) ).
fof(arity_RealDef_Oreal___Rings_Osemiring__0,axiom,
semiring_0(real) ).
fof(arity_RealDef_Oreal___Rat_Ofield__char__0,axiom,
field_char_0(real) ).
fof(arity_RealDef_Oreal___Groups_Ogroup__add,axiom,
group_add(real) ).
fof(arity_RealDef_Oreal___Rings_Omult__zero,axiom,
mult_zero(real) ).
fof(arity_RealDef_Oreal___Rings_Ocomm__ring,axiom,
comm_ring(real) ).
fof(arity_RealDef_Oreal___Orderings_Oorder,axiom,
order(real) ).
fof(arity_RealDef_Oreal___Int_Oring__char__0,axiom,
ring_char_0(real) ).
fof(arity_RealDef_Oreal___Int_Onumber__ring,axiom,
number_ring(real) ).
fof(arity_RealDef_Oreal___Rings_Osemiring,axiom,
semiring(real) ).
fof(arity_RealDef_Oreal___Fields_Oinverse,axiom,
inverse(real) ).
fof(arity_RealDef_Oreal___Orderings_Oord,axiom,
ord(real) ).
fof(arity_RealDef_Oreal___Groups_Ouminus,axiom,
uminus(real) ).
fof(arity_RealDef_Oreal___Groups_Osgn__if,axiom,
sgn_if(real) ).
fof(arity_RealDef_Oreal___Groups_Oabs__if,axiom,
abs_if(real) ).
fof(arity_RealDef_Oreal___Rings_Oring__1,axiom,
ring_1(real) ).
fof(arity_RealDef_Oreal___Groups_Ominus,axiom,
minus(real) ).
fof(arity_RealDef_Oreal___Fields_Ofield,axiom,
field(real) ).
fof(arity_RealDef_Oreal___Power_Opower,axiom,
power(real) ).
fof(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
fof(arity_RealDef_Oreal___Groups_Oplus,axiom,
plus(real) ).
fof(arity_RealDef_Oreal___SEQ_Obanach,axiom,
banach(real) ).
fof(arity_RealDef_Oreal___Rings_Oring,axiom,
ring(real) ).
fof(arity_RealDef_Oreal___Rings_Oidom,axiom,
idom(real) ).
fof(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
fof(arity_RealDef_Oreal___Groups_Oone,axiom,
one(real) ).
fof(arity_RealDef_Oreal___Rings_Odvd,axiom,
dvd(real) ).
fof(arity_sum___Finite__Set_Ofinite,axiom,
! [T_2,T_1] :
( ( finite_finite(T_1)
& finite_finite(T_2) )
=> finite_finite(sum_sum(T_2,T_1)) ) ).
fof(arity_Limits_Ofilter___Complete__Lattice_Ocomplete__lattice,axiom,
! [T_1] : comple44059174attice(filter(T_1)) ).
fof(arity_Limits_Ofilter___Orderings_Opreorder,axiom,
! [T_1] : preorder(filter(T_1)) ).
fof(arity_Limits_Ofilter___Orderings_Oorder,axiom,
! [T_1] : order(filter(T_1)) ).
fof(arity_Limits_Ofilter___Orderings_Oord,axiom,
! [T_1] : ord(filter(T_1)) ).
fof(arity_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__cro,axiom,
semiri456707255roduct(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Oreal__normed__div__algebra,axiom,
real_n1866405975lgebra(complex) ).
fof(arity_Complex_Ocomplex___Fields_Odivision__ring__inverse__zero,axiom,
divisi14063676e_zero(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Oreal__normed__algebra__1,axiom,
real_n2089651433ebra_1(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Oreal__normed__algebra,axiom,
real_normed_algebra(complex) ).
fof(arity_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(complex) ).
fof(arity_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Otopological__space,axiom,
topological_space(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Oreal__normed__field,axiom,
real_normed_field(complex) ).
fof(arity_Complex_Ocomplex___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Oreal__div__algebra,axiom,
real_div_algebra(complex) ).
fof(arity_Complex_Ocomplex___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Oreal__algebra__1,axiom,
real_algebra_1(complex) ).
fof(arity_Complex_Ocomplex___Fields_Ofield__inverse__zero,axiom,
field_inverse_zero(complex) ).
fof(arity_Complex_Ocomplex___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Osgn__div__norm,axiom,
sgn_div_norm(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Oreal__algebra,axiom,
real_algebra(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Ometric__space,axiom,
metric_space(complex) ).
fof(arity_Complex_Ocomplex___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(complex) ).
fof(arity_Complex_Ocomplex___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(complex) ).
fof(arity_Complex_Ocomplex___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Oreal__vector,axiom,
real_vector(complex) ).
fof(arity_Complex_Ocomplex___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(complex) ).
fof(arity_Complex_Ocomplex___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(complex) ).
fof(arity_Complex_Ocomplex___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Oreal__field,axiom,
real_field(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Odist__norm,axiom,
dist_norm(complex) ).
fof(arity_Complex_Ocomplex___Fields_Odivision__ring,axiom,
division_ring(complex) ).
fof(arity_Complex_Ocomplex___Rings_Ocomm__semiring,axiom,
comm_semiring(complex) ).
fof(arity_Complex_Ocomplex___RealVector_Ot2__space,axiom,
t2_space(complex) ).
fof(arity_Complex_Ocomplex___Nat_Osemiring__char__0,axiom,
semiring_char_0(complex) ).
fof(arity_Complex_Ocomplex___Int_Onumber__semiring,axiom,
number_semiring(complex) ).
fof(arity_Complex_Ocomplex___Groups_Oab__group__add,axiom,
ab_group_add(complex) ).
fof(arity_Complex_Ocomplex___Rings_Ozero__neq__one,axiom,
zero_neq_one(complex) ).
fof(arity_Complex_Ocomplex___Groups_Omonoid__mult,axiom,
monoid_mult(complex) ).
fof(arity_Complex_Ocomplex___Rings_Ocomm__ring__1,axiom,
comm_ring_1(complex) ).
fof(arity_Complex_Ocomplex___Groups_Omonoid__add,axiom,
monoid_add(complex) ).
fof(arity_Complex_Ocomplex___Rings_Osemiring__1,axiom,
semiring_1(complex) ).
fof(arity_Complex_Ocomplex___Rings_Osemiring__0,axiom,
semiring_0(complex) ).
fof(arity_Complex_Ocomplex___Rat_Ofield__char__0,axiom,
field_char_0(complex) ).
fof(arity_Complex_Ocomplex___Groups_Ogroup__add,axiom,
group_add(complex) ).
fof(arity_Complex_Ocomplex___Rings_Omult__zero,axiom,
mult_zero(complex) ).
fof(arity_Complex_Ocomplex___Rings_Ocomm__ring,axiom,
comm_ring(complex) ).
fof(arity_Complex_Ocomplex___Int_Oring__char__0,axiom,
ring_char_0(complex) ).
fof(arity_Complex_Ocomplex___Int_Onumber__ring,axiom,
number_ring(complex) ).
fof(arity_Complex_Ocomplex___Rings_Osemiring,axiom,
semiring(complex) ).
fof(arity_Complex_Ocomplex___Fields_Oinverse,axiom,
inverse(complex) ).
fof(arity_Complex_Ocomplex___Groups_Ouminus,axiom,
uminus(complex) ).
fof(arity_Complex_Ocomplex___Rings_Oring__1,axiom,
ring_1(complex) ).
fof(arity_Complex_Ocomplex___Groups_Ominus,axiom,
minus(complex) ).
fof(arity_Complex_Ocomplex___Fields_Ofield,axiom,
field(complex) ).
fof(arity_Complex_Ocomplex___Power_Opower,axiom,
power(complex) ).
fof(arity_Complex_Ocomplex___Groups_Ozero,axiom,
zero(complex) ).
fof(arity_Complex_Ocomplex___Groups_Oplus,axiom,
plus(complex) ).
fof(arity_Complex_Ocomplex___SEQ_Obanach,axiom,
banach(complex) ).
fof(arity_Complex_Ocomplex___Rings_Oring,axiom,
ring(complex) ).
fof(arity_Complex_Ocomplex___Rings_Oidom,axiom,
idom(complex) ).
fof(arity_Complex_Ocomplex___Int_Onumber,axiom,
number(complex) ).
fof(arity_Complex_Ocomplex___Groups_Oone,axiom,
one(complex) ).
fof(arity_Complex_Ocomplex___Rings_Odvd,axiom,
dvd(complex) ).
fof(arity_prod___Finite__Set_Ofinite,axiom,
! [T_2,T_1] :
( ( finite_finite(T_1)
& finite_finite(T_2) )
=> finite_finite(product_prod(T_2,T_1)) ) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__ab__semigroup__add__imp__l,axiom,
ordere236663937imp_le(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Olinordered__semiring,axiom,
linordered_semiring(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Olinordered__semidom,axiom,
linordered_semidom(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Oordered__semiring,axiom,
ordered_semiring(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Divides_Osemiring__div,axiom,
semiring_div(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Ocomm__semiring,axiom,
comm_semiring(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Nat_Osemiring__char__0,axiom,
semiring_char_0(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Ozero__neq__one,axiom,
zero_neq_one(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Orderings_Opreorder,axiom,
preorder(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Orderings_Olinorder,axiom,
linorder(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Omonoid__mult,axiom,
monoid_mult(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Omonoid__add,axiom,
monoid_add(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Osemiring__1,axiom,
semiring_1(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Osemiring__0,axiom,
semiring_0(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Omult__zero,axiom,
mult_zero(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Orderings_Oorder,axiom,
order(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Osemiring,axiom,
semiring(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Orderings_Oord,axiom,
ord(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Ominus,axiom,
minus(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Power_Opower,axiom,
power(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Ozero,axiom,
zero(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Oplus,axiom,
plus(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Int_Onumber,axiom,
number(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Groups_Oone,axiom,
one(code_code_numeral) ).
fof(arity_Code__Numeral_Ocode__numeral___Rings_Odvd,axiom,
dvd(code_code_numeral) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__cancel__ab__semigroup_,axiom,
ordere223160158up_add(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__ab__semigroup__add__im,axiom,
ordere236663937imp_le(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Olinordered__comm__semiring__stri,axiom,
linord893533164strict(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Olinordered__semiring,axiom,
linordered_semiring(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Olinordered__semidom,axiom,
linordered_semidom(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Oordered__semiring,axiom,
ordered_semiring(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Divides_Osemiring__div,axiom,
semiring_div(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ocomm__semiring,axiom,
comm_semiring(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Nat_Osemiring__char__0,axiom,
semiring_char_0(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Ozero__neq__one,axiom,
zero_neq_one(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Orderings_Opreorder,axiom,
preorder(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Orderings_Olinorder,axiom,
linorder(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Omonoid__mult,axiom,
monoid_mult(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Omonoid__add,axiom,
monoid_add(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Osemiring__1,axiom,
semiring_1(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Osemiring__0,axiom,
semiring_0(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Omult__zero,axiom,
mult_zero(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Orderings_Oorder,axiom,
order(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Osemiring,axiom,
semiring(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Orderings_Oord,axiom,
ord(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ominus,axiom,
minus(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Power_Opower,axiom,
power(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Ozero,axiom,
zero(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oplus,axiom,
plus(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Int_Onumber,axiom,
number(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Groups_Oone,axiom,
one(quickcheck_code_int) ).
fof(arity_Quickcheck__Narrowing_Ocode__int___Rings_Odvd,axiom,
dvd(quickcheck_code_int) ).
%----Helper facts (27)
fof(help_ti_idem,axiom,
! [T,A] : ti(T,ti(T,A)) = ti(T,A) ).
fof(help_If_1_1_T,axiom,
! [X_a,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),fTrue),X),Y) = ti(X_a,X) ).
fof(help_If_2_1_T,axiom,
! [X_a,X,Y] : hAPP(X_a,X_a,hAPP(X_a,fun(X_a,X_a),hAPP(bool,fun(X_a,fun(X_a,X_a)),if(X_a),fFalse),X),Y) = ti(X_a,Y) ).
fof(help_If_3_1_T,axiom,
! [P] :
( ti(bool,P) = fTrue
| ti(bool,P) = fFalse ) ).
fof(help_fEx_1_1_U,axiom,
! [X_a,P,X] :
( ~ hBOOL(hAPP(X_a,bool,P,X))
| hBOOL(hAPP(fun(X_a,bool),bool,fEx(X_a),P)) ) ).
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_COMBB_1_1_U,axiom,
! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_b,X_c),fun(fun(X_a,X_b),fun(X_a,X_c)),combb(X_b,X_c,X_a),P),Q),R) = hAPP(X_b,X_c,P,hAPP(X_a,X_b,Q,R)) ).
fof(help_COMBC_1_1_U,axiom,
! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(X_b,fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(X_b,fun(X_a,X_c)),combc(X_a,X_b,X_c),P),Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),Q) ).
fof(help_COMBI_1_1_U,axiom,
! [X_a,P] : hAPP(X_a,X_a,combi(X_a),P) = ti(X_a,P) ).
fof(help_COMBK_1_1_U,axiom,
! [X_b,X_a,P,Q] : hAPP(X_b,X_a,hAPP(X_a,fun(X_b,X_a),combk(X_a,X_b),P),Q) = ti(X_a,P) ).
fof(help_COMBS_1_1_U,axiom,
! [X_a,X_c,X_b,P,Q,R] : hAPP(X_a,X_c,hAPP(fun(X_a,X_b),fun(X_a,X_c),hAPP(fun(X_a,fun(X_b,X_c)),fun(fun(X_a,X_b),fun(X_a,X_c)),combs(X_a,X_b,X_c),P),Q),R) = hAPP(X_b,X_c,hAPP(X_a,fun(X_b,X_c),P,R),hAPP(X_a,X_b,Q,R)) ).
fof(help_fTrue_1_1_U,axiom,
hBOOL(fTrue) ).
fof(help_fTrue_1_1_T,axiom,
! [P] :
( ti(bool,P) = fTrue
| ti(bool,P) = fFalse ) ).
fof(help_fconj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q)) ) ).
fof(help_fconj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q))
| hBOOL(P) ) ).
fof(help_fconj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fconj,P),Q))
| hBOOL(Q) ) ).
fof(help_fdisj_1_1_U,axiom,
! [Q,P] :
( ~ hBOOL(P)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) ) ).
fof(help_fdisj_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q)) ) ).
fof(help_fdisj_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fdisj,P),Q))
| hBOOL(P)
| hBOOL(Q) ) ).
fof(help_fFalse_1_1_U,axiom,
~ hBOOL(fFalse) ).
fof(help_fFalse_1_1_T,axiom,
! [P] :
( ti(bool,P) = fTrue
| ti(bool,P) = fFalse ) ).
fof(help_fequal_1_1_T,axiom,
! [X_a,X,Y] :
( ~ hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y))
| ti(X_a,X) = ti(X_a,Y) ) ).
fof(help_fequal_2_1_T,axiom,
! [X_a,X,Y] :
( ti(X_a,X) != ti(X_a,Y)
| hBOOL(hAPP(X_a,bool,hAPP(X_a,fun(X_a,bool),fequal(X_a),X),Y)) ) ).
fof(help_fimplies_1_1_U,axiom,
! [Q,P] :
( hBOOL(P)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) ) ).
fof(help_fimplies_2_1_U,axiom,
! [P,Q] :
( ~ hBOOL(Q)
| hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q)) ) ).
fof(help_fimplies_3_1_U,axiom,
! [P,Q] :
( ~ hBOOL(hAPP(bool,bool,hAPP(bool,fun(bool,bool),fimplies,P),Q))
| ~ hBOOL(P)
| hBOOL(Q) ) ).
%----Conjectures (1)
fof(conj_0,conjecture,
hAPP(int,int,hAPP(int,fun(int,int),times_times(int),hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),a),b))),hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,int)),product_Pair(int,int),p),q))) = hAPP(product_prod(int,int),int,twoSqu1094008024sum2sq,hAPP(int,product_prod(int,int),hAPP(int,fun(int,product_prod(int,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),p)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),b),q))),hAPP(int,int,hAPP(int,fun(int,int),minus_minus(int),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),a),q)),hAPP(int,int,hAPP(int,fun(int,int),times_times(int),b),p)))) ).
%------------------------------------------------------------------------------