TPTP Problem File: ITP241^1.p
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%------------------------------------------------------------------------------
% File : ITP241^1 : TPTP v8.2.0. Released v8.1.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Pred 00822_049130
% Version : [Des22] axioms.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source : [Des22]
% Names : 0069_VEBT_Pred_00822_049130 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 11233 (5702 unt; 988 typ; 0 def)
% Number of atoms : 27840 (12143 equ; 0 cnn)
% Maximal formula atoms : 71 ( 2 avg)
% Number of connectives : 115572 (2836 ~; 515 |;1734 &;99970 @)
% ( 0 <=>;10517 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 6 avg)
% Number of types : 85 ( 84 usr)
% Number of type conns : 4523 (4523 >; 0 *; 0 +; 0 <<)
% Number of symbols : 907 ( 904 usr; 61 con; 0-8 aty)
% Number of variables : 26014 (2358 ^;22889 !; 767 ?;26014 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 23:44:42.332
%------------------------------------------------------------------------------
% Could-be-implicit typings (84)
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% Explicit typings (904)
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bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
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bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
gbinomial_complex: complex > nat > complex ).
thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
gbinomial_int: int > nat > int ).
thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
gbinomial_nat: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
gbinomial_rat: rat > nat > rat ).
thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
gbinomial_real: real > nat > real ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > option_num ).
thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > option_num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > option_num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
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thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
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thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocnj,type,
cnj: complex > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocomplex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
thf(sy_c_Complex_Orcis,type,
rcis: real > real > complex ).
thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: product_prod_int_int > int ).
thf(sy_c_Divides_Oadjust__mod,type,
adjust_mod: int > int > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
unique6319869463603278526ux_int: product_prod_int_int > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
unique5052692396658037445od_int: num > num > product_prod_int_int ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
comm_s2602460028002588243omplex: complex > nat > complex ).
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Rat_OAbs__Rat,type,
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thf(sy_c_Rat_OFract,type,
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thf(sy_c_Rat_OFrct,type,
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thf(sy_c_Rat_ORep__Rat,type,
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thf(sy_c_Rat_Onormalize,type,
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thf(sy_c_Rat_Oof__int,type,
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thf(sy_c_Rat_Opcr__rat,type,
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thf(sy_c_Rat_Opositive,type,
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thf(sy_c_Rat_Oquotient__of,type,
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thf(sy_c_Rat_Oratrel,type,
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thf(sy_c_Real_OReal,type,
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thf(sy_c_Real_Ocauchy,type,
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thf(sy_c_Real_Opcr__real,type,
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thf(sy_c_Real_Orealrel,type,
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thf(sy_c_Real_Orep__real,type,
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thf(sy_c_Real_Ovanishes,type,
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thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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thf(sy_c_Series_Osummable_001t__Int__Oint,type,
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set_ord_atMost_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
set_ord_atMost_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
set_ord_atMost_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
set_ord_atMost_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
set_or5849166863359141190n_real: real > set_real ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Extended____Nat__Oenat,type,
set_or8419480210114673929d_enat: extended_enat > set_Extended_enat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
set_ord_lessThan_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
set_ord_lessThan_rat: rat > set_rat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
set_or5984915006950818249n_real: real > set_real ).
thf(sy_c_String_Oascii__of,type,
ascii_of: char > char ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
unique3096191561947761185of_nat: nat > char ).
thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
topolo4899668324122417113eq_int: ( nat > int ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
topolo1459490580787246023eq_num: ( nat > num ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
topolo7278393974255667507et_nat: ( nat > set_nat ) > $o ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(sy_c_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
arcosh_real: real > real ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
arsinh_real: real > real ).
thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
artanh_real: real > real ).
thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
cos_complex: complex > complex ).
thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
cos_real: real > real ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
cosh_complex: complex > complex ).
thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
cosh_real: real > real ).
thf(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
cot_complex: complex > complex ).
thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
cot_real: real > real ).
thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
diffs_complex: ( nat > complex ) > nat > complex ).
thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
diffs_int: ( nat > int ) > nat > int ).
thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
diffs_rat: ( nat > rat ) > nat > rat ).
thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
diffs_real: ( nat > real ) > nat > real ).
thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
exp_complex: complex > complex ).
thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
exp_real: real > real ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
powr_real: real > real > real ).
thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
sin_complex: complex > complex ).
thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
sin_real: real > real ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
sinh_complex: complex > complex ).
thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
sinh_real: real > real ).
thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
tan_complex: complex > complex ).
thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
tan_real: real > real ).
thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
tanh_complex: complex > complex ).
thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
tanh_real: real > real ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_VEBT__Insert_Ovebt__insert,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
vEBT_VEBT_add: option_nat > option_nat > option_nat ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
vEBT_VEBT_greater: option_nat > option_nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
vEBT_VEBT_less: option_nat > option_nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
vEBT_VEBT_max_in_set: set_nat > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
vEBT_VEBT_min_in_set: set_nat > nat > $o ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
vEBT_VEBT_power: option_nat > option_nat > option_nat ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
vEBT_vebt_mint: vEBT_VEBT > option_nat ).
thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
thf(sy_c_VEBT__Pred_Ovebt__pred,type,
vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
wf_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: list_o > set_list_o > $o ).
thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
member_list_int: list_int > set_list_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Rat__Orat,type,
member_rat: rat > set_rat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(sy_v_deg____,type,
deg: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_maxy____,type,
maxy: nat ).
thf(sy_v_mi____,type,
mi: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_pr____,type,
pr: nat ).
thf(sy_v_res____,type,
res: nat ).
thf(sy_v_summary____,type,
summary: vEBT_VEBT ).
thf(sy_v_treeList____,type,
treeList: list_VEBT_VEBT ).
thf(sy_v_xa____,type,
xa: nat ).
thf(sy_v_za____,type,
za: nat ).
% Relevant facts (10205)
thf(fact_0_maxt__corr__help,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ Maxi ) )
=> ( ( vEBT_vebt_member @ T @ X )
=> ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).
% maxt_corr_help
thf(fact_1_max__in__set__def,axiom,
( vEBT_VEBT_max_in_set
= ( ^ [Xs: set_nat,X2: nat] :
( ( member_nat @ X2 @ Xs )
& ! [Y: nat] :
( ( member_nat @ Y @ Xs )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ) ).
% max_in_set_def
thf(fact_2_min__in__set__def,axiom,
( vEBT_VEBT_min_in_set
= ( ^ [Xs: set_nat,X2: nat] :
( ( member_nat @ X2 @ Xs )
& ! [Y: nat] :
( ( member_nat @ Y @ Xs )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ).
% min_in_set_def
thf(fact_3_True,axiom,
( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= pr ) ).
% True
thf(fact_4__092_060open_062Some_Amaxy_A_061_Avebt__maxt_A_ItreeList_A_B_Apr_J_092_060close_062,axiom,
( ( some_nat @ maxy )
= ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ pr ) ) ) ).
% \<open>Some maxy = vebt_maxt (treeList ! pr)\<close>
thf(fact_5__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% \<open>2 \<le> deg\<close>
thf(fact_6_abe,axiom,
vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% abe
thf(fact_7__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= na ) ).
% \<open>deg div 2 = n\<close>
thf(fact_8_False,axiom,
za != mi ).
% False
thf(fact_9_bit__split__inv,axiom,
! [X: nat,D: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
= X ) ).
% bit_split_inv
thf(fact_10_abc,axiom,
vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% abc
thf(fact_11__092_060open_062high_Az_A_Ideg_Adiv_A2_J_A_092_060le_062_Apr_092_060close_062,axiom,
ord_less_eq_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ pr ).
% \<open>high z (deg div 2) \<le> pr\<close>
thf(fact_12_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_13_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_14_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_15_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_16_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_17_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_18_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_19_numeral__Bit0__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit0_div_2
thf(fact_20_divide__numeral__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_21_divide__numeral__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_22_divide__numeral__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_23__092_060open_062pr_A_060_Ahigh_Az_A_Ideg_Adiv_A2_J_A_092_060Longrightarrow_062_AFalse_092_060close_062,axiom,
~ ( ord_less_nat @ pr @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% \<open>pr < high z (deg div 2) \<Longrightarrow> False\<close>
thf(fact_24_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y2: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y2 ) )
= ( X22 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_25_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_26_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera6690914467698888265omplex @ M )
= ( numera6690914467698888265omplex @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_27_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_28_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_rat @ M )
= ( numeral_numeral_rat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_29_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_30_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_31_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_32_order__refl,axiom,
! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% order_refl
thf(fact_33_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_34_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_35_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_36_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_37_dual__order_Orefl,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% dual_order.refl
thf(fact_38_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_39_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_40_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_41_maxt__member,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ Maxi ) )
=> ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% maxt_member
thf(fact_42_member__correct,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_vebt_member @ T @ X )
= ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).
% member_correct
thf(fact_43__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxy_O_ASome_Amaxy_A_061_Avebt__maxt_A_ItreeList_A_B_Apr_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Maxy: nat] :
( ( some_nat @ Maxy )
!= ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ pr ) ) ) ).
% \<open>\<And>thesis. (\<And>maxy. Some maxy = vebt_maxt (treeList ! pr) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_44_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_45_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_46_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_47_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_48_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_49_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_50_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_51__C5_Ohyps_C_I9_J,axiom,
ord_less_eq_nat @ mi @ ma ).
% "5.hyps"(9)
thf(fact_52_less__shift,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% less_shift
thf(fact_53_lesseq__shift,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% lesseq_shift
thf(fact_54_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_55_verit__comp__simplify1_I1_J,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_56_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_57_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_58_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_59_lt__ex,axiom,
! [X: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).
% lt_ex
thf(fact_60_lt__ex,axiom,
! [X: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).
% lt_ex
thf(fact_61_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_62_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_63_gt__ex,axiom,
! [X: rat] :
? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% gt_ex
thf(fact_64_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_65_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_66_dense,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ? [Z: real] :
( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Z @ Y4 ) ) ) ).
% dense
thf(fact_67_dense,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ? [Z: rat] :
( ( ord_less_rat @ X @ Z )
& ( ord_less_rat @ Z @ Y4 ) ) ) ).
% dense
thf(fact_68_less__imp__neq,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( X != Y4 ) ) ).
% less_imp_neq
thf(fact_69_less__imp__neq,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ( X != Y4 ) ) ).
% less_imp_neq
thf(fact_70_less__imp__neq,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( X != Y4 ) ) ).
% less_imp_neq
thf(fact_71_less__imp__neq,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( X != Y4 ) ) ).
% less_imp_neq
thf(fact_72_less__imp__neq,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( X != Y4 ) ) ).
% less_imp_neq
thf(fact_73_order_Oasym,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order.asym
thf(fact_74_order_Oasym,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order.asym
thf(fact_75_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_76_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_77_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_78_ord__eq__less__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_79_ord__eq__less__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_80_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_81_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_82_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_83_ord__less__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_84_ord__less__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_85_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_86_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_87_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_88_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_89_antisym__conv3,axiom,
! [Y4: real,X: real] :
( ~ ( ord_less_real @ Y4 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv3
thf(fact_90_antisym__conv3,axiom,
! [Y4: rat,X: rat] :
( ~ ( ord_less_rat @ Y4 @ X )
=> ( ( ~ ( ord_less_rat @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv3
thf(fact_91_antisym__conv3,axiom,
! [Y4: num,X: num] :
( ~ ( ord_less_num @ Y4 @ X )
=> ( ( ~ ( ord_less_num @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv3
thf(fact_92_antisym__conv3,axiom,
! [Y4: nat,X: nat] :
( ~ ( ord_less_nat @ Y4 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv3
thf(fact_93_antisym__conv3,axiom,
! [Y4: int,X: int] :
( ~ ( ord_less_int @ Y4 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv3
thf(fact_94_linorder__cases,axiom,
! [X: real,Y4: real] :
( ~ ( ord_less_real @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less_real @ Y4 @ X ) ) ) ).
% linorder_cases
thf(fact_95_linorder__cases,axiom,
! [X: rat,Y4: rat] :
( ~ ( ord_less_rat @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less_rat @ Y4 @ X ) ) ) ).
% linorder_cases
thf(fact_96_linorder__cases,axiom,
! [X: num,Y4: num] :
( ~ ( ord_less_num @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less_num @ Y4 @ X ) ) ) ).
% linorder_cases
thf(fact_97_linorder__cases,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_nat @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_cases
thf(fact_98_linorder__cases,axiom,
! [X: int,Y4: int] :
( ~ ( ord_less_int @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less_int @ Y4 @ X ) ) ) ).
% linorder_cases
thf(fact_99_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_100_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_101_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_102_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_103_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_104_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_105_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_106_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_107_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_108_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_109_Collect__mem__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_111_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_112_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_113_Collect__cong,axiom,
! [P: real > $o,Q: real > $o] :
( ! [X3: real] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_real @ P )
= ( collect_real @ Q ) ) ) ).
% Collect_cong
thf(fact_114_Collect__cong,axiom,
! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec3392354462482085612at_nat @ P )
= ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_115_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_116_dual__order_Oasym,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_117_dual__order_Oasym,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ A @ B ) ) ).
% dual_order.asym
thf(fact_118_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_119_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_120_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_121_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_122_dual__order_Oirrefl,axiom,
! [A: rat] :
~ ( ord_less_rat @ A @ A ) ).
% dual_order.irrefl
thf(fact_123_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_124_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_125_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_126_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_127_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A3: real,B2: real] :
( ( ord_less_real @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: real] : ( P @ A3 @ A3 )
=> ( ! [A3: real,B2: real] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_128_linorder__less__wlog,axiom,
! [P: rat > rat > $o,A: rat,B: rat] :
( ! [A3: rat,B2: rat] :
( ( ord_less_rat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: rat] : ( P @ A3 @ A3 )
=> ( ! [A3: rat,B2: rat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_129_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A3: num,B2: num] :
( ( ord_less_num @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: num] : ( P @ A3 @ A3 )
=> ( ! [A3: num,B2: num] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_130_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_131_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_132_order_Ostrict__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_133_order_Ostrict__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_134_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_135_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_136_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_137_not__less__iff__gr__or__eq,axiom,
! [X: real,Y4: real] :
( ( ~ ( ord_less_real @ X @ Y4 ) )
= ( ( ord_less_real @ Y4 @ X )
| ( X = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_138_not__less__iff__gr__or__eq,axiom,
! [X: rat,Y4: rat] :
( ( ~ ( ord_less_rat @ X @ Y4 ) )
= ( ( ord_less_rat @ Y4 @ X )
| ( X = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_139_not__less__iff__gr__or__eq,axiom,
! [X: num,Y4: num] :
( ( ~ ( ord_less_num @ X @ Y4 ) )
= ( ( ord_less_num @ Y4 @ X )
| ( X = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_140_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( ( ord_less_nat @ Y4 @ X )
| ( X = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_141_not__less__iff__gr__or__eq,axiom,
! [X: int,Y4: int] :
( ( ~ ( ord_less_int @ X @ Y4 ) )
= ( ( ord_less_int @ Y4 @ X )
| ( X = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_142_dual__order_Ostrict__trans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_143_dual__order_Ostrict__trans,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_144_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_145_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_146_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_147_order_Ostrict__implies__not__eq,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_148_order_Ostrict__implies__not__eq,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_149_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_150_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_151_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_152_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_153_dual__order_Ostrict__implies__not__eq,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_154_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_155_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_156_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_157_linorder__neqE,axiom,
! [X: real,Y4: real] :
( ( X != Y4 )
=> ( ~ ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ Y4 @ X ) ) ) ).
% linorder_neqE
thf(fact_158_linorder__neqE,axiom,
! [X: rat,Y4: rat] :
( ( X != Y4 )
=> ( ~ ( ord_less_rat @ X @ Y4 )
=> ( ord_less_rat @ Y4 @ X ) ) ) ).
% linorder_neqE
thf(fact_159_linorder__neqE,axiom,
! [X: num,Y4: num] :
( ( X != Y4 )
=> ( ~ ( ord_less_num @ X @ Y4 )
=> ( ord_less_num @ Y4 @ X ) ) ) ).
% linorder_neqE
thf(fact_160_linorder__neqE,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
=> ( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neqE
thf(fact_161_linorder__neqE,axiom,
! [X: int,Y4: int] :
( ( X != Y4 )
=> ( ~ ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ Y4 @ X ) ) ) ).
% linorder_neqE
thf(fact_162_order__less__asym,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X ) ) ).
% order_less_asym
thf(fact_163_order__less__asym,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ~ ( ord_less_rat @ Y4 @ X ) ) ).
% order_less_asym
thf(fact_164_order__less__asym,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X ) ) ).
% order_less_asym
thf(fact_165_order__less__asym,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X ) ) ).
% order_less_asym
thf(fact_166_order__less__asym,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X ) ) ).
% order_less_asym
thf(fact_167_linorder__neq__iff,axiom,
! [X: real,Y4: real] :
( ( X != Y4 )
= ( ( ord_less_real @ X @ Y4 )
| ( ord_less_real @ Y4 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_168_linorder__neq__iff,axiom,
! [X: rat,Y4: rat] :
( ( X != Y4 )
= ( ( ord_less_rat @ X @ Y4 )
| ( ord_less_rat @ Y4 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_169_linorder__neq__iff,axiom,
! [X: num,Y4: num] :
( ( X != Y4 )
= ( ( ord_less_num @ X @ Y4 )
| ( ord_less_num @ Y4 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_170_linorder__neq__iff,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
= ( ( ord_less_nat @ X @ Y4 )
| ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_171_linorder__neq__iff,axiom,
! [X: int,Y4: int] :
( ( X != Y4 )
= ( ( ord_less_int @ X @ Y4 )
| ( ord_less_int @ Y4 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_172_order__less__asym_H,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_173_order__less__asym_H,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( ord_less_rat @ B @ A ) ) ).
% order_less_asym'
thf(fact_174_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_175_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_176_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_177_order__less__trans,axiom,
! [X: real,Y4: real,Z2: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_178_order__less__trans,axiom,
! [X: rat,Y4: rat,Z2: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ( ( ord_less_rat @ Y4 @ Z2 )
=> ( ord_less_rat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_179_order__less__trans,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_180_order__less__trans,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_181_order__less__trans,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_182_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_183_ord__eq__less__subst,axiom,
! [A: rat,F: real > rat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_184_ord__eq__less__subst,axiom,
! [A: num,F: real > num,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_185_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_186_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_187_ord__eq__less__subst,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_188_ord__eq__less__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_189_ord__eq__less__subst,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_190_ord__eq__less__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_191_ord__eq__less__subst,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_192_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_193_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > rat,C: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_194_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_195_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_196_ord__less__eq__subst,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_197_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_198_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_199_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_200_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_201_ord__less__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_202_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_203_order__less__irrefl,axiom,
! [X: rat] :
~ ( ord_less_rat @ X @ X ) ).
% order_less_irrefl
thf(fact_204_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_205_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_206_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_207_order__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_208_order__less__subst1,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_209_order__less__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_210_order__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_211_order__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_212_order__less__subst1,axiom,
! [A: rat,F: real > rat,B: real,C: real] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_213_order__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_214_order__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_215_order__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_216_order__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C: int] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_217_order__less__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_218_order__less__subst2,axiom,
! [A: real,B: real,F: real > rat,C: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_219_order__less__subst2,axiom,
! [A: real,B: real,F: real > num,C: num] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_220_order__less__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_221_order__less__subst2,axiom,
! [A: real,B: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_222_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_223_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_224_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_225_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_226_order__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_227_order__less__not__sym,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X ) ) ).
% order_less_not_sym
thf(fact_228_order__less__not__sym,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ~ ( ord_less_rat @ Y4 @ X ) ) ).
% order_less_not_sym
thf(fact_229_order__less__not__sym,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X ) ) ).
% order_less_not_sym
thf(fact_230_order__less__not__sym,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X ) ) ).
% order_less_not_sym
thf(fact_231_order__less__not__sym,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X ) ) ).
% order_less_not_sym
thf(fact_232_order__less__imp__triv,axiom,
! [X: real,Y4: real,P: $o] :
( ( ord_less_real @ X @ Y4 )
=> ( ( ord_less_real @ Y4 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_233_order__less__imp__triv,axiom,
! [X: rat,Y4: rat,P: $o] :
( ( ord_less_rat @ X @ Y4 )
=> ( ( ord_less_rat @ Y4 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_234_order__less__imp__triv,axiom,
! [X: num,Y4: num,P: $o] :
( ( ord_less_num @ X @ Y4 )
=> ( ( ord_less_num @ Y4 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_235_order__less__imp__triv,axiom,
! [X: nat,Y4: nat,P: $o] :
( ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_nat @ Y4 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_236_order__less__imp__triv,axiom,
! [X: int,Y4: int,P: $o] :
( ( ord_less_int @ X @ Y4 )
=> ( ( ord_less_int @ Y4 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_237_linorder__less__linear,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
| ( X = Y4 )
| ( ord_less_real @ Y4 @ X ) ) ).
% linorder_less_linear
thf(fact_238_linorder__less__linear,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ Y4 )
| ( X = Y4 )
| ( ord_less_rat @ Y4 @ X ) ) ).
% linorder_less_linear
thf(fact_239_linorder__less__linear,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
| ( X = Y4 )
| ( ord_less_num @ Y4 @ X ) ) ).
% linorder_less_linear
thf(fact_240_linorder__less__linear,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 )
| ( ord_less_nat @ Y4 @ X ) ) ).
% linorder_less_linear
thf(fact_241_linorder__less__linear,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
| ( X = Y4 )
| ( ord_less_int @ Y4 @ X ) ) ).
% linorder_less_linear
thf(fact_242_order__less__imp__not__eq,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( X != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_243_order__less__imp__not__eq,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ( X != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_244_order__less__imp__not__eq,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( X != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_245_order__less__imp__not__eq,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( X != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_246_order__less__imp__not__eq,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( X != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_247_order__less__imp__not__eq2,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( Y4 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_248_order__less__imp__not__eq2,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ( Y4 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_249_order__less__imp__not__eq2,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( Y4 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_250_order__less__imp__not__eq2,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( Y4 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_251_order__less__imp__not__eq2,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( Y4 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_252_order__less__imp__not__less,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ~ ( ord_less_real @ Y4 @ X ) ) ).
% order_less_imp_not_less
thf(fact_253_order__less__imp__not__less,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ~ ( ord_less_rat @ Y4 @ X ) ) ).
% order_less_imp_not_less
thf(fact_254_order__less__imp__not__less,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ~ ( ord_less_num @ Y4 @ X ) ) ).
% order_less_imp_not_less
thf(fact_255_order__less__imp__not__less,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X ) ) ).
% order_less_imp_not_less
thf(fact_256_order__less__imp__not__less,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ~ ( ord_less_int @ Y4 @ X ) ) ).
% order_less_imp_not_less
thf(fact_257_order__le__imp__less__or__eq,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ( ord_less_real @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_258_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ( ord_less_set_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_259_order__le__imp__less__or__eq,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ Y4 )
=> ( ( ord_less_rat @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_260_order__le__imp__less__or__eq,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ord_less_num @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_261_order__le__imp__less__or__eq,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_262_order__le__imp__less__or__eq,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ord_less_int @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_263_linorder__le__less__linear,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
| ( ord_less_real @ Y4 @ X ) ) ).
% linorder_le_less_linear
thf(fact_264_linorder__le__less__linear,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ Y4 )
| ( ord_less_rat @ Y4 @ X ) ) ).
% linorder_le_less_linear
thf(fact_265_linorder__le__less__linear,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
| ( ord_less_num @ Y4 @ X ) ) ).
% linorder_le_less_linear
thf(fact_266_linorder__le__less__linear,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
| ( ord_less_nat @ Y4 @ X ) ) ).
% linorder_le_less_linear
thf(fact_267_linorder__le__less__linear,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
| ( ord_less_int @ Y4 @ X ) ) ).
% linorder_le_less_linear
thf(fact_268_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_269_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_270_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_271_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_272_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_273_order__less__le__subst2,axiom,
! [A: real,B: real,F: real > rat,C: rat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_274_order__less__le__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_275_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_276_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_277_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > rat,C: rat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_278_order__less__le__subst1,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_279_order__less__le__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_280_order__less__le__subst1,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_281_order__less__le__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_282_order__less__le__subst1,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_283_order__less__le__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_284_order__less__le__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_285_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_286_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_287_order__less__le__subst1,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_288_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > real,C: real] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_289_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_290_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_291_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_292_order__le__less__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_293_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > real,C: real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_real @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_294_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_rat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_295_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_296_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_297_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_298_order__le__less__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_299_order__le__less__subst1,axiom,
! [A: real,F: rat > real,B: rat,C: rat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_300_order__le__less__subst1,axiom,
! [A: real,F: num > real,B: num,C: num] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_301_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_302_order__le__less__subst1,axiom,
! [A: real,F: int > real,B: int,C: int] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_303_order__le__less__subst1,axiom,
! [A: rat,F: real > rat,B: real,C: real] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y3: real] :
( ( ord_less_real @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_304_order__le__less__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_rat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_305_order__le__less__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_306_order__le__less__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_307_order__le__less__subst1,axiom,
! [A: rat,F: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_308_order__less__le__trans,axiom,
! [X: real,Y4: real,Z2: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_309_order__less__le__trans,axiom,
! [X: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_310_order__less__le__trans,axiom,
! [X: rat,Y4: rat,Z2: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ( ( ord_less_eq_rat @ Y4 @ Z2 )
=> ( ord_less_rat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_311_order__less__le__trans,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_312_order__less__le__trans,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_313_order__less__le__trans,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_314_order__le__less__trans,axiom,
! [X: real,Y4: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ( ord_less_real @ Y4 @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_315_order__le__less__trans,axiom,
! [X: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ( ord_less_set_nat @ Y4 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_316_order__le__less__trans,axiom,
! [X: rat,Y4: rat,Z2: rat] :
( ( ord_less_eq_rat @ X @ Y4 )
=> ( ( ord_less_rat @ Y4 @ Z2 )
=> ( ord_less_rat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_317_order__le__less__trans,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ord_less_num @ Y4 @ Z2 )
=> ( ord_less_num @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_318_order__le__less__trans,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_319_order__le__less__trans,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ord_less_int @ Y4 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_320_order__neq__le__trans,axiom,
! [A: real,B: real] :
( ( A != B )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_321_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_322_order__neq__le__trans,axiom,
! [A: rat,B: rat] :
( ( A != B )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_323_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_324_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_325_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_326_order__le__neq__trans,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( A != B )
=> ( ord_less_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_327_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_328_order__le__neq__trans,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( A != B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_329_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_330_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_331_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_332_order__less__imp__le,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_eq_real @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_333_order__less__imp__le,axiom,
! [X: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_334_order__less__imp__le,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ( ord_less_eq_rat @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_335_order__less__imp__le,axiom,
! [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
=> ( ord_less_eq_num @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_336_order__less__imp__le,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_337_order__less__imp__le,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_eq_int @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_338_linorder__not__less,axiom,
! [X: real,Y4: real] :
( ( ~ ( ord_less_real @ X @ Y4 ) )
= ( ord_less_eq_real @ Y4 @ X ) ) ).
% linorder_not_less
thf(fact_339_linorder__not__less,axiom,
! [X: rat,Y4: rat] :
( ( ~ ( ord_less_rat @ X @ Y4 ) )
= ( ord_less_eq_rat @ Y4 @ X ) ) ).
% linorder_not_less
thf(fact_340_linorder__not__less,axiom,
! [X: num,Y4: num] :
( ( ~ ( ord_less_num @ X @ Y4 ) )
= ( ord_less_eq_num @ Y4 @ X ) ) ).
% linorder_not_less
thf(fact_341_linorder__not__less,axiom,
! [X: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( ord_less_eq_nat @ Y4 @ X ) ) ).
% linorder_not_less
thf(fact_342_linorder__not__less,axiom,
! [X: int,Y4: int] :
( ( ~ ( ord_less_int @ X @ Y4 ) )
= ( ord_less_eq_int @ Y4 @ X ) ) ).
% linorder_not_less
thf(fact_343_linorder__not__le,axiom,
! [X: real,Y4: real] :
( ( ~ ( ord_less_eq_real @ X @ Y4 ) )
= ( ord_less_real @ Y4 @ X ) ) ).
% linorder_not_le
thf(fact_344_linorder__not__le,axiom,
! [X: rat,Y4: rat] :
( ( ~ ( ord_less_eq_rat @ X @ Y4 ) )
= ( ord_less_rat @ Y4 @ X ) ) ).
% linorder_not_le
thf(fact_345_linorder__not__le,axiom,
! [X: num,Y4: num] :
( ( ~ ( ord_less_eq_num @ X @ Y4 ) )
= ( ord_less_num @ Y4 @ X ) ) ).
% linorder_not_le
thf(fact_346_linorder__not__le,axiom,
! [X: nat,Y4: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y4 ) )
= ( ord_less_nat @ Y4 @ X ) ) ).
% linorder_not_le
thf(fact_347_linorder__not__le,axiom,
! [X: int,Y4: int] :
( ( ~ ( ord_less_eq_int @ X @ Y4 ) )
= ( ord_less_int @ Y4 @ X ) ) ).
% linorder_not_le
thf(fact_348_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_349_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_350_order__less__le,axiom,
( ord_less_rat
= ( ^ [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_351_order__less__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_352_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_353_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_354_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_355_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_356_order__le__less,axiom,
( ord_less_eq_rat
= ( ^ [X2: rat,Y: rat] :
( ( ord_less_rat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_357_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_358_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_359_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_360_dual__order_Ostrict__implies__order,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_eq_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_361_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_362_dual__order_Ostrict__implies__order,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_363_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_364_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_365_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_366_order_Ostrict__implies__order,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_367_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_368_order_Ostrict__implies__order,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_369_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_370_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_371_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_372_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ~ ( ord_less_eq_real @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_373_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_374_dual__order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [B3: rat,A4: rat] :
( ( ord_less_eq_rat @ B3 @ A4 )
& ~ ( ord_less_eq_rat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_375_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ~ ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_376_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_377_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_378_dual__order_Ostrict__trans2,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_379_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_380_dual__order_Ostrict__trans2,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_381_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_382_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_383_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_384_dual__order_Ostrict__trans1,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_385_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_386_dual__order_Ostrict__trans1,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_387_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_388_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_389_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_390_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_eq_real @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_391_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_392_dual__order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [B3: rat,A4: rat] :
( ( ord_less_eq_rat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_393_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_394_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_395_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_396_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A4: real] :
( ( ord_less_real @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_397_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_398_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A4: rat] :
( ( ord_less_rat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_399_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A4: num] :
( ( ord_less_num @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_400_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_401_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_402_dense__le__bounded,axiom,
! [X: real,Y4: real,Z2: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y4 )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y4 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_403_dense__le__bounded,axiom,
! [X: rat,Y4: rat,Z2: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ( ! [W: rat] :
( ( ord_less_rat @ X @ W )
=> ( ( ord_less_rat @ W @ Y4 )
=> ( ord_less_eq_rat @ W @ Z2 ) ) )
=> ( ord_less_eq_rat @ Y4 @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_404_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y4: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y4 @ W ) ) )
=> ( ord_less_eq_real @ Y4 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_405_dense__ge__bounded,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( ord_less_rat @ Z2 @ X )
=> ( ! [W: rat] :
( ( ord_less_rat @ Z2 @ W )
=> ( ( ord_less_rat @ W @ X )
=> ( ord_less_eq_rat @ Y4 @ W ) ) )
=> ( ord_less_eq_rat @ Y4 @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_406_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_407_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_408_order_Ostrict__iff__not,axiom,
( ord_less_rat
= ( ^ [A4: rat,B3: rat] :
( ( ord_less_eq_rat @ A4 @ B3 )
& ~ ( ord_less_eq_rat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_409_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ~ ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_410_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_411_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_412_order_Ostrict__trans2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_413_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_414_order_Ostrict__trans2,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_415_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_416_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_417_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_418_order_Ostrict__trans1,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_419_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_420_order_Ostrict__trans1,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_421_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_422_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_423_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_424_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_425_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_426_order_Ostrict__iff__order,axiom,
( ord_less_rat
= ( ^ [A4: rat,B3: rat] :
( ( ord_less_eq_rat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_427_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_428_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_429_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_430_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_431_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_432_order_Oorder__iff__strict,axiom,
( ord_less_eq_rat
= ( ^ [A4: rat,B3: rat] :
( ( ord_less_rat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_433_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B3: num] :
( ( ord_less_num @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_434_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_435_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_436_not__le__imp__less,axiom,
! [Y4: real,X: real] :
( ~ ( ord_less_eq_real @ Y4 @ X )
=> ( ord_less_real @ X @ Y4 ) ) ).
% not_le_imp_less
thf(fact_437_not__le__imp__less,axiom,
! [Y4: rat,X: rat] :
( ~ ( ord_less_eq_rat @ Y4 @ X )
=> ( ord_less_rat @ X @ Y4 ) ) ).
% not_le_imp_less
thf(fact_438_not__le__imp__less,axiom,
! [Y4: num,X: num] :
( ~ ( ord_less_eq_num @ Y4 @ X )
=> ( ord_less_num @ X @ Y4 ) ) ).
% not_le_imp_less
thf(fact_439_not__le__imp__less,axiom,
! [Y4: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y4 @ X )
=> ( ord_less_nat @ X @ Y4 ) ) ).
% not_le_imp_less
thf(fact_440_not__le__imp__less,axiom,
! [Y4: int,X: int] :
( ~ ( ord_less_eq_int @ Y4 @ X )
=> ( ord_less_int @ X @ Y4 ) ) ).
% not_le_imp_less
thf(fact_441_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
& ~ ( ord_less_eq_real @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_442_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ~ ( ord_less_eq_set_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_443_less__le__not__le,axiom,
( ord_less_rat
= ( ^ [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
& ~ ( ord_less_eq_rat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_444_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
& ~ ( ord_less_eq_num @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_445_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_446_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ~ ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_447_dense__le,axiom,
! [Y4: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y4 @ Z2 ) ) ).
% dense_le
thf(fact_448_dense__le,axiom,
! [Y4: rat,Z2: rat] :
( ! [X3: rat] :
( ( ord_less_rat @ X3 @ Y4 )
=> ( ord_less_eq_rat @ X3 @ Z2 ) )
=> ( ord_less_eq_rat @ Y4 @ Z2 ) ) ).
% dense_le
thf(fact_449_dense__ge,axiom,
! [Z2: real,Y4: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y4 @ X3 ) )
=> ( ord_less_eq_real @ Y4 @ Z2 ) ) ).
% dense_ge
thf(fact_450_dense__ge,axiom,
! [Z2: rat,Y4: rat] :
( ! [X3: rat] :
( ( ord_less_rat @ Z2 @ X3 )
=> ( ord_less_eq_rat @ Y4 @ X3 ) )
=> ( ord_less_eq_rat @ Y4 @ Z2 ) ) ).
% dense_ge
thf(fact_451_antisym__conv2,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ( ~ ( ord_less_real @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_452_antisym__conv2,axiom,
! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ( ~ ( ord_less_set_nat @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_453_antisym__conv2,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ Y4 )
=> ( ( ~ ( ord_less_rat @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_454_antisym__conv2,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ~ ( ord_less_num @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_455_antisym__conv2,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_456_antisym__conv2,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ~ ( ord_less_int @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_457_antisym__conv1,axiom,
! [X: real,Y4: real] :
( ~ ( ord_less_real @ X @ Y4 )
=> ( ( ord_less_eq_real @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_458_antisym__conv1,axiom,
! [X: set_nat,Y4: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y4 )
=> ( ( ord_less_eq_set_nat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_459_antisym__conv1,axiom,
! [X: rat,Y4: rat] :
( ~ ( ord_less_rat @ X @ Y4 )
=> ( ( ord_less_eq_rat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_460_antisym__conv1,axiom,
! [X: num,Y4: num] :
( ~ ( ord_less_num @ X @ Y4 )
=> ( ( ord_less_eq_num @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_461_antisym__conv1,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_462_antisym__conv1,axiom,
! [X: int,Y4: int] :
( ~ ( ord_less_int @ X @ Y4 )
=> ( ( ord_less_eq_int @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_463_nless__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_464_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_465_nless__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_466_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_467_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_468_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_469_leI,axiom,
! [X: real,Y4: real] :
( ~ ( ord_less_real @ X @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X ) ) ).
% leI
thf(fact_470_leI,axiom,
! [X: rat,Y4: rat] :
( ~ ( ord_less_rat @ X @ Y4 )
=> ( ord_less_eq_rat @ Y4 @ X ) ) ).
% leI
thf(fact_471_leI,axiom,
! [X: num,Y4: num] :
( ~ ( ord_less_num @ X @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X ) ) ).
% leI
thf(fact_472_leI,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ).
% leI
thf(fact_473_leI,axiom,
! [X: int,Y4: int] :
( ~ ( ord_less_int @ X @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X ) ) ).
% leI
thf(fact_474_leD,axiom,
! [Y4: real,X: real] :
( ( ord_less_eq_real @ Y4 @ X )
=> ~ ( ord_less_real @ X @ Y4 ) ) ).
% leD
thf(fact_475_leD,axiom,
! [Y4: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X )
=> ~ ( ord_less_set_nat @ X @ Y4 ) ) ).
% leD
thf(fact_476_leD,axiom,
! [Y4: rat,X: rat] :
( ( ord_less_eq_rat @ Y4 @ X )
=> ~ ( ord_less_rat @ X @ Y4 ) ) ).
% leD
thf(fact_477_leD,axiom,
! [Y4: num,X: num] :
( ( ord_less_eq_num @ Y4 @ X )
=> ~ ( ord_less_num @ X @ Y4 ) ) ).
% leD
thf(fact_478_leD,axiom,
! [Y4: nat,X: nat] :
( ( ord_less_eq_nat @ Y4 @ X )
=> ~ ( ord_less_nat @ X @ Y4 ) ) ).
% leD
thf(fact_479_leD,axiom,
! [Y4: int,X: int] :
( ( ord_less_eq_int @ Y4 @ X )
=> ~ ( ord_less_int @ X @ Y4 ) ) ).
% leD
thf(fact_480_verit__comp__simplify1_I3_J,axiom,
! [B4: real,A5: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
= ( ord_less_real @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_481_verit__comp__simplify1_I3_J,axiom,
! [B4: rat,A5: rat] :
( ( ~ ( ord_less_eq_rat @ B4 @ A5 ) )
= ( ord_less_rat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_482_verit__comp__simplify1_I3_J,axiom,
! [B4: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B4 @ A5 ) )
= ( ord_less_num @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_483_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_484_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_485_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_486_order__antisym__conv,axiom,
! [Y4: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_487_order__antisym__conv,axiom,
! [Y4: rat,X: rat] :
( ( ord_less_eq_rat @ Y4 @ X )
=> ( ( ord_less_eq_rat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_488_order__antisym__conv,axiom,
! [Y4: num,X: num] :
( ( ord_less_eq_num @ Y4 @ X )
=> ( ( ord_less_eq_num @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_489_order__antisym__conv,axiom,
! [Y4: nat,X: nat] :
( ( ord_less_eq_nat @ Y4 @ X )
=> ( ( ord_less_eq_nat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_490_order__antisym__conv,axiom,
! [Y4: int,X: int] :
( ( ord_less_eq_int @ Y4 @ X )
=> ( ( ord_less_eq_int @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_491_linorder__le__cases,axiom,
! [X: rat,Y4: rat] :
( ~ ( ord_less_eq_rat @ X @ Y4 )
=> ( ord_less_eq_rat @ Y4 @ X ) ) ).
% linorder_le_cases
thf(fact_492_linorder__le__cases,axiom,
! [X: num,Y4: num] :
( ~ ( ord_less_eq_num @ X @ Y4 )
=> ( ord_less_eq_num @ Y4 @ X ) ) ).
% linorder_le_cases
thf(fact_493_linorder__le__cases,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ).
% linorder_le_cases
thf(fact_494_linorder__le__cases,axiom,
! [X: int,Y4: int] :
( ~ ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ Y4 @ X ) ) ).
% linorder_le_cases
thf(fact_495_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_496_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_497_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_498_ord__le__eq__subst,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_499_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_500_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_501_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_502_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_503_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_504_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_505_ord__eq__le__subst,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_506_ord__eq__le__subst,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_507_ord__eq__le__subst,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_508_ord__eq__le__subst,axiom,
! [A: int,F: rat > int,B: rat,C: rat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_509_ord__eq__le__subst,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_510_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_511_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_512_ord__eq__le__subst,axiom,
! [A: int,F: num > int,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_513_ord__eq__le__subst,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_514_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_515_linorder__linear,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ Y4 )
| ( ord_less_eq_rat @ Y4 @ X ) ) ).
% linorder_linear
thf(fact_516_linorder__linear,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
| ( ord_less_eq_num @ Y4 @ X ) ) ).
% linorder_linear
thf(fact_517_linorder__linear,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X ) ) ).
% linorder_linear
thf(fact_518_linorder__linear,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
| ( ord_less_eq_int @ Y4 @ X ) ) ).
% linorder_linear
thf(fact_519_verit__la__disequality,axiom,
! [A: rat,B: rat] :
( ( A = B )
| ~ ( ord_less_eq_rat @ A @ B )
| ~ ( ord_less_eq_rat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_520_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_521_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_522_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_523_order__eq__refl,axiom,
! [X: set_nat,Y4: set_nat] :
( ( X = Y4 )
=> ( ord_less_eq_set_nat @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_524_order__eq__refl,axiom,
! [X: rat,Y4: rat] :
( ( X = Y4 )
=> ( ord_less_eq_rat @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_525_order__eq__refl,axiom,
! [X: num,Y4: num] :
( ( X = Y4 )
=> ( ord_less_eq_num @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_526_order__eq__refl,axiom,
! [X: nat,Y4: nat] :
( ( X = Y4 )
=> ( ord_less_eq_nat @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_527_order__eq__refl,axiom,
! [X: int,Y4: int] :
( ( X = Y4 )
=> ( ord_less_eq_int @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_528_order__subst2,axiom,
! [A: rat,B: rat,F: rat > rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_529_order__subst2,axiom,
! [A: rat,B: rat,F: rat > num,C: num] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_530_order__subst2,axiom,
! [A: rat,B: rat,F: rat > nat,C: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_531_order__subst2,axiom,
! [A: rat,B: rat,F: rat > int,C: int] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_532_order__subst2,axiom,
! [A: num,B: num,F: num > rat,C: rat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_533_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_534_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_535_order__subst2,axiom,
! [A: num,B: num,F: num > int,C: int] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_536_order__subst2,axiom,
! [A: nat,B: nat,F: nat > rat,C: rat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_rat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_537_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_538_order__subst1,axiom,
! [A: rat,F: rat > rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_539_order__subst1,axiom,
! [A: rat,F: num > rat,B: num,C: num] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_540_order__subst1,axiom,
! [A: rat,F: nat > rat,B: nat,C: nat] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_541_order__subst1,axiom,
! [A: rat,F: int > rat,B: int,C: int] :
( ( ord_less_eq_rat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_542_order__subst1,axiom,
! [A: num,F: rat > num,B: rat,C: rat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_543_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_544_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_545_order__subst1,axiom,
! [A: num,F: int > num,B: int,C: int] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_546_order__subst1,axiom,
! [A: nat,F: rat > nat,B: rat,C: rat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ! [X3: rat,Y3: rat] :
( ( ord_less_eq_rat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_547_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_548_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : Y6 = Z3 )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_549_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: rat,Z3: rat] : Y6 = Z3 )
= ( ^ [A4: rat,B3: rat] :
( ( ord_less_eq_rat @ A4 @ B3 )
& ( ord_less_eq_rat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_550_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z3: num] : Y6 = Z3 )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_551_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_552_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : Y6 = Z3 )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_553_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_554_antisym,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_555_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_556_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_557_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_558_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_559_dual__order_Otrans,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_560_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_561_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_562_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_563_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_564_dual__order_Oantisym,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_565_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_566_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_567_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_568_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : Y6 = Z3 )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_569_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: rat,Z3: rat] : Y6 = Z3 )
= ( ^ [A4: rat,B3: rat] :
( ( ord_less_eq_rat @ B3 @ A4 )
& ( ord_less_eq_rat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_570_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: num,Z3: num] : Y6 = Z3 )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_571_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_572_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : Y6 = Z3 )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_573_linorder__wlog,axiom,
! [P: rat > rat > $o,A: rat,B: rat] :
( ! [A3: rat,B2: rat] :
( ( ord_less_eq_rat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: rat,B2: rat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_574_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A3: num,B2: num] :
( ( ord_less_eq_num @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: num,B2: num] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_575_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_576_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_577_order__trans,axiom,
! [X: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_578_order__trans,axiom,
! [X: rat,Y4: rat,Z2: rat] :
( ( ord_less_eq_rat @ X @ Y4 )
=> ( ( ord_less_eq_rat @ Y4 @ Z2 )
=> ( ord_less_eq_rat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_579_order__trans,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ Z2 )
=> ( ord_less_eq_num @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_580_order__trans,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_581_order__trans,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_582_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_583_order_Otrans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% order.trans
thf(fact_584_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_585_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_586_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_587_order__antisym,axiom,
! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_588_order__antisym,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ Y4 )
=> ( ( ord_less_eq_rat @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_589_order__antisym,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_590_order__antisym,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_591_order__antisym,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_592_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_593_ord__le__eq__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_594_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_595_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_596_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_597_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_598_ord__eq__le__trans,axiom,
! [A: rat,B: rat,C: rat] :
( ( A = B )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_599_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_600_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_601_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_602_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : Y6 = Z3 )
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ( ord_less_eq_set_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_603_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: rat,Z3: rat] : Y6 = Z3 )
= ( ^ [X2: rat,Y: rat] :
( ( ord_less_eq_rat @ X2 @ Y )
& ( ord_less_eq_rat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_604_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z3: num] : Y6 = Z3 )
= ( ^ [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
& ( ord_less_eq_num @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_605_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_606_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z3: int] : Y6 = Z3 )
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_607_le__cases3,axiom,
! [X: rat,Y4: rat,Z2: rat] :
( ( ( ord_less_eq_rat @ X @ Y4 )
=> ~ ( ord_less_eq_rat @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_rat @ Y4 @ X )
=> ~ ( ord_less_eq_rat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_rat @ X @ Z2 )
=> ~ ( ord_less_eq_rat @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_rat @ Z2 @ Y4 )
=> ~ ( ord_less_eq_rat @ Y4 @ X ) )
=> ( ( ( ord_less_eq_rat @ Y4 @ Z2 )
=> ~ ( ord_less_eq_rat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_rat @ Z2 @ X )
=> ~ ( ord_less_eq_rat @ X @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_608_le__cases3,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ( ord_less_eq_num @ X @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ X )
=> ~ ( ord_less_eq_num @ X @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ X ) )
=> ( ( ( ord_less_eq_num @ Y4 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X )
=> ~ ( ord_less_eq_num @ X @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_609_le__cases3,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_610_le__cases3,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ X ) )
=> ( ( ( ord_less_eq_int @ Y4 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_611_nle__le,axiom,
! [A: rat,B: rat] :
( ( ~ ( ord_less_eq_rat @ A @ B ) )
= ( ( ord_less_eq_rat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_612_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_613_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_614_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_615_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_616_verit__comp__simplify1_I2_J,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_617_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_618_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_619_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_620_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_621_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_622_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_623_maxt__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ X ) )
=> ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% maxt_corr
thf(fact_624_maxt__sound,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
=> ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ X ) ) ) ) ).
% maxt_sound
thf(fact_625__092_060open_062high_Ares_A_Ideg_Adiv_A2_J_A_061_Apr_092_060close_062,axiom,
( ( vEBT_VEBT_high @ res @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= pr ) ).
% \<open>high res (deg div 2) = pr\<close>
thf(fact_626_greater__shift,axiom,
( ord_less_nat
= ( ^ [Y: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% greater_shift
thf(fact_627_mint__corr__help,axiom,
! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= ( some_nat @ Mini ) )
=> ( ( vEBT_vebt_member @ T @ X )
=> ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).
% mint_corr_help
thf(fact_628_aaa,axiom,
( ( ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% aaa
thf(fact_629__092_060open_062z_A_060_A2_A_094_Adeg_092_060close_062,axiom,
ord_less_nat @ za @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% \<open>z < 2 ^ deg\<close>
thf(fact_630__C33_C,axiom,
~ ? [U: nat] :
( ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U )
& ( ord_less_nat @ U @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% "33"
thf(fact_631__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% \<open>invar_vebt (treeList ! high x (deg div 2)) n\<close>
thf(fact_632_mint__member,axiom,
! [T: vEBT_VEBT,N: nat,Maxi: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= ( some_nat @ Maxi ) )
=> ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% mint_member
thf(fact_633_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_634_abh,axiom,
vEBT_vebt_member @ summary @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% abh
thf(fact_635__092_060open_062z_A_092_060noteq_062_Ama_092_060close_062,axiom,
za != ma ).
% \<open>z \<noteq> ma\<close>
thf(fact_636__092_060open_062mi_A_092_060noteq_062_Ama_092_060close_062,axiom,
mi != ma ).
% \<open>mi \<noteq> ma\<close>
thf(fact_637_power__shift,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ( power_power_nat @ X @ Y4 )
= Z2 )
= ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y4 ) )
= ( some_nat @ Z2 ) ) ) ).
% power_shift
thf(fact_638_set__vebt__set__vebt_H__valid,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_set_vebt @ T )
= ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_639__C5_Ohyps_C_I10_J,axiom,
ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% "5.hyps"(10)
thf(fact_640__092_060open_062x_A_092_060le_062_Ama_092_060close_062,axiom,
ord_less_eq_nat @ xa @ ma ).
% \<open>x \<le> ma\<close>
thf(fact_641__092_060open_062res_A_060_Ax_092_060close_062,axiom,
ord_less_nat @ res @ xa ).
% \<open>res < x\<close>
thf(fact_642__092_060open_062vebt__member_Asummary_Apr_092_060close_062,axiom,
vEBT_vebt_member @ summary @ pr ).
% \<open>vebt_member summary pr\<close>
thf(fact_643__092_060open_062res_A_060_Ama_092_060close_062,axiom,
ord_less_nat @ res @ ma ).
% \<open>res < ma\<close>
thf(fact_644__092_060open_062mi_A_060_Ares_092_060close_062,axiom,
ord_less_nat @ mi @ res ).
% \<open>mi < res\<close>
thf(fact_645_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X2: nat,N2: nat] : ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% high_def
thf(fact_646__C5_Ohyps_C_I2_J,axiom,
vEBT_invar_vebt @ summary @ m ).
% "5.hyps"(2)
thf(fact_647_mint__sound,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
=> ( ( vEBT_vebt_mint @ T )
= ( some_nat @ X ) ) ) ) ).
% mint_sound
thf(fact_648_mint__corr,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= ( some_nat @ X ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% mint_corr
thf(fact_649_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_650_misiz,axiom,
! [T: vEBT_VEBT,N: nat,M: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( some_nat @ M )
= ( vEBT_vebt_mint @ T ) )
=> ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% misiz
thf(fact_651_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_652_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_653_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_654_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_655_pred__member,axiom,
! [T: vEBT_VEBT,X: nat,Y4: nat] :
( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y4 )
= ( ( vEBT_vebt_member @ T @ Y4 )
& ( ord_less_nat @ Y4 @ X )
& ! [Z4: nat] :
( ( ( vEBT_vebt_member @ T @ Z4 )
& ( ord_less_nat @ Z4 @ X ) )
=> ( ord_less_eq_nat @ Z4 @ Y4 ) ) ) ) ).
% pred_member
thf(fact_656__092_060open_062vebt__pred_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Apr_092_060close_062,axiom,
( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( some_nat @ pr ) ) ).
% \<open>vebt_pred summary (high x (deg div 2)) = Some pr\<close>
thf(fact_657__092_060open_062is__pred__in__set_A_Iset__vebt_H_Asummary_J_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_Apr_092_060close_062,axiom,
vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ pr ).
% \<open>is_pred_in_set (set_vebt' summary) (high x (deg div 2)) pr\<close>
thf(fact_658_post__member__pre__member,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y4: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ Y4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y4 )
=> ( ( vEBT_vebt_member @ T @ Y4 )
| ( X = Y4 ) ) ) ) ) ) ).
% post_member_pre_member
thf(fact_659__C20_C,axiom,
( ( za = mi )
| ( za = ma )
| ( ( ord_less_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) )
& ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% "20"
thf(fact_660__092_060open_062high_Ax_An_A_060_A2_A_094_Am_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
& ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% \<open>high x n < 2 ^ m \<and> low x n < 2 ^ n\<close>
thf(fact_661__092_060open_062high_Az_A_Ideg_Adiv_A2_J_A_060_A2_A_094_Am_092_060close_062,axiom,
ord_less_nat @ ( vEBT_VEBT_high @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).
% \<open>high z (deg div 2) < 2 ^ m\<close>
thf(fact_662_i1,axiom,
( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
= none_nat )
| ~ ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% i1
thf(fact_663__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062pr_O_Avebt__pred_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Apr_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Pr: nat] :
( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
!= ( some_nat @ Pr ) ) ).
% \<open>\<And>thesis. (\<And>pr. vebt_pred summary (high x (deg div 2)) = Some pr \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_664__092_060open_062pr_A_060_A2_A_094_Am_092_060close_062,axiom,
ord_less_nat @ pr @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).
% \<open>pr < 2 ^ m\<close>
thf(fact_665__092_060open_062both__member__options_A_ItreeList_A_B_Ahigh_Ares_A_Ideg_Adiv_A2_J_J_A_Ilow_Ares_A_Ideg_Adiv_A2_J_J_092_060close_062,axiom,
vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ res @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ res @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% \<open>both_member_options (treeList ! high res (deg div 2)) (low res (deg div 2))\<close>
thf(fact_666__C5_Ohyps_C_I11_J,axiom,
( ( mi != ma )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ na )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ na )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
=> ( ( ord_less_nat @ mi @ X5 )
& ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).
% "5.hyps"(11)
thf(fact_667_fgh,axiom,
( ( vEBT_VEBT_set_vebt @ ( nth_VEBT_VEBT @ treeList @ pr ) )
!= bot_bot_set_nat ) ).
% fgh
thf(fact_668_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_669_both__member__options__equiv__member,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
= ( vEBT_vebt_member @ T @ X ) ) ) ).
% both_member_options_equiv_member
thf(fact_670_valid__member__both__member__options,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
=> ( vEBT_vebt_member @ T @ X ) ) ) ).
% valid_member_both_member_options
thf(fact_671_maxbmo,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( ( vEBT_vebt_maxt @ T )
= ( some_nat @ X ) )
=> ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).
% maxbmo
thf(fact_672__C5_Ohyps_C_I4_J,axiom,
( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% "5.hyps"(4)
thf(fact_673__092_060open_062both__member__options_Asummary_Apr_092_060close_062,axiom,
vEBT_V8194947554948674370ptions @ summary @ pr ).
% \<open>both_member_options summary pr\<close>
thf(fact_674__092_060open_062_092_060exists_062maxy_O_Aboth__member__options_A_ItreeList_A_B_Apr_J_Amaxy_092_060close_062,axiom,
? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ pr ) @ X_1 ) ).
% \<open>\<exists>maxy. both_member_options (treeList ! pr) maxy\<close>
thf(fact_675_mint__corr__help__empty,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_mint @ T )
= none_nat )
=> ( ( vEBT_VEBT_set_vebt @ T )
= bot_bot_set_nat ) ) ) ).
% mint_corr_help_empty
thf(fact_676_maxt__corr__help__empty,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ( vEBT_vebt_maxt @ T )
= none_nat )
=> ( ( vEBT_VEBT_set_vebt @ T )
= bot_bot_set_nat ) ) ) ).
% maxt_corr_help_empty
thf(fact_677__C5_Ohyps_C_I3_J,axiom,
! [X: nat,Px: nat] :
( ( ( vEBT_vebt_pred @ summary @ X )
= ( some_nat @ Px ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X @ Px ) ) ).
% "5.hyps"(3)
thf(fact_678_valid__insert__both__member__options__pres,axiom,
! [T: vEBT_VEBT,N: nat,X: nat,Y4: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ Y4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y4 ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_679_valid__insert__both__member__options__add,axiom,
! [T: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_680__092_060open_062length_AtreeList_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_Am_092_060close_062,axiom,
( ( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
& ( vEBT_invar_vebt @ summary @ m ) ) ).
% \<open>length treeList = 2 ^ m \<and> invar_vebt summary m\<close>
thf(fact_681__C5_Ohyps_C_I5_J,axiom,
( m
= ( suc @ na ) ) ).
% "5.hyps"(5)
thf(fact_682__C5_Ohyps_C_I6_J,axiom,
( deg
= ( plus_plus_nat @ na @ m ) ) ).
% "5.hyps"(6)
thf(fact_683_local_Opower__def,axiom,
( vEBT_VEBT_power
= ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% local.power_def
thf(fact_684__C5_Ohyps_C_I1_J,axiom,
! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
=> ( ( vEBT_invar_vebt @ X5 @ na )
& ! [Xa: nat,Xb: nat] :
( ( ( vEBT_vebt_pred @ X5 @ Xa )
= ( some_nat @ Xb ) )
= ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ X5 ) @ Xa @ Xb ) ) ) ) ).
% "5.hyps"(1)
thf(fact_685__C5_Ohyps_C_I7_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ summary @ I ) ) ) ).
% "5.hyps"(7)
thf(fact_686_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_le72135733267957522d_enat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_687_bot_Oextremum__uniqueI,axiom,
! [A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
=> ( A = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_uniqueI
thf(fact_688_bot_Oextremum__uniqueI,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
=> ( A = bot_bot_set_int ) ) ).
% bot.extremum_uniqueI
thf(fact_689_bot_Oextremum__uniqueI,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
=> ( A = bot_bot_set_real ) ) ).
% bot.extremum_uniqueI
thf(fact_690_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_691_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_692_bot_Oextremum__unique,axiom,
! [A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
= ( A = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_unique
thf(fact_693_bot_Oextremum__unique,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% bot.extremum_unique
thf(fact_694_bot_Oextremum__unique,axiom,
! [A: set_real] :
( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
= ( A = bot_bot_set_real ) ) ).
% bot.extremum_unique
thf(fact_695_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_696_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_697_bot_Oextremum,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).
% bot.extremum
thf(fact_698_bot_Oextremum,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% bot.extremum
thf(fact_699_bot_Oextremum,axiom,
! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% bot.extremum
thf(fact_700_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_701_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_702_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_703_bot_Oextremum__strict,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).
% bot.extremum_strict
thf(fact_704_bot_Oextremum__strict,axiom,
! [A: set_int] :
~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% bot.extremum_strict
thf(fact_705_bot_Oextremum__strict,axiom,
! [A: set_real] :
~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% bot.extremum_strict
thf(fact_706_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_707_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_708_bot_Onot__eq__extremum,axiom,
! [A: extended_enat] :
( ( A != bot_bo4199563552545308370d_enat )
= ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_709_bot_Onot__eq__extremum,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
= ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% bot.not_eq_extremum
thf(fact_710_bot_Onot__eq__extremum,axiom,
! [A: set_real] :
( ( A != bot_bot_set_real )
= ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% bot.not_eq_extremum
thf(fact_711_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_712_is__pred__in__set__def,axiom,
( vEBT_is_pred_in_set
= ( ^ [Xs: set_nat,X2: nat,Y: nat] :
( ( member_nat @ Y @ Xs )
& ( ord_less_nat @ Y @ X2 )
& ! [Z4: nat] :
( ( member_nat @ Z4 @ Xs )
=> ( ( ord_less_nat @ Z4 @ X2 )
=> ( ord_less_eq_nat @ Z4 @ Y ) ) ) ) ) ) ).
% is_pred_in_set_def
thf(fact_713_power__divide,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_divide
thf(fact_714_power__divide,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
= ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_divide
thf(fact_715_power__divide,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
= ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% power_divide
thf(fact_716_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_717_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_718_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_719_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ N2 ) ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) ) ) ).
% in_children_def
thf(fact_720__092_060open_062max__in__set_A_Iset__vebt_H_A_ItreeList_A_B_Athe_A_Ivebt__pred_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_Amaxy_092_060close_062,axiom,
vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ maxy ).
% \<open>max_in_set (set_vebt' (treeList ! the (vebt_pred summary (high x (deg div 2))))) maxy\<close>
thf(fact_721__092_060open_062Some_Amaxy_A_061_Avebt__maxt_A_ItreeList_A_B_Athe_A_Ivebt__pred_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
( ( some_nat @ maxy )
= ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% \<open>Some maxy = vebt_maxt (treeList ! the (vebt_pred summary (high x (deg div 2))))\<close>
thf(fact_722_scmem,axiom,
vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ maxy ).
% scmem
thf(fact_723__092_060open_062invar__vebt_A_ItreeList_A_B_Athe_A_Ivebt__pred_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_An_092_060close_062,axiom,
vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ na ).
% \<open>invar_vebt (treeList ! the (vebt_pred summary (high x (deg div 2)))) n\<close>
thf(fact_724_both__member__options__ding,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_725_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= bot_bot_set_nat ) ).
% buildup_gives_empty
thf(fact_726_minNullmin,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ T )
=> ( ( vEBT_vebt_mint @ T )
= none_nat ) ) ).
% minNullmin
thf(fact_727_minminNull,axiom,
! [T: vEBT_VEBT] :
( ( ( vEBT_vebt_mint @ T )
= none_nat )
=> ( vEBT_VEBT_minNull @ T ) ) ).
% minminNull
thf(fact_728_not__None__eq,axiom,
! [X: option_nat] :
( ( X != none_nat )
= ( ? [Y: nat] :
( X
= ( some_nat @ Y ) ) ) ) ).
% not_None_eq
thf(fact_729_not__None__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( X != none_P5556105721700978146at_nat )
= ( ? [Y: product_prod_nat_nat] :
( X
= ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% not_None_eq
thf(fact_730_not__None__eq,axiom,
! [X: option_num] :
( ( X != none_num )
= ( ? [Y: num] :
( X
= ( some_num @ Y ) ) ) ) ).
% not_None_eq
thf(fact_731_not__Some__eq,axiom,
! [X: option_nat] :
( ( ! [Y: nat] :
( X
!= ( some_nat @ Y ) ) )
= ( X = none_nat ) ) ).
% not_Some_eq
thf(fact_732_not__Some__eq,axiom,
! [X: option4927543243414619207at_nat] :
( ( ! [Y: product_prod_nat_nat] :
( X
!= ( some_P7363390416028606310at_nat @ Y ) ) )
= ( X = none_P5556105721700978146at_nat ) ) ).
% not_Some_eq
thf(fact_733_not__Some__eq,axiom,
! [X: option_num] :
( ( ! [Y: num] :
( X
!= ( some_num @ Y ) ) )
= ( X = none_num ) ) ).
% not_Some_eq
thf(fact_734_even__odd__cases,axiom,
! [X: nat] :
( ! [N3: nat] :
( X
!= ( plus_plus_nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X
!= ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_735_deg__deg__n,axiom,
! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_736_not__min__Null__member,axiom,
! [T: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T )
=> ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% not_min_Null_member
thf(fact_737_min__Null__member,axiom,
! [T: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T )
=> ~ ( vEBT_vebt_member @ T @ X ) ) ).
% min_Null_member
thf(fact_738_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S ) ) ) ).
% deg_SUcn_Node
thf(fact_739_inthall,axiom,
! [Xs2: list_real,P: real > $o,N: nat] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
=> ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_740_inthall,axiom,
! [Xs2: list_complex,P: complex > $o,N: nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_741_inthall,axiom,
! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N: nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_742_inthall,axiom,
! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_743_inthall,axiom,
! [Xs2: list_o,P: $o > $o,N: nat] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
=> ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_744_inthall,axiom,
! [Xs2: list_nat,P: nat > $o,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_745_inthall,axiom,
! [Xs2: list_int,P: int > $o,N: nat] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
=> ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% inthall
thf(fact_746__C5_Ohyps_C_I8_J,axiom,
( ( mi = ma )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ).
% "5.hyps"(8)
thf(fact_747_option_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( some_nat @ X22 )
= ( some_nat @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_748_option_Oinject,axiom,
! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ( some_P7363390416028606310at_nat @ X22 )
= ( some_P7363390416028606310at_nat @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_749_option_Oinject,axiom,
! [X22: num,Y2: num] :
( ( ( some_num @ X22 )
= ( some_num @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_750_pow__sum,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% pow_sum
thf(fact_751_high__bound__aux,axiom,
! [Ma: nat,N: nat,M: nat] :
( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_752_add__numeral__left,axiom,
! [V: num,W2: num,Z2: complex] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W2 ) @ Z2 ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_753_add__numeral__left,axiom,
! [V: num,W2: num,Z2: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_754_add__numeral__left,axiom,
! [V: num,W2: num,Z2: rat] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_755_add__numeral__left,axiom,
! [V: num,W2: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_756_add__numeral__left,axiom,
! [V: num,W2: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_757_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_758_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_759_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_760_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_761_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_762_option_Ocollapse,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( ( some_nat @ ( the_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_763_option_Ocollapse,axiom,
! [Option: option4927543243414619207at_nat] :
( ( Option != none_P5556105721700978146at_nat )
=> ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_764_option_Ocollapse,axiom,
! [Option: option_num] :
( ( Option != none_num )
=> ( ( some_num @ ( the_num @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_765_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_766_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_767_div2__Suc__Suc,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_768_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= M ) ).
% add_self_div_2
thf(fact_769__092_060open_062vebt__member_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Az_A_092_060and_062_Az_A_060_Ax_092_060close_062,axiom,
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ za )
& ( ord_less_nat @ za @ xa ) ) ).
% \<open>vebt_member (Node (Some (mi, ma)) deg treeList summary) z \<and> z < x\<close>
thf(fact_770_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_771_is__num__normalize_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_772_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_773_option_Osel,axiom,
! [X22: nat] :
( ( the_nat @ ( some_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_774_option_Osel,axiom,
! [X22: product_prod_nat_nat] :
( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_775_option_Osel,axiom,
! [X22: num] :
( ( the_num @ ( some_num @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_776_option_Oexpand,axiom,
! [Option: option_nat,Option2: option_nat] :
( ( ( Option = none_nat )
= ( Option2 = none_nat ) )
=> ( ( ( Option != none_nat )
=> ( ( Option2 != none_nat )
=> ( ( the_nat @ Option )
= ( the_nat @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_777_option_Oexpand,axiom,
! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
( ( ( Option = none_P5556105721700978146at_nat )
= ( Option2 = none_P5556105721700978146at_nat ) )
=> ( ( ( Option != none_P5556105721700978146at_nat )
=> ( ( Option2 != none_P5556105721700978146at_nat )
=> ( ( the_Pr8591224930841456533at_nat @ Option )
= ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_778_option_Oexpand,axiom,
! [Option: option_num,Option2: option_num] :
( ( ( Option = none_num )
= ( Option2 = none_num ) )
=> ( ( ( Option != none_num )
=> ( ( Option2 != none_num )
=> ( ( the_num @ Option )
= ( the_num @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_779_vebt__pred_Osimps_I4_J,axiom,
! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
= none_nat ) ).
% vebt_pred.simps(4)
thf(fact_780_option_Oexhaust__sel,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( Option
= ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_781_option_Oexhaust__sel,axiom,
! [Option: option4927543243414619207at_nat] :
( ( Option != none_P5556105721700978146at_nat )
=> ( Option
= ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_782_option_Oexhaust__sel,axiom,
! [Option: option_num] :
( ( Option != none_num )
=> ( Option
= ( some_num @ ( the_num @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_783_numeral__Bit0,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_Bit0
thf(fact_784_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_785_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_Bit0
thf(fact_786_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_787_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_788_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_789_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_nat > $o,Y4: option_nat] :
( ( ( X = none_nat )
=> ( P @ X @ Y4 ) )
=> ( ( ( Y4 = none_nat )
=> ( P @ X @ Y4 ) )
=> ( ! [A3: nat,B2: nat] :
( ( X
= ( some_nat @ A3 ) )
=> ( ( Y4
= ( some_nat @ B2 ) )
=> ( P @ X @ Y4 ) ) )
=> ( P @ X @ Y4 ) ) ) ) ).
% combine_options_cases
thf(fact_790_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y4: option4927543243414619207at_nat] :
( ( ( X = none_nat )
=> ( P @ X @ Y4 ) )
=> ( ( ( Y4 = none_P5556105721700978146at_nat )
=> ( P @ X @ Y4 ) )
=> ( ! [A3: nat,B2: product_prod_nat_nat] :
( ( X
= ( some_nat @ A3 ) )
=> ( ( Y4
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( P @ X @ Y4 ) ) )
=> ( P @ X @ Y4 ) ) ) ) ).
% combine_options_cases
thf(fact_791_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_num > $o,Y4: option_num] :
( ( ( X = none_nat )
=> ( P @ X @ Y4 ) )
=> ( ( ( Y4 = none_num )
=> ( P @ X @ Y4 ) )
=> ( ! [A3: nat,B2: num] :
( ( X
= ( some_nat @ A3 ) )
=> ( ( Y4
= ( some_num @ B2 ) )
=> ( P @ X @ Y4 ) ) )
=> ( P @ X @ Y4 ) ) ) ) ).
% combine_options_cases
thf(fact_792_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y4: option_nat] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y4 ) )
=> ( ( ( Y4 = none_nat )
=> ( P @ X @ Y4 ) )
=> ( ! [A3: product_prod_nat_nat,B2: nat] :
( ( X
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ( ( Y4
= ( some_nat @ B2 ) )
=> ( P @ X @ Y4 ) ) )
=> ( P @ X @ Y4 ) ) ) ) ).
% combine_options_cases
thf(fact_793_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y4: option4927543243414619207at_nat] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y4 ) )
=> ( ( ( Y4 = none_P5556105721700978146at_nat )
=> ( P @ X @ Y4 ) )
=> ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( X
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ( ( Y4
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( P @ X @ Y4 ) ) )
=> ( P @ X @ Y4 ) ) ) ) ).
% combine_options_cases
thf(fact_794_combine__options__cases,axiom,
! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y4: option_num] :
( ( ( X = none_P5556105721700978146at_nat )
=> ( P @ X @ Y4 ) )
=> ( ( ( Y4 = none_num )
=> ( P @ X @ Y4 ) )
=> ( ! [A3: product_prod_nat_nat,B2: num] :
( ( X
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ( ( Y4
= ( some_num @ B2 ) )
=> ( P @ X @ Y4 ) ) )
=> ( P @ X @ Y4 ) ) ) ) ).
% combine_options_cases
thf(fact_795_combine__options__cases,axiom,
! [X: option_num,P: option_num > option_nat > $o,Y4: option_nat] :
( ( ( X = none_num )
=> ( P @ X @ Y4 ) )
=> ( ( ( Y4 = none_nat )
=> ( P @ X @ Y4 ) )
=> ( ! [A3: num,B2: nat] :
( ( X
= ( some_num @ A3 ) )
=> ( ( Y4
= ( some_nat @ B2 ) )
=> ( P @ X @ Y4 ) ) )
=> ( P @ X @ Y4 ) ) ) ) ).
% combine_options_cases
thf(fact_796_combine__options__cases,axiom,
! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y4: option4927543243414619207at_nat] :
( ( ( X = none_num )
=> ( P @ X @ Y4 ) )
=> ( ( ( Y4 = none_P5556105721700978146at_nat )
=> ( P @ X @ Y4 ) )
=> ( ! [A3: num,B2: product_prod_nat_nat] :
( ( X
= ( some_num @ A3 ) )
=> ( ( Y4
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( P @ X @ Y4 ) ) )
=> ( P @ X @ Y4 ) ) ) ) ).
% combine_options_cases
thf(fact_797_combine__options__cases,axiom,
! [X: option_num,P: option_num > option_num > $o,Y4: option_num] :
( ( ( X = none_num )
=> ( P @ X @ Y4 ) )
=> ( ( ( Y4 = none_num )
=> ( P @ X @ Y4 ) )
=> ( ! [A3: num,B2: num] :
( ( X
= ( some_num @ A3 ) )
=> ( ( Y4
= ( some_num @ B2 ) )
=> ( P @ X @ Y4 ) ) )
=> ( P @ X @ Y4 ) ) ) ) ).
% combine_options_cases
thf(fact_798_split__option__all,axiom,
( ( ^ [P2: option_nat > $o] :
! [X4: option_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
& ! [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_799_split__option__all,axiom,
( ( ^ [P2: option4927543243414619207at_nat > $o] :
! [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option4927543243414619207at_nat > $o] :
( ( P3 @ none_P5556105721700978146at_nat )
& ! [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_800_split__option__all,axiom,
( ( ^ [P2: option_num > $o] :
! [X4: option_num] : ( P2 @ X4 ) )
= ( ^ [P3: option_num > $o] :
( ( P3 @ none_num )
& ! [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_801_split__option__ex,axiom,
( ( ^ [P2: option_nat > $o] :
? [X4: option_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
| ? [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_802_split__option__ex,axiom,
( ( ^ [P2: option4927543243414619207at_nat > $o] :
? [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
= ( ^ [P3: option4927543243414619207at_nat > $o] :
( ( P3 @ none_P5556105721700978146at_nat )
| ? [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_803_split__option__ex,axiom,
( ( ^ [P2: option_num > $o] :
? [X4: option_num] : ( P2 @ X4 ) )
= ( ^ [P3: option_num > $o] :
( ( P3 @ none_num )
| ? [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_804_option_Oexhaust,axiom,
! [Y4: option_nat] :
( ( Y4 != none_nat )
=> ~ ! [X23: nat] :
( Y4
!= ( some_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_805_option_Oexhaust,axiom,
! [Y4: option4927543243414619207at_nat] :
( ( Y4 != none_P5556105721700978146at_nat )
=> ~ ! [X23: product_prod_nat_nat] :
( Y4
!= ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_806_option_Oexhaust,axiom,
! [Y4: option_num] :
( ( Y4 != none_num )
=> ~ ! [X23: num] :
( Y4
!= ( some_num @ X23 ) ) ) ).
% option.exhaust
thf(fact_807_option_OdiscI,axiom,
! [Option: option_nat,X22: nat] :
( ( Option
= ( some_nat @ X22 ) )
=> ( Option != none_nat ) ) ).
% option.discI
thf(fact_808_option_OdiscI,axiom,
! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
( ( Option
= ( some_P7363390416028606310at_nat @ X22 ) )
=> ( Option != none_P5556105721700978146at_nat ) ) ).
% option.discI
thf(fact_809_option_OdiscI,axiom,
! [Option: option_num,X22: num] :
( ( Option
= ( some_num @ X22 ) )
=> ( Option != none_num ) ) ).
% option.discI
thf(fact_810_option_Odistinct_I1_J,axiom,
! [X22: nat] :
( none_nat
!= ( some_nat @ X22 ) ) ).
% option.distinct(1)
thf(fact_811_option_Odistinct_I1_J,axiom,
! [X22: product_prod_nat_nat] :
( none_P5556105721700978146at_nat
!= ( some_P7363390416028606310at_nat @ X22 ) ) ).
% option.distinct(1)
thf(fact_812_option_Odistinct_I1_J,axiom,
! [X22: num] :
( none_num
!= ( some_num @ X22 ) ) ).
% option.distinct(1)
thf(fact_813_snd,axiom,
( res
= ( the_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% snd
thf(fact_814_invar__vebt_Ointros_I3_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_815__092_060open_062res_A_061_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Apr_A_L_Amaxy_092_060close_062,axiom,
( res
= ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ pr ) @ maxy ) ) ).
% \<open>res = 2 ^ (deg div 2) * pr + maxy\<close>
thf(fact_816_invar__vebt_Ointros_I2_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_817__092_060open_062vebt__member_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ares_092_060close_062,axiom,
vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ res ).
% \<open>vebt_member (Node (Some (mi, ma)) deg treeList summary) res\<close>
thf(fact_818__092_060open_062both__member__options_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ares_092_060close_062,axiom,
vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ res ).
% \<open>both_member_options (Node (Some (mi, ma)) deg treeList summary) res\<close>
thf(fact_819_member__inv,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
& ( ( X = Mi )
| ( X = Ma )
| ( ( ord_less_nat @ X @ Ma )
& ( ord_less_nat @ Mi @ X )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_820_add__def,axiom,
( vEBT_VEBT_add
= ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% add_def
thf(fact_821_thisvalid,axiom,
vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% thisvalid
thf(fact_822_pred__list__to__short,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_eq_nat @ X @ Ma )
=> ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= none_nat ) ) ) ) ).
% pred_list_to_short
thf(fact_823_set__n__deg__not__0,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_824_add__shift,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ( plus_plus_nat @ X @ Y4 )
= Z2 )
= ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y4 ) )
= ( some_nat @ Z2 ) ) ) ).
% add_shift
thf(fact_825_mul__shift,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ( times_times_nat @ X @ Y4 )
= Z2 )
= ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y4 ) )
= ( some_nat @ Z2 ) ) ) ).
% mul_shift
thf(fact_826__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
ord_less_eq_nat @ one_one_nat @ na ).
% \<open>1 \<le> n\<close>
thf(fact_827_mul__def,axiom,
( vEBT_VEBT_mul
= ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% mul_def
thf(fact_828_VEBT_Oinject_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
= ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% VEBT.inject(1)
thf(fact_829_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
& ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_830_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_831_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_832_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_833_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_834_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_835_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z2: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ Z2 ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_836_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_837_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z2: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_838_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_839_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W2: num,Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Z2 ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_840_high__inv,axiom,
! [X: nat,N: nat,Y4: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
= Y4 ) ) ).
% high_inv
thf(fact_841_low__inv,axiom,
! [X: nat,N: nat,Y4: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_842_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).
% bit_concat_def
thf(fact_843_power__one,axiom,
! [N: nat] :
( ( power_power_rat @ one_one_rat @ N )
= one_one_rat ) ).
% power_one
thf(fact_844_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_845_power__one,axiom,
! [N: nat] :
( ( power_power_real @ one_one_real @ N )
= one_one_real ) ).
% power_one
thf(fact_846_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_847_power__one,axiom,
! [N: nat] :
( ( power_power_complex @ one_one_complex @ N )
= one_one_complex ) ).
% power_one
thf(fact_848_insert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_849_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_850_power__one__right,axiom,
! [A: real] :
( ( power_power_real @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_851_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_852_power__one__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_853_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_854_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
& ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_855_pred__max,axiom,
! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( some_nat @ Ma ) ) ) ) ).
% pred_max
thf(fact_856_distrib__left__numeral,axiom,
! [V: num,B: complex,C: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
= ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_857_distrib__left__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_858_distrib__left__numeral,axiom,
! [V: num,B: rat,C: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_859_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_860_distrib__left__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_861_distrib__right__numeral,axiom,
! [A: complex,B: complex,V: num] :
( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
= ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_862_distrib__right__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_863_distrib__right__numeral,axiom,
! [A: rat,B: rat,V: num] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_864_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_865_distrib__right__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_866_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_complex
= ( numera6690914467698888265omplex @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_867_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_868_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_rat
= ( numeral_numeral_rat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_869_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_870_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_871_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera6690914467698888265omplex @ N )
= one_one_complex )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_872_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_873_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_rat @ N )
= one_one_rat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_874_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_875_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_876_power__inject__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( power_power_real @ A @ M )
= ( power_power_real @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_877_power__inject__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ( power_power_rat @ A @ M )
= ( power_power_rat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_878_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_879_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_880_summaxma,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
=> ( ( Mi != Ma )
=> ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
= ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% summaxma
thf(fact_881_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_882_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_883_le__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_884_le__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_885_divide__le__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_886_divide__le__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_887_less__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_888_less__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_889_divide__less__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_890_divide__less__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_891_power__strict__increasing__iff,axiom,
! [B: real,X: nat,Y4: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y4 ) )
= ( ord_less_nat @ X @ Y4 ) ) ) ).
% power_strict_increasing_iff
thf(fact_892_power__strict__increasing__iff,axiom,
! [B: rat,X: nat,Y4: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y4 ) )
= ( ord_less_nat @ X @ Y4 ) ) ) ).
% power_strict_increasing_iff
thf(fact_893_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y4: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y4 ) )
= ( ord_less_nat @ X @ Y4 ) ) ) ).
% power_strict_increasing_iff
thf(fact_894_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y4: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y4 ) )
= ( ord_less_nat @ X @ Y4 ) ) ) ).
% power_strict_increasing_iff
thf(fact_895_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_896_power__add__numeral2,axiom,
! [A: complex,M: num,N: num,B: complex] :
( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_897_power__add__numeral2,axiom,
! [A: real,M: num,N: num,B: real] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_898_power__add__numeral2,axiom,
! [A: rat,M: num,N: num,B: rat] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_899_power__add__numeral2,axiom,
! [A: nat,M: num,N: num,B: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_900_power__add__numeral2,axiom,
! [A: int,M: num,N: num,B: int] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
= ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% power_add_numeral2
thf(fact_901_power__add__numeral,axiom,
! [A: complex,M: num,N: num] :
( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_902_power__add__numeral,axiom,
! [A: real,M: num,N: num] :
( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_903_power__add__numeral,axiom,
! [A: rat,M: num,N: num] :
( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_904_power__add__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_905_power__add__numeral,axiom,
! [A: int,M: num,N: num] :
( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% power_add_numeral
thf(fact_906_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_907_one__add__one,axiom,
( ( plus_plus_complex @ one_one_complex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_908_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_909_one__add__one,axiom,
( ( plus_plus_rat @ one_one_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_910_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_911_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_912_power__increasing__iff,axiom,
! [B: real,X: nat,Y4: nat] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y4 ) )
= ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% power_increasing_iff
thf(fact_913_power__increasing__iff,axiom,
! [B: rat,X: nat,Y4: nat] :
( ( ord_less_rat @ one_one_rat @ B )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y4 ) )
= ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% power_increasing_iff
thf(fact_914_power__increasing__iff,axiom,
! [B: nat,X: nat,Y4: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y4 ) )
= ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% power_increasing_iff
thf(fact_915_power__increasing__iff,axiom,
! [B: int,X: nat,Y4: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y4 ) )
= ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% power_increasing_iff
thf(fact_916_nested__mint,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( N
= ( suc @ ( suc @ Va ) ) )
=> ( ~ ( ord_less_nat @ Ma @ Mi )
=> ( ( Ma != Mi )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% nested_mint
thf(fact_917_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_918_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_919_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_920_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_921_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_922_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_923_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_924_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_925_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_926_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_927_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_928_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_929_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_930_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_931_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_932_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_933_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_934_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_935_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_936_fst,axiom,
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% fst
thf(fact_937__C2_C,axiom,
( ( ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
=> ( ( ( ord_less_nat @ mi @ xa )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( some_nat @ mi ) ) )
& ( ~ ( ord_less_nat @ mi @ xa )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= none_nat ) ) ) )
& ( ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
!= none_nat )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% "2"
thf(fact_938_left__right__inverse__power,axiom,
! [X: complex,Y4: complex,N: nat] :
( ( ( times_times_complex @ X @ Y4 )
= one_one_complex )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y4 @ N ) )
= one_one_complex ) ) ).
% left_right_inverse_power
thf(fact_939_left__right__inverse__power,axiom,
! [X: real,Y4: real,N: nat] :
( ( ( times_times_real @ X @ Y4 )
= one_one_real )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ N ) )
= one_one_real ) ) ).
% left_right_inverse_power
thf(fact_940_left__right__inverse__power,axiom,
! [X: rat,Y4: rat,N: nat] :
( ( ( times_times_rat @ X @ Y4 )
= one_one_rat )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y4 @ N ) )
= one_one_rat ) ) ).
% left_right_inverse_power
thf(fact_941_left__right__inverse__power,axiom,
! [X: nat,Y4: nat,N: nat] :
( ( ( times_times_nat @ X @ Y4 )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y4 @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_942_left__right__inverse__power,axiom,
! [X: int,Y4: int,N: nat] :
( ( ( times_times_int @ X @ Y4 )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y4 @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_943_power__less__power__Suc,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_944_power__less__power__Suc,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_945_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_946_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_947_power__gt1__lemma,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_948_power__gt1__lemma,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_949_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_950_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_951_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_952_power__commutes,axiom,
! [A: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_commutes
thf(fact_953_power__commutes,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_commutes
thf(fact_954_power__commutes,axiom,
! [A: rat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% power_commutes
thf(fact_955_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_956_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_957_power__mult__distrib,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_958_power__mult__distrib,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
= ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_959_power__mult__distrib,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
= ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_960_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_961_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_962_power__commuting__commutes,axiom,
! [X: complex,Y4: complex,N: nat] :
( ( ( times_times_complex @ X @ Y4 )
= ( times_times_complex @ Y4 @ X ) )
=> ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y4 )
= ( times_times_complex @ Y4 @ ( power_power_complex @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_963_power__commuting__commutes,axiom,
! [X: real,Y4: real,N: nat] :
( ( ( times_times_real @ X @ Y4 )
= ( times_times_real @ Y4 @ X ) )
=> ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y4 )
= ( times_times_real @ Y4 @ ( power_power_real @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_964_power__commuting__commutes,axiom,
! [X: rat,Y4: rat,N: nat] :
( ( ( times_times_rat @ X @ Y4 )
= ( times_times_rat @ Y4 @ X ) )
=> ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y4 )
= ( times_times_rat @ Y4 @ ( power_power_rat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_965_power__commuting__commutes,axiom,
! [X: nat,Y4: nat,N: nat] :
( ( ( times_times_nat @ X @ Y4 )
= ( times_times_nat @ Y4 @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y4 )
= ( times_times_nat @ Y4 @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_966_power__commuting__commutes,axiom,
! [X: int,Y4: int,N: nat] :
( ( ( times_times_int @ X @ Y4 )
= ( times_times_int @ Y4 @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y4 )
= ( times_times_int @ Y4 @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_967_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_968_power__mult,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_969_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_970_power__mult,axiom,
! [A: complex,M: nat,N: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_971_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_972_le__numeral__extra_I4_J,axiom,
ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% le_numeral_extra(4)
thf(fact_973_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_974_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_975_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_976_less__numeral__extra_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% less_numeral_extra(4)
thf(fact_977_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_978_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_979_left__add__mult__distrib,axiom,
! [I2: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_980_div__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% div_mult2_eq
thf(fact_981_mult__numeral__1__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_982_mult__numeral__1__right,axiom,
! [A: real] :
( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_983_mult__numeral__1__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_984_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_985_mult__numeral__1__right,axiom,
! [A: int] :
( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_986_mult__numeral__1,axiom,
! [A: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_987_mult__numeral__1,axiom,
! [A: real] :
( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_988_mult__numeral__1,axiom,
! [A: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_989_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_990_mult__numeral__1,axiom,
! [A: int] :
( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_991_power__Suc2,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_992_power__Suc2,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_993_power__Suc2,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_994_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_995_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_996_power__Suc,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% power_Suc
thf(fact_997_power__Suc,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% power_Suc
thf(fact_998_power__Suc,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% power_Suc
thf(fact_999_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_1000_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_1001_power__add,axiom,
! [A: complex,M: nat,N: nat] :
( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% power_add
thf(fact_1002_power__add,axiom,
! [A: real,M: nat,N: nat] :
( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% power_add
thf(fact_1003_power__add,axiom,
! [A: rat,M: nat,N: nat] :
( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_add
thf(fact_1004_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_1005_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_1006_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_1007_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% one_le_numeral
thf(fact_1008_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_1009_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_1010_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_1011_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% not_numeral_less_one
thf(fact_1012_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1013_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_1014_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% one_plus_numeral_commute
thf(fact_1015_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_1016_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% one_plus_numeral_commute
thf(fact_1017_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_1018_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_1019_numeral__One,axiom,
( ( numera6690914467698888265omplex @ one )
= one_one_complex ) ).
% numeral_One
thf(fact_1020_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_1021_numeral__One,axiom,
( ( numeral_numeral_rat @ one )
= one_one_rat ) ).
% numeral_One
thf(fact_1022_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_1023_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_1024_one__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% one_le_power
thf(fact_1025_one__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1026_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1027_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_1028_less__mult__imp__div__less,axiom,
! [M: nat,I2: nat,N: nat] :
( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_1029_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1030_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1031_power__one__over,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
= ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% power_one_over
thf(fact_1032_power__one__over,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% power_one_over
thf(fact_1033_power__one__over,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
= ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% power_one_over
thf(fact_1034_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1035_power__odd__eq,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_1036_power__odd__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_1037_power__odd__eq,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_1038_power__odd__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_1039_power__odd__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% power_odd_eq
thf(fact_1040_power__gt1,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1041_power__gt1,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1042_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1043_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1044_power__less__imp__less__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1045_power__less__imp__less__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1046_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1047_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1048_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_real @ one_one_real @ A )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_1049_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: rat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_rat @ one_one_rat @ A )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_1050_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_1051_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_1052_power__increasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_real @ one_one_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_1053_power__increasing,axiom,
! [N: nat,N4: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_1054_power__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_1055_power__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_1056_left__add__twice,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1057_left__add__twice,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1058_left__add__twice,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1059_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1060_left__add__twice,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_1061_mult__2__right,axiom,
! [Z2: complex] :
( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_1062_mult__2__right,axiom,
! [Z2: real] :
( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( plus_plus_real @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_1063_mult__2__right,axiom,
! [Z2: rat] :
( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_1064_mult__2__right,axiom,
! [Z2: nat] :
( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_1065_mult__2__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2_right
thf(fact_1066_mult__2,axiom,
! [Z2: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_1067_mult__2,axiom,
! [Z2: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_real @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_1068_mult__2,axiom,
! [Z2: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_1069_mult__2,axiom,
! [Z2: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_1070_mult__2,axiom,
! [Z2: int] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
= ( plus_plus_int @ Z2 @ Z2 ) ) ).
% mult_2
thf(fact_1071_power2__eq__square,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_complex @ A @ A ) ) ).
% power2_eq_square
thf(fact_1072_power2__eq__square,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ A @ A ) ) ).
% power2_eq_square
thf(fact_1073_power2__eq__square,axiom,
! [A: rat] :
( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_rat @ A @ A ) ) ).
% power2_eq_square
thf(fact_1074_power2__eq__square,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_nat @ A @ A ) ) ).
% power2_eq_square
thf(fact_1075_power2__eq__square,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_int @ A @ A ) ) ).
% power2_eq_square
thf(fact_1076_power4__eq__xxxx,axiom,
! [X: complex] :
( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_1077_power4__eq__xxxx,axiom,
! [X: real] :
( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_1078_power4__eq__xxxx,axiom,
! [X: rat] :
( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_1079_power4__eq__xxxx,axiom,
! [X: nat] :
( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_1080_power4__eq__xxxx,axiom,
! [X: int] :
( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% power4_eq_xxxx
thf(fact_1081_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1082_power__even__eq,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_1083_power__even__eq,axiom,
! [A: real,N: nat] :
( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_1084_power__even__eq,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_1085_power__even__eq,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power_even_eq
thf(fact_1086_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
=> ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
=> ( ( divide_divide_nat @ M @ N )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_1087_power__le__imp__le__exp,axiom,
! [A: real,M: nat,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1088_power__le__imp__le__exp,axiom,
! [A: rat,M: nat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1089_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1090_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1091_one__power2,axiom,
( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat ) ).
% one_power2
thf(fact_1092_one__power2,axiom,
( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_power2
thf(fact_1093_one__power2,axiom,
( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real ) ).
% one_power2
thf(fact_1094_one__power2,axiom,
( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_power2
thf(fact_1095_one__power2,axiom,
( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex ) ).
% one_power2
thf(fact_1096_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1097_power2__sum,axiom,
! [X: complex,Y4: complex] :
( ( power_power_complex @ ( plus_plus_complex @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% power2_sum
thf(fact_1098_power2__sum,axiom,
! [X: real,Y4: real] :
( ( power_power_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% power2_sum
thf(fact_1099_power2__sum,axiom,
! [X: rat,Y4: rat] :
( ( power_power_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% power2_sum
thf(fact_1100_power2__sum,axiom,
! [X: nat,Y4: nat] :
( ( power_power_nat @ ( plus_plus_nat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% power2_sum
thf(fact_1101_power2__sum,axiom,
! [X: int,Y4: int] :
( ( power_power_int @ ( plus_plus_int @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% power2_sum
thf(fact_1102_ex__power__ivl2,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1103_ex__power__ivl1,axiom,
! [B: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N3: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1104_invar__vebt_Ointros_I4_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less_nat @ Mi @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_1105_invar__vebt_Ointros_I5_J,axiom,
! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus_nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq_nat @ Mi @ Ma )
=> ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less_nat @ Mi @ X3 )
& ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_1106_mintlistlength,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
=> ( ( Mi != Ma )
=> ( ( ord_less_nat @ Mi @ Ma )
& ? [M4: nat] :
( ( ( some_nat @ M4 )
= ( vEBT_vebt_mint @ Summary ) )
& ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% mintlistlength
thf(fact_1107_sum__squares__bound,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y4 ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_1108_sum__squares__bound,axiom,
! [X: rat,Y4: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y4 ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_squares_bound
thf(fact_1109_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1110_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_1111__C1_C,axiom,
( ( ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
& ( ~ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( if_option_nat
@ ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ mi @ xa ) @ ( some_nat @ mi ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% "1"
thf(fact_1112_div__exp__eq,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_1113_div__exp__eq,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% div_exp_eq
thf(fact_1114_field__less__half__sum,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_1115_field__less__half__sum,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ Y4 )
=> ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% field_less_half_sum
thf(fact_1116_vebt__maxt_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some_nat @ Ma ) ) ).
% vebt_maxt.simps(3)
thf(fact_1117_vebt__mint_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
= ( some_nat @ Mi ) ) ).
% vebt_mint.simps(3)
thf(fact_1118_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1119_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1120_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% set_vebt'_def
thf(fact_1121_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1122_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_1123_power__minus__is__div,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
= ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% power_minus_is_div
thf(fact_1124_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_1125_bits__div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% bits_div_by_1
thf(fact_1126_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1127_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1128_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1129_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1130_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1131_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1132_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1133_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_1134_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1135_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1136_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1137_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_1138_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_1139_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_1140_right__diff__distrib__numeral,axiom,
! [V: num,B: complex,C: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1141_right__diff__distrib__numeral,axiom,
! [V: num,B: real,C: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1142_right__diff__distrib__numeral,axiom,
! [V: num,B: rat,C: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1143_right__diff__distrib__numeral,axiom,
! [V: num,B: int,C: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1144_left__diff__distrib__numeral,axiom,
! [A: complex,B: complex,V: num] :
( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
= ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1145_left__diff__distrib__numeral,axiom,
! [A: real,B: real,V: num] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
= ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1146_left__diff__distrib__numeral,axiom,
! [A: rat,B: rat,V: num] :
( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1147_left__diff__distrib__numeral,axiom,
! [A: int,B: int,V: num] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
= ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1148_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1149_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1150_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1151_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1152_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1153_power__mult__numeral,axiom,
! [A: nat,M: num,N: num] :
( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_1154_power__mult__numeral,axiom,
! [A: real,M: num,N: num] :
( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_1155_power__mult__numeral,axiom,
! [A: int,M: num,N: num] :
( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_1156_power__mult__numeral,axiom,
! [A: complex,M: num,N: num] :
( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
= ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% power_mult_numeral
thf(fact_1157_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1158_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1159_pred__lesseq__max,axiom,
! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_eq_nat @ X @ Ma )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
@ none_nat ) ) ) ) ).
% pred_lesseq_max
thf(fact_1160_pred__less__length__list,axiom,
! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( ord_less_eq_nat @ X @ Ma )
=> ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% pred_less_length_list
thf(fact_1161_add__diff__add,axiom,
! [A: real,C: real,B: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_1162_add__diff__add,axiom,
! [A: rat,C: rat,B: rat,D: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
= ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% add_diff_add
thf(fact_1163_add__diff__add,axiom,
! [A: int,C: int,B: int,D: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
= ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% add_diff_add
thf(fact_1164_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1165_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1166_diff__less__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1167_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1168_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1169_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1170_diff__le__mono,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1171_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1172_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1173_diff__le__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_1174_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1175_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1176_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1177_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1178_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1179_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1180_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
= ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_code(2)
thf(fact_1181_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_1182_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit0 @ N ) )
= ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_code(2)
thf(fact_1183_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_1184_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_1185_mult__diff__mult,axiom,
! [X: real,Y4: real,A: real,B: real] :
( ( minus_minus_real @ ( times_times_real @ X @ Y4 ) @ ( times_times_real @ A @ B ) )
= ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y4 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_1186_mult__diff__mult,axiom,
! [X: rat,Y4: rat,A: rat,B: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ Y4 ) @ ( times_times_rat @ A @ B ) )
= ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y4 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_1187_mult__diff__mult,axiom,
! [X: int,Y4: int,A: int,B: int] :
( ( minus_minus_int @ ( times_times_int @ X @ Y4 ) @ ( times_times_int @ A @ B ) )
= ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y4 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_1188_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% set_vebt_def
thf(fact_1189_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1190_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1191_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1192_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1193_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1194_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1195_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1196_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1197_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1198_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1199_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1200_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1201_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1202_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_1203_power__numeral__even,axiom,
! [Z2: complex,W2: num] :
( ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
= ( times_times_complex @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_numeral_even
thf(fact_1204_power__numeral__even,axiom,
! [Z2: real,W2: num] :
( ( power_power_real @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
= ( times_times_real @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_numeral_even
thf(fact_1205_power__numeral__even,axiom,
! [Z2: rat,W2: num] :
( ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
= ( times_times_rat @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_numeral_even
thf(fact_1206_power__numeral__even,axiom,
! [Z2: nat,W2: num] :
( ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
= ( times_times_nat @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_numeral_even
thf(fact_1207_power__numeral__even,axiom,
! [Z2: int,W2: num] :
( ( power_power_int @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
= ( times_times_int @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_numeral_even
thf(fact_1208_four__x__squared,axiom,
! [X: real] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% four_x_squared
thf(fact_1209_L2__set__mult__ineq__lemma,axiom,
! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_1210_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L2: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_1211_div__mult2__numeral__eq,axiom,
! [A: int,K: num,L2: num] :
( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_1212_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1213_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1214_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1215_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1216_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1217_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1218_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1219_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1220_Suc__inject,axiom,
! [X: nat,Y4: nat] :
( ( ( suc @ X )
= ( suc @ Y4 ) )
=> ( X = Y4 ) ) ).
% Suc_inject
thf(fact_1221_linorder__neqE__nat,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
=> ( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1222_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1223_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1224_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1225_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_1226_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1227_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1228_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1229_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1230_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1231_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1232_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1233_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1234_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1235_size__neq__size__imp__neq,axiom,
! [X: list_VEBT_VEBT,Y4: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ X )
!= ( size_s6755466524823107622T_VEBT @ Y4 ) )
=> ( X != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_1236_size__neq__size__imp__neq,axiom,
! [X: list_o,Y4: list_o] :
( ( ( size_size_list_o @ X )
!= ( size_size_list_o @ Y4 ) )
=> ( X != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_1237_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y4: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y4 ) )
=> ( X != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_1238_size__neq__size__imp__neq,axiom,
! [X: list_int,Y4: list_int] :
( ( ( size_size_list_int @ X )
!= ( size_size_list_int @ Y4 ) )
=> ( X != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_1239_size__neq__size__imp__neq,axiom,
! [X: num,Y4: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y4 ) )
=> ( X != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_1240_power2__commute,axiom,
! [X: complex,Y4: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_complex @ ( minus_minus_complex @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_1241_power2__commute,axiom,
! [X: real,Y4: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ ( minus_minus_real @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_1242_power2__commute,axiom,
! [X: rat,Y4: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ ( minus_minus_rat @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_1243_power2__commute,axiom,
! [X: int,Y4: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ ( minus_minus_int @ Y4 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_commute
thf(fact_1244_nat__less__add__iff2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1245_nat__less__add__iff1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1246_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1247_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1248_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1249_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1250_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1251_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1252_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1253_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1254_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1255_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1256_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1257_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1258_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1259_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1260_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1261_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1262_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1263_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1264_power2__diff,axiom,
! [X: complex,Y4: complex] :
( ( power_power_complex @ ( minus_minus_complex @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% power2_diff
thf(fact_1265_power2__diff,axiom,
! [X: real,Y4: real] :
( ( power_power_real @ ( minus_minus_real @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% power2_diff
thf(fact_1266_power2__diff,axiom,
! [X: rat,Y4: rat] :
( ( power_power_rat @ ( minus_minus_rat @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% power2_diff
thf(fact_1267_power2__diff,axiom,
! [X: int,Y4: int] :
( ( power_power_int @ ( minus_minus_int @ X @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y4 ) ) ) ).
% power2_diff
thf(fact_1268_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1269_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1270_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1271_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1272_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1273_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1274_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1275_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1276_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1277_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1278_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1279_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1280_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1281_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1282_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1283_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1284_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1285_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1286_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1287_less__add__eq__less,axiom,
! [K: nat,L2: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M @ L2 )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1288_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1289_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1290_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1291_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1292_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1293_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_1294_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1295_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1296_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1297_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1298_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1299_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1300_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1301_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N3: nat] :
( L2
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1302_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_1303_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1304_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1305_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1306_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1307_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1308_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1309_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% mult_le_mono
thf(fact_1310_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1311_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1312_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1313_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1314_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1315_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1316_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
= ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_1317_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: num > num > num,A: num,B: num] :
( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
= ( some_num @ ( F @ A @ B ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_1318_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
! [F: nat > nat > nat,A: nat,B: nat] :
( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
= ( some_nat @ ( F @ A @ B ) ) ) ).
% VEBT_internal.option_shift.simps(3)
thf(fact_1319_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
= none_P5556105721700978146at_nat ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_1320_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [Uu: num > num > num,Uv: option_num] :
( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
= none_num ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_1321_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
! [Uu: nat > nat > nat,Uv: option_nat] :
( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
= none_nat ) ).
% VEBT_internal.option_shift.simps(1)
thf(fact_1322_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1323_lift__Suc__mono__less__iff,axiom,
! [F: nat > rat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1324_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1325_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1326_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1327_lift__Suc__mono__less,axiom,
! [F: nat > real,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1328_lift__Suc__mono__less,axiom,
! [F: nat > rat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1329_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1330_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1331_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1332_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1333_lift__Suc__mono__le,axiom,
! [F: nat > rat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1334_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1335_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1336_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1337_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1338_lift__Suc__antimono__le,axiom,
! [F: nat > rat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1339_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1340_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1341_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1342_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1343_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1344_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1345_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1346_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1347_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1348_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1349_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1350_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1351_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1352_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1353_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1354_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1355_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1356_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1357_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1358_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1359_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1360_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1361_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1362_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1363_vebt__pred_Osimps_I7_J,axiom,
! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( ord_less_nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
= ( some_nat @ Ma ) ) )
& ( ~ ( ord_less_nat @ Ma @ X )
=> ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
@ none_nat ) ) ) ) ).
% vebt_pred.simps(7)
thf(fact_1364_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
= none_P5556105721700978146at_nat ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_1365_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw: num > num > num,V: num] :
( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
= none_num ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_1366_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
! [Uw: nat > nat > nat,V: nat] :
( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
= none_nat ) ).
% VEBT_internal.option_shift.simps(2)
thf(fact_1367_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y4: option4927543243414619207at_nat] :
( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb2 )
= Y4 )
=> ( ( ( Xa2 = none_P5556105721700978146at_nat )
=> ( Y4 != none_P5556105721700978146at_nat ) )
=> ( ( ? [V2: product_prod_nat_nat] :
( Xa2
= ( some_P7363390416028606310at_nat @ V2 ) )
=> ( ( Xb2 = none_P5556105721700978146at_nat )
=> ( Y4 != none_P5556105721700978146at_nat ) ) )
=> ~ ! [A3: product_prod_nat_nat] :
( ( Xa2
= ( some_P7363390416028606310at_nat @ A3 ) )
=> ! [B2: product_prod_nat_nat] :
( ( Xb2
= ( some_P7363390416028606310at_nat @ B2 ) )
=> ( Y4
!= ( some_P7363390416028606310at_nat @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_1368_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: num > num > num,Xa2: option_num,Xb2: option_num,Y4: option_num] :
( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb2 )
= Y4 )
=> ( ( ( Xa2 = none_num )
=> ( Y4 != none_num ) )
=> ( ( ? [V2: num] :
( Xa2
= ( some_num @ V2 ) )
=> ( ( Xb2 = none_num )
=> ( Y4 != none_num ) ) )
=> ~ ! [A3: num] :
( ( Xa2
= ( some_num @ A3 ) )
=> ! [B2: num] :
( ( Xb2
= ( some_num @ B2 ) )
=> ( Y4
!= ( some_num @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_1369_VEBT__internal_Ooption__shift_Oelims,axiom,
! [X: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y4: option_nat] :
( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb2 )
= Y4 )
=> ( ( ( Xa2 = none_nat )
=> ( Y4 != none_nat ) )
=> ( ( ? [V2: nat] :
( Xa2
= ( some_nat @ V2 ) )
=> ( ( Xb2 = none_nat )
=> ( Y4 != none_nat ) ) )
=> ~ ! [A3: nat] :
( ( Xa2
= ( some_nat @ A3 ) )
=> ! [B2: nat] :
( ( Xb2
= ( some_nat @ B2 ) )
=> ( Y4
!= ( some_nat @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.option_shift.elims
thf(fact_1370_vebt__mint_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
= none_nat ) ).
% vebt_mint.simps(2)
thf(fact_1371_vebt__maxt_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
= none_nat ) ).
% vebt_maxt.simps(2)
thf(fact_1372_field__sum__of__halves,axiom,
! [X: real] :
( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= X ) ).
% field_sum_of_halves
thf(fact_1373_field__sum__of__halves,axiom,
! [X: rat] :
( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
= X ) ).
% field_sum_of_halves
thf(fact_1374_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
= ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less_nat @ X @ Mi )
& ( ~ ( ord_less_nat @ X @ Mi )
=> ( ~ ( ord_less_nat @ Ma @ X )
& ( ~ ( ord_less_nat @ Ma @ X )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_1375_le__add__diff__inverse2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1376_le__add__diff__inverse2,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1377_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1378_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1379_le__add__diff__inverse,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1380_le__add__diff__inverse,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1381_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1382_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1383_set__conv__nth,axiom,
( set_complex2
= ( ^ [Xs: list_complex] :
( collect_complex
@ ^ [Uu2: complex] :
? [I4: nat] :
( ( Uu2
= ( nth_complex @ Xs @ I4 ) )
& ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1384_set__conv__nth,axiom,
( set_real2
= ( ^ [Xs: list_real] :
( collect_real
@ ^ [Uu2: real] :
? [I4: nat] :
( ( Uu2
= ( nth_real @ Xs @ I4 ) )
& ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1385_set__conv__nth,axiom,
( set_Pr5648618587558075414at_nat
= ( ^ [Xs: list_P6011104703257516679at_nat] :
( collec3392354462482085612at_nat
@ ^ [Uu2: product_prod_nat_nat] :
? [I4: nat] :
( ( Uu2
= ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
& ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1386_set__conv__nth,axiom,
( set_list_nat2
= ( ^ [Xs: list_list_nat] :
( collect_list_nat
@ ^ [Uu2: list_nat] :
? [I4: nat] :
( ( Uu2
= ( nth_list_nat @ Xs @ I4 ) )
& ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1387_set__conv__nth,axiom,
( set_VEBT_VEBT2
= ( ^ [Xs: list_VEBT_VEBT] :
( collect_VEBT_VEBT
@ ^ [Uu2: vEBT_VEBT] :
? [I4: nat] :
( ( Uu2
= ( nth_VEBT_VEBT @ Xs @ I4 ) )
& ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1388_set__conv__nth,axiom,
( set_o2
= ( ^ [Xs: list_o] :
( collect_o
@ ^ [Uu2: $o] :
? [I4: nat] :
( ( Uu2
= ( nth_o @ Xs @ I4 ) )
& ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1389_set__conv__nth,axiom,
( set_nat2
= ( ^ [Xs: list_nat] :
( collect_nat
@ ^ [Uu2: nat] :
? [I4: nat] :
( ( Uu2
= ( nth_nat @ Xs @ I4 ) )
& ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1390_set__conv__nth,axiom,
( set_int2
= ( ^ [Xs: list_int] :
( collect_int
@ ^ [Uu2: int] :
? [I4: nat] :
( ( Uu2
= ( nth_int @ Xs @ I4 ) )
& ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_1391_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_1392_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_1393_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_1394_div__by__1,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ one_one_complex )
= A ) ).
% div_by_1
thf(fact_1395_div__by__1,axiom,
! [A: real] :
( ( divide_divide_real @ A @ one_one_real )
= A ) ).
% div_by_1
thf(fact_1396_div__by__1,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ one_one_rat )
= A ) ).
% div_by_1
thf(fact_1397_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_1398_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_1399_add__diff__cancel__right_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1400_add__diff__cancel__right_H,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1401_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1402_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1403_add__diff__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1404_add__diff__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( minus_minus_rat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1405_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1406_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1407_add__diff__cancel__left_H,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1408_add__diff__cancel__left_H,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1409_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1410_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1411_buildup__nothing__in__min__max,axiom,
! [N: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_1412_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1413_add__left__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1414_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1415_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1416_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1417_add__right__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1418_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1419_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1420_real__divide__square__eq,axiom,
! [R2: real,A: real] :
( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
= ( divide_divide_real @ A @ R2 ) ) ).
% real_divide_square_eq
thf(fact_1421_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1422_add__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1423_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1424_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1425_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1426_add__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1427_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1428_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1429_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1430_add__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1431_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1432_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1433_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1434_add__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
= ( ord_less_rat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1435_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1436_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1437_mult_Oright__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.right_neutral
thf(fact_1438_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_1439_mult_Oright__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.right_neutral
thf(fact_1440_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1441_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1442_mult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% mult_1
thf(fact_1443_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_1444_mult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% mult_1
thf(fact_1445_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1446_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1447_add__diff__cancel,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1448_add__diff__cancel,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1449_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1450_diff__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1451_diff__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1452_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1453_add__diff__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( minus_minus_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1454_add__diff__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
= ( minus_minus_rat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1455_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1456_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1457_real__arch__pow,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y4 @ ( power_power_real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1458_add__diff__assoc__enat,axiom,
! [Z2: extended_enat,Y4: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z2 @ Y4 )
=> ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y4 @ Z2 ) )
= ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y4 ) @ Z2 ) ) ) ).
% add_diff_assoc_enat
thf(fact_1459_linorder__neqE__linordered__idom,axiom,
! [X: real,Y4: real] :
( ( X != Y4 )
=> ( ~ ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ Y4 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1460_linorder__neqE__linordered__idom,axiom,
! [X: rat,Y4: rat] :
( ( X != Y4 )
=> ( ~ ( ord_less_rat @ X @ Y4 )
=> ( ord_less_rat @ Y4 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1461_linorder__neqE__linordered__idom,axiom,
! [X: int,Y4: int] :
( ( X != Y4 )
=> ( ~ ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ Y4 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_1462_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1463_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1464_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1465_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1466_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1467_mult_Oassoc,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1468_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1469_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_1470_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1471_mult_Ocommute,axiom,
( times_times_rat
= ( ^ [A4: rat,B3: rat] : ( times_times_rat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1472_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1473_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_1474_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1475_mult_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] :
( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
= ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1476_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1477_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_1478_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_1479_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_1480_one__reorient,axiom,
! [X: rat] :
( ( one_one_rat = X )
= ( X = one_one_rat ) ) ).
% one_reorient
thf(fact_1481_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_1482_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_1483_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1484_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1485_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1486_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1487_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( I2 = J )
& ( K = L2 ) )
=> ( ( plus_plus_real @ I2 @ K )
= ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1488_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( I2 = J )
& ( K = L2 ) )
=> ( ( plus_plus_rat @ I2 @ K )
= ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1489_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( I2 = J )
& ( K = L2 ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1490_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( I2 = J )
& ( K = L2 ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1491_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1492_group__cancel_Oadd1,axiom,
! [A2: rat,K: rat,A: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( plus_plus_rat @ A2 @ B )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1493_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1494_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1495_group__cancel_Oadd2,axiom,
! [B5: real,K: real,B: real,A: real] :
( ( B5
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B5 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1496_group__cancel_Oadd2,axiom,
! [B5: rat,K: rat,B: rat,A: rat] :
( ( B5
= ( plus_plus_rat @ K @ B ) )
=> ( ( plus_plus_rat @ A @ B5 )
= ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1497_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1498_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1499_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_1500_add_Oassoc,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.assoc
thf(fact_1501_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_1502_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_1503_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_1504_add_Oleft__cancel,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_1505_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_1506_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_1507_add_Oright__cancel,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_1508_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_1509_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B3: real] : ( plus_plus_real @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_1510_add_Ocommute,axiom,
( plus_plus_rat
= ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_1511_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_1512_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_1513_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1514_add_Oleft__commute,axiom,
! [B: rat,A: rat,C: rat] :
( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
= ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1515_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1516_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1517_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1518_add__left__imp__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( plus_plus_rat @ A @ B )
= ( plus_plus_rat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1519_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1520_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1521_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1522_add__right__imp__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( ( plus_plus_rat @ B @ A )
= ( plus_plus_rat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1523_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1524_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1525_subset__code_I1_J,axiom,
! [Xs2: list_real,B5: set_real] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B5 )
= ( ! [X2: real] :
( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
=> ( member_real @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1526_subset__code_I1_J,axiom,
! [Xs2: list_complex,B5: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B5 )
= ( ! [X2: complex] :
( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
=> ( member_complex @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1527_subset__code_I1_J,axiom,
! [Xs2: list_P6011104703257516679at_nat,B5: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ B5 )
= ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( member8440522571783428010at_nat @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1528_subset__code_I1_J,axiom,
! [Xs2: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B5 )
= ( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( member_VEBT_VEBT @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1529_subset__code_I1_J,axiom,
! [Xs2: list_int,B5: set_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B5 )
= ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
=> ( member_int @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1530_subset__code_I1_J,axiom,
! [Xs2: list_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B5 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_1531_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_VEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_1532_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_o] :
( ( size_size_list_o @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_1533_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_1534_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_int] :
( ( size_size_list_int @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_1535_neq__if__length__neq,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
!= ( size_s6755466524823107622T_VEBT @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_1536_neq__if__length__neq,axiom,
! [Xs2: list_o,Ys: list_o] :
( ( ( size_size_list_o @ Xs2 )
!= ( size_size_list_o @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_1537_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_1538_neq__if__length__neq,axiom,
! [Xs2: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs2 )
!= ( size_size_list_int @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_1539_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1540_add__le__imp__le__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1541_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1542_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1543_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1544_add__le__imp__le__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1545_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1546_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1547_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C2: nat] :
( B3
= ( plus_plus_nat @ A4 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_1548_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1549_add__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1550_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1551_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1552_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_1553_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1554_add__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1555_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1556_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1557_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1558_add__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1559_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1560_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1561_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( ord_less_eq_real @ K @ L2 ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1562_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( ord_less_eq_rat @ K @ L2 ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1563_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1564_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L2 ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1565_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( I2 = J )
& ( ord_less_eq_real @ K @ L2 ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1566_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( I2 = J )
& ( ord_less_eq_rat @ K @ L2 ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1567_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1568_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K @ L2 ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1569_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1570_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1571_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1572_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1573_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1574_diff__eq__diff__less__eq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C @ D ) )
=> ( ( ord_less_eq_rat @ A @ B )
= ( ord_less_eq_rat @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1575_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1576_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1577_diff__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1578_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_1579_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1580_diff__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1581_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_1582_diff__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_1583_diff__mono,axiom,
! [A: rat,B: rat,D: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ D @ C )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_1584_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_1585_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1586_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( ord_less_rat @ K @ L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1587_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1588_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K @ L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1589_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( I2 = J )
& ( ord_less_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1590_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( I2 = J )
& ( ord_less_rat @ K @ L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1591_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1592_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( I2 = J )
& ( ord_less_int @ K @ L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1593_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_real @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1594_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1595_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1596_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K = L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1597_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1598_add__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1599_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1600_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1601_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1602_add__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1603_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1604_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1605_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1606_add__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1607_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1608_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1609_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1610_add__less__imp__less__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1611_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1612_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1613_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1614_add__less__imp__less__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
=> ( ord_less_rat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1615_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1616_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1617_diff__strict__mono,axiom,
! [A: real,B: real,D: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ D @ C )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1618_diff__strict__mono,axiom,
! [A: rat,B: rat,D: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ D @ C )
=> ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1619_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_1620_diff__eq__diff__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_real @ A @ B )
= ( ord_less_real @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1621_diff__eq__diff__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( minus_minus_rat @ A @ B )
= ( minus_minus_rat @ C @ D ) )
=> ( ( ord_less_rat @ A @ B )
= ( ord_less_rat @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1622_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_1623_diff__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1624_diff__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1625_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1626_diff__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1627_diff__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1628_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1629_comm__monoid__mult__class_Omult__1,axiom,
! [A: complex] :
( ( times_times_complex @ one_one_complex @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1630_comm__monoid__mult__class_Omult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1631_comm__monoid__mult__class_Omult__1,axiom,
! [A: rat] :
( ( times_times_rat @ one_one_rat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1632_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1633_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1634_mult_Ocomm__neutral,axiom,
! [A: complex] :
( ( times_times_complex @ A @ one_one_complex )
= A ) ).
% mult.comm_neutral
thf(fact_1635_mult_Ocomm__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.comm_neutral
thf(fact_1636_mult_Ocomm__neutral,axiom,
! [A: rat] :
( ( times_times_rat @ A @ one_one_rat )
= A ) ).
% mult.comm_neutral
thf(fact_1637_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_1638_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_1639_ring__class_Oring__distribs_I2_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1640_ring__class_Oring__distribs_I2_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1641_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1642_ring__class_Oring__distribs_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1643_ring__class_Oring__distribs_I1_J,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1644_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1645_comm__semiring__class_Odistrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1646_comm__semiring__class_Odistrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1647_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1648_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1649_distrib__left,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
= ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_1650_distrib__left,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% distrib_left
thf(fact_1651_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_1652_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_1653_distrib__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_1654_distrib__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1655_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1656_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_1657_combine__common__factor,axiom,
! [A: real,E: real,B: real,C: real] :
( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
= ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1658_combine__common__factor,axiom,
! [A: rat,E: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
= ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1659_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1660_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1661_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1662_left__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1663_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_1664_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1665_right__diff__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1666_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_1667_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1668_left__diff__distrib_H,axiom,
! [B: rat,C: rat,A: rat] :
( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
= ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1669_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1670_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_1671_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1672_right__diff__distrib_H,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1673_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1674_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1675_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1676_group__cancel_Osub1,axiom,
! [A2: rat,K: rat,A: rat,B: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( minus_minus_rat @ A2 @ B )
= ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1677_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1678_diff__eq__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( minus_minus_real @ A @ B )
= C )
= ( A
= ( plus_plus_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_1679_diff__eq__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ( minus_minus_rat @ A @ B )
= C )
= ( A
= ( plus_plus_rat @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_1680_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_1681_eq__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( A
= ( minus_minus_real @ C @ B ) )
= ( ( plus_plus_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_1682_eq__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( A
= ( minus_minus_rat @ C @ B ) )
= ( ( plus_plus_rat @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_1683_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_1684_add__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_1685_add__diff__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_1686_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_1687_diff__diff__eq2,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1688_diff__diff__eq2,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1689_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1690_diff__add__eq,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_1691_diff__add__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_1692_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_1693_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1694_diff__add__eq__diff__diff__swap,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1695_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1696_add__implies__diff,axiom,
! [C: real,B: real,A: real] :
( ( ( plus_plus_real @ C @ B )
= A )
=> ( C
= ( minus_minus_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1697_add__implies__diff,axiom,
! [C: rat,B: rat,A: rat] :
( ( ( plus_plus_rat @ C @ B )
= A )
=> ( C
= ( minus_minus_rat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1698_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1699_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1700_diff__diff__eq,axiom,
! [A: real,B: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1701_diff__diff__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1702_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1703_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1704_length__induct,axiom,
! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
( ! [Xs3: list_VEBT_VEBT] :
( ! [Ys2: list_VEBT_VEBT] :
( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_1705_length__induct,axiom,
! [P: list_o > $o,Xs2: list_o] :
( ! [Xs3: list_o] :
( ! [Ys2: list_o] :
( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_1706_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_1707_length__induct,axiom,
! [P: list_int > $o,Xs2: list_int] :
( ! [Xs3: list_int] :
( ! [Ys2: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_1708_lambda__one,axiom,
( ( ^ [X2: complex] : X2 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_1709_lambda__one,axiom,
( ( ^ [X2: real] : X2 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_1710_lambda__one,axiom,
( ( ^ [X2: rat] : X2 )
= ( times_times_rat @ one_one_rat ) ) ).
% lambda_one
thf(fact_1711_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_1712_lambda__one,axiom,
( ( ^ [X2: int] : X2 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_1713_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_eq_real @ I2 @ J )
& ( ord_less_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1714_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_eq_rat @ I2 @ J )
& ( ord_less_rat @ K @ L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1715_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1716_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_int @ K @ L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1717_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: real,J: real,K: real,L2: real] :
( ( ( ord_less_real @ I2 @ J )
& ( ord_less_eq_real @ K @ L2 ) )
=> ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1718_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: rat,J: rat,K: rat,L2: rat] :
( ( ( ord_less_rat @ I2 @ J )
& ( ord_less_eq_rat @ K @ L2 ) )
=> ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1719_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L2: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L2 ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1720_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J: int,K: int,L2: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L2 ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1721_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1722_add__le__less__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1723_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1724_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1725_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1726_add__less__le__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1727_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1728_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1729_less__1__mult,axiom,
! [M: real,N: real] :
( ( ord_less_real @ one_one_real @ M )
=> ( ( ord_less_real @ one_one_real @ N )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1730_less__1__mult,axiom,
! [M: rat,N: rat] :
( ( ord_less_rat @ one_one_rat @ M )
=> ( ( ord_less_rat @ one_one_rat @ N )
=> ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1731_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1732_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1733_diff__le__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1734_diff__le__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1735_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1736_le__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1737_le__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1738_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1739_add__le__imp__le__diff,axiom,
! [I2: real,K: real,N: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1740_add__le__imp__le__diff,axiom,
! [I2: rat,K: rat,N: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
=> ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1741_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1742_add__le__imp__le__diff,axiom,
! [I2: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1743_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1744_add__le__add__imp__diff__le,axiom,
! [I2: real,K: real,N: real,J: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N )
=> ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1745_add__le__add__imp__diff__le,axiom,
! [I2: rat,K: rat,N: rat,J: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
=> ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
=> ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1746_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1747_add__le__add__imp__diff__le,axiom,
! [I2: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1748_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1749_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1750_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1751_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1752_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1753_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1754_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1755_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1756_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1757_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_1758_add__mono1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% add_mono1
thf(fact_1759_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1760_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_1761_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_1762_less__add__one,axiom,
! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% less_add_one
thf(fact_1763_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1764_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1765_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: real,B: real] :
( ~ ( ord_less_real @ A @ B )
=> ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1766_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: rat,B: rat] :
( ~ ( ord_less_rat @ A @ B )
=> ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1767_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1768_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1769_less__diff__eq,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
= ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_1770_less__diff__eq,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
= ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_1771_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_1772_diff__less__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
= ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_1773_diff__less__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_1774_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_1775_square__diff__square__factored,axiom,
! [X: real,Y4: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) )
= ( times_times_real @ ( plus_plus_real @ X @ Y4 ) @ ( minus_minus_real @ X @ Y4 ) ) ) ).
% square_diff_square_factored
thf(fact_1776_square__diff__square__factored,axiom,
! [X: rat,Y4: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) )
= ( times_times_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( minus_minus_rat @ X @ Y4 ) ) ) ).
% square_diff_square_factored
thf(fact_1777_square__diff__square__factored,axiom,
! [X: int,Y4: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y4 ) @ ( minus_minus_int @ X @ Y4 ) ) ) ).
% square_diff_square_factored
thf(fact_1778_eq__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( C
= ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1779_eq__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( C
= ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1780_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1781_eq__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
= ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1782_eq__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
= ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1783_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1784_nth__equalityI,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
= ( size_s6755466524823107622T_VEBT @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
= ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1785_nth__equalityI,axiom,
! [Xs2: list_o,Ys: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
=> ( ( nth_o @ Xs2 @ I3 )
= ( nth_o @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1786_nth__equalityI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1787_nth__equalityI,axiom,
! [Xs2: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ Xs2 @ I3 )
= ( nth_int @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_1788_Skolem__list__nth,axiom,
! [K: nat,P: nat > vEBT_VEBT > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X6: vEBT_VEBT] : ( P @ I4 @ X6 ) ) )
= ( ? [Xs: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1789_Skolem__list__nth,axiom,
! [K: nat,P: nat > $o > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X6: $o] : ( P @ I4 @ X6 ) ) )
= ( ? [Xs: list_o] :
( ( ( size_size_list_o @ Xs )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1790_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X6: nat] : ( P @ I4 @ X6 ) ) )
= ( ? [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1791_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X6: int] : ( P @ I4 @ X6 ) ) )
= ( ? [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1792_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_VEBT_VEBT,Z3: list_VEBT_VEBT] : Y6 = Z3 )
= ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs )
= ( size_s6755466524823107622T_VEBT @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
=> ( ( nth_VEBT_VEBT @ Xs @ I4 )
= ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1793_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_o,Z3: list_o] : Y6 = Z3 )
= ( ^ [Xs: list_o,Ys3: list_o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ Xs @ I4 )
= ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1794_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_nat,Z3: list_nat] : Y6 = Z3 )
= ( ^ [Xs: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I4 )
= ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1795_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_int,Z3: list_int] : Y6 = Z3 )
= ( ^ [Xs: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I4 )
= ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1796_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).
% vebt_member.simps(2)
thf(fact_1797_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_1798_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_1799_ordered__ring__class_Ole__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1800_ordered__ring__class_Ole__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1801_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1802_ordered__ring__class_Ole__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1803_ordered__ring__class_Ole__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1804_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1805_less__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1806_less__add__iff2,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1807_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1808_less__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
= ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1809_less__add__iff1,axiom,
! [A: rat,E: rat,C: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
= ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1810_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1811_square__diff__one__factored,axiom,
! [X: complex] :
( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
= ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% square_diff_one_factored
thf(fact_1812_square__diff__one__factored,axiom,
! [X: real] :
( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
= ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% square_diff_one_factored
thf(fact_1813_square__diff__one__factored,axiom,
! [X: rat] :
( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
= ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% square_diff_one_factored
thf(fact_1814_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_1815_nth__mem,axiom,
! [N: nat,Xs2: list_real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
=> ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1816_nth__mem,axiom,
! [N: nat,Xs2: list_complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1817_nth__mem,axiom,
! [N: nat,Xs2: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1818_nth__mem,axiom,
! [N: nat,Xs2: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1819_nth__mem,axiom,
! [N: nat,Xs2: list_o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
=> ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1820_nth__mem,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1821_nth__mem,axiom,
! [N: nat,Xs2: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
=> ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_1822_list__ball__nth,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1823_list__ball__nth,axiom,
! [N: nat,Xs2: list_o,P: $o > $o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1824_list__ball__nth,axiom,
! [N: nat,Xs2: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1825_list__ball__nth,axiom,
! [N: nat,Xs2: list_int,P: int > $o] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1826_in__set__conv__nth,axiom,
! [X: real,Xs2: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
& ( ( nth_real @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1827_in__set__conv__nth,axiom,
! [X: complex,Xs2: list_complex] :
( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
& ( ( nth_complex @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1828_in__set__conv__nth,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1829_in__set__conv__nth,axiom,
! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
& ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1830_in__set__conv__nth,axiom,
! [X: $o,Xs2: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
& ( ( nth_o @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1831_in__set__conv__nth,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1832_in__set__conv__nth,axiom,
! [X: int,Xs2: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
& ( ( nth_int @ Xs2 @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1833_all__nth__imp__all__set,axiom,
! [Xs2: list_real,P: real > $o,X: real] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
=> ( P @ ( nth_real @ Xs2 @ I3 ) ) )
=> ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1834_all__nth__imp__all__set,axiom,
! [Xs2: list_complex,P: complex > $o,X: complex] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( P @ ( nth_complex @ Xs2 @ I3 ) ) )
=> ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1835_all__nth__imp__all__set,axiom,
! [Xs2: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( P @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I3 ) ) )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1836_all__nth__imp__all__set,axiom,
! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
=> ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1837_all__nth__imp__all__set,axiom,
! [Xs2: list_o,P: $o > $o,X: $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
=> ( P @ ( nth_o @ Xs2 @ I3 ) ) )
=> ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1838_all__nth__imp__all__set,axiom,
! [Xs2: list_nat,P: nat > $o,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1839_all__nth__imp__all__set,axiom,
! [Xs2: list_int,P: int > $o,X: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
=> ( P @ ( nth_int @ Xs2 @ I3 ) ) )
=> ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1840_all__set__conv__all__nth,axiom,
! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( P @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1841_all__set__conv__all__nth,axiom,
! [Xs2: list_o,P: $o > $o] :
( ( ! [X2: $o] :
( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
=> ( P @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
=> ( P @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1842_all__set__conv__all__nth,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1843_all__set__conv__all__nth,axiom,
! [Xs2: list_int,P: int > $o] :
( ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
=> ( P @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
=> ( P @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1844_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_1845_real__average__minus__first,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_first
thf(fact_1846_real__average__minus__second,axiom,
! [B: real,A: real] :
( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
= ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% real_average_minus_second
thf(fact_1847_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_1848_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T2: vEBT_VEBT,X2: nat] :
( ( vEBT_V5719532721284313246member @ T2 @ X2 )
| ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).
% both_member_options_def
thf(fact_1849_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_1850_times__divide__eq__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_1851_times__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_1852_times__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_1853_divide__divide__eq__right,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_1854_divide__divide__eq__right,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
= ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_1855_divide__divide__eq__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_1856_divide__divide__eq__left,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
= ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_1857_divide__divide__eq__left,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_1858_divide__divide__eq__left,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
= ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_1859_times__divide__eq__left,axiom,
! [B: complex,C: complex,A: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
= ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_1860_times__divide__eq__left,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
= ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_1861_times__divide__eq__left,axiom,
! [B: rat,C: rat,A: rat] :
( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_1862_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S2 ) @ X )
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_1863_valid__0__not,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_0_not
thf(fact_1864_valid__tree__deg__neq__0,axiom,
! [T: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% valid_tree_deg_neq_0
thf(fact_1865_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_1866_deg__not__0,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% deg_not_0
thf(fact_1867_buildup__gives__valid,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% buildup_gives_valid
thf(fact_1868_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1869_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1870_mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1871_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1872_mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1873_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1874_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1875_mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1876_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1877_mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1878_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1879_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1880_mult__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_1881_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_1882_mult__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% mult_eq_0_iff
thf(fact_1883_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1884_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1885_mult__zero__right,axiom,
! [A: complex] :
( ( times_times_complex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_1886_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_1887_mult__zero__right,axiom,
! [A: rat] :
( ( times_times_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% mult_zero_right
thf(fact_1888_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1889_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1890_mult__zero__left,axiom,
! [A: complex] :
( ( times_times_complex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_1891_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_1892_mult__zero__left,axiom,
! [A: rat] :
( ( times_times_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% mult_zero_left
thf(fact_1893_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1894_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1895_add__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add_0
thf(fact_1896_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_1897_add__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add_0
thf(fact_1898_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1899_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_1900_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y4: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y4 ) )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1901_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y4: nat] :
( ( ( plus_plus_nat @ X @ Y4 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1902_add__cancel__right__right,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ A @ B ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_right
thf(fact_1903_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_1904_add__cancel__right__right,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ A @ B ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_right
thf(fact_1905_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1906_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1907_add__cancel__right__left,axiom,
! [A: complex,B: complex] :
( ( A
= ( plus_plus_complex @ B @ A ) )
= ( B = zero_zero_complex ) ) ).
% add_cancel_right_left
thf(fact_1908_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_1909_add__cancel__right__left,axiom,
! [A: rat,B: rat] :
( ( A
= ( plus_plus_rat @ B @ A ) )
= ( B = zero_zero_rat ) ) ).
% add_cancel_right_left
thf(fact_1910_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1911_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1912_add__cancel__left__right,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_right
thf(fact_1913_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_1914_add__cancel__left__right,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_right
thf(fact_1915_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1916_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1917_add__cancel__left__left,axiom,
! [B: complex,A: complex] :
( ( ( plus_plus_complex @ B @ A )
= A )
= ( B = zero_zero_complex ) ) ).
% add_cancel_left_left
thf(fact_1918_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_1919_add__cancel__left__left,axiom,
! [B: rat,A: rat] :
( ( ( plus_plus_rat @ B @ A )
= A )
= ( B = zero_zero_rat ) ) ).
% add_cancel_left_left
thf(fact_1920_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1921_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1922_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_1923_double__zero__sym,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( plus_plus_rat @ A @ A ) )
= ( A = zero_zero_rat ) ) ).
% double_zero_sym
thf(fact_1924_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1925_add_Oright__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.right_neutral
thf(fact_1926_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_1927_add_Oright__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.right_neutral
thf(fact_1928_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1929_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_1930_divide__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divide_eq_0_iff
thf(fact_1931_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_1932_divide__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_eq_0_iff
thf(fact_1933_divide__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ C @ A )
= ( divide1717551699836669952omplex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_1934_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_1935_divide__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( divide_divide_rat @ C @ A )
= ( divide_divide_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_1936_divide__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_1937_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_1938_divide__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( divide_divide_rat @ A @ C )
= ( divide_divide_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_1939_division__ring__divide__zero,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% division_ring_divide_zero
thf(fact_1940_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_1941_division__ring__divide__zero,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% division_ring_divide_zero
thf(fact_1942_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_1943_bits__div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% bits_div_0
thf(fact_1944_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_1945_bits__div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% bits_div_by_0
thf(fact_1946_div__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
= zero_zero_complex ) ).
% div_0
thf(fact_1947_div__0,axiom,
! [A: real] :
( ( divide_divide_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% div_0
thf(fact_1948_div__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ zero_zero_rat @ A )
= zero_zero_rat ) ).
% div_0
thf(fact_1949_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_1950_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_1951_div__by__0,axiom,
! [A: complex] :
( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
= zero_zero_complex ) ).
% div_by_0
thf(fact_1952_div__by__0,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% div_by_0
thf(fact_1953_div__by__0,axiom,
! [A: rat] :
( ( divide_divide_rat @ A @ zero_zero_rat )
= zero_zero_rat ) ).
% div_by_0
thf(fact_1954_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_1955_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_1956_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1957_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1958_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1959_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1960_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1961_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1962_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1963_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1964_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1965_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1966_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1967_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1968_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1969_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1970_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1971_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1972_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1973_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_1974_le__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1975_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1976_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_1977_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_1978_le__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1979_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1980_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_1981_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_1982_add__le__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel2
thf(fact_1983_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1984_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1985_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_1986_add__le__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% add_le_same_cancel1
thf(fact_1987_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1988_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1989_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1990_diff__ge__0__iff__ge,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1991_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1992_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_1993_add__less__same__cancel1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel1
thf(fact_1994_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1995_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1996_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_1997_add__less__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% add_less_same_cancel2
thf(fact_1998_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1999_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_2000_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_2001_less__add__same__cancel1,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel1
thf(fact_2002_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_2003_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_2004_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_2005_less__add__same__cancel2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% less_add_same_cancel2
thf(fact_2006_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_2007_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_2008_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_2009_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_2010_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_2011_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_2012_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_2013_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_2014_diff__gt__0__iff__gt,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_real @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_2015_diff__gt__0__iff__gt,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
= ( ord_less_rat @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_2016_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_2017_mult__cancel__right2,axiom,
! [A: complex,C: complex] :
( ( ( times_times_complex @ A @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_2018_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_2019_mult__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ( times_times_rat @ A @ C )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_right2
thf(fact_2020_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_2021_mult__cancel__right1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_2022_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_2023_mult__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_right1
thf(fact_2024_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_2025_mult__cancel__left2,axiom,
! [C: complex,A: complex] :
( ( ( times_times_complex @ C @ A )
= C )
= ( ( C = zero_zero_complex )
| ( A = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_2026_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_2027_mult__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ( times_times_rat @ C @ A )
= C )
= ( ( C = zero_zero_rat )
| ( A = one_one_rat ) ) ) ).
% mult_cancel_left2
thf(fact_2028_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_2029_mult__cancel__left1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_2030_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_2031_mult__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( C
= ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( B = one_one_rat ) ) ) ).
% mult_cancel_left1
thf(fact_2032_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_2033_sum__squares__eq__zero__iff,axiom,
! [X: real,Y4: real] :
( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y4 = zero_zero_real ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_2034_sum__squares__eq__zero__iff,axiom,
! [X: rat,Y4: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y4 = zero_zero_rat ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_2035_sum__squares__eq__zero__iff,axiom,
! [X: int,Y4: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_2036_mult__divide__mult__cancel__left__if,axiom,
! [C: complex,A: complex,B: complex] :
( ( ( C = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= zero_zero_complex ) )
& ( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2037_mult__divide__mult__cancel__left__if,axiom,
! [C: real,A: real,B: real] :
( ( ( C = zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= zero_zero_real ) )
& ( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2038_mult__divide__mult__cancel__left__if,axiom,
! [C: rat,A: rat,B: rat] :
( ( ( C = zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= zero_zero_rat ) )
& ( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_2039_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2040_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2041_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_2042_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2043_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2044_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_2045_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2046_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2047_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_2048_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2049_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2050_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_2051_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2052_div__mult__mult1__if,axiom,
! [C: int,A: int,B: int] :
( ( ( C = zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= zero_zero_int ) )
& ( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_2053_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_2054_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_2055_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_2056_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_2057_nonzero__mult__div__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2058_nonzero__mult__div__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2059_nonzero__mult__div__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2060_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2061_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_2062_nonzero__mult__div__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2063_nonzero__mult__div__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2064_nonzero__mult__div__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2065_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2066_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_2067_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_2068_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_2069_diff__numeral__special_I9_J,axiom,
( ( minus_minus_rat @ one_one_rat @ one_one_rat )
= zero_zero_rat ) ).
% diff_numeral_special(9)
thf(fact_2070_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_2071_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_2072_divide__eq__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_2073_divide__eq__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_2074_divide__eq__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_2075_one__eq__divide__iff,axiom,
! [A: complex,B: complex] :
( ( one_one_complex
= ( divide1717551699836669952omplex @ A @ B ) )
= ( ( B != zero_zero_complex )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_2076_one__eq__divide__iff,axiom,
! [A: real,B: real] :
( ( one_one_real
= ( divide_divide_real @ A @ B ) )
= ( ( B != zero_zero_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_2077_one__eq__divide__iff,axiom,
! [A: rat,B: rat] :
( ( one_one_rat
= ( divide_divide_rat @ A @ B ) )
= ( ( B != zero_zero_rat )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_2078_divide__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% divide_self
thf(fact_2079_divide__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% divide_self
thf(fact_2080_divide__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% divide_self
thf(fact_2081_divide__self__if,axiom,
! [A: complex] :
( ( ( A = zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= zero_zero_complex ) )
& ( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ) ).
% divide_self_if
thf(fact_2082_divide__self__if,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= zero_zero_real ) )
& ( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ) ).
% divide_self_if
thf(fact_2083_divide__self__if,axiom,
! [A: rat] :
( ( ( A = zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= zero_zero_rat ) )
& ( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ) ).
% divide_self_if
thf(fact_2084_divide__eq__eq__1,axiom,
! [B: real,A: real] :
( ( ( divide_divide_real @ B @ A )
= one_one_real )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_2085_divide__eq__eq__1,axiom,
! [B: rat,A: rat] :
( ( ( divide_divide_rat @ B @ A )
= one_one_rat )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% divide_eq_eq_1
thf(fact_2086_eq__divide__eq__1,axiom,
! [B: real,A: real] :
( ( one_one_real
= ( divide_divide_real @ B @ A ) )
= ( ( A != zero_zero_real )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_2087_eq__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( one_one_rat
= ( divide_divide_rat @ B @ A ) )
= ( ( A != zero_zero_rat )
& ( A = B ) ) ) ).
% eq_divide_eq_1
thf(fact_2088_one__divide__eq__0__iff,axiom,
! [A: real] :
( ( ( divide_divide_real @ one_one_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% one_divide_eq_0_iff
thf(fact_2089_one__divide__eq__0__iff,axiom,
! [A: rat] :
( ( ( divide_divide_rat @ one_one_rat @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% one_divide_eq_0_iff
thf(fact_2090_zero__eq__1__divide__iff,axiom,
! [A: real] :
( ( zero_zero_real
= ( divide_divide_real @ one_one_real @ A ) )
= ( A = zero_zero_real ) ) ).
% zero_eq_1_divide_iff
thf(fact_2091_zero__eq__1__divide__iff,axiom,
! [A: rat] :
( ( zero_zero_rat
= ( divide_divide_rat @ one_one_rat @ A ) )
= ( A = zero_zero_rat ) ) ).
% zero_eq_1_divide_iff
thf(fact_2092_div__self,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ A )
= one_one_complex ) ) ).
% div_self
thf(fact_2093_div__self,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ A )
= one_one_real ) ) ).
% div_self
thf(fact_2094_div__self,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ A )
= one_one_rat ) ) ).
% div_self
thf(fact_2095_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_2096_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_2097_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
= zero_zero_rat ) ).
% power_0_Suc
thf(fact_2098_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_2099_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
= zero_zero_real ) ).
% power_0_Suc
thf(fact_2100_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_2101_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
= zero_zero_complex ) ).
% power_0_Suc
thf(fact_2102_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
= zero_zero_rat ) ).
% power_zero_numeral
thf(fact_2103_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
= zero_zero_nat ) ).
% power_zero_numeral
thf(fact_2104_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
= zero_zero_real ) ).
% power_zero_numeral
thf(fact_2105_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
= zero_zero_int ) ).
% power_zero_numeral
thf(fact_2106_power__zero__numeral,axiom,
! [K: num] :
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
= zero_zero_complex ) ).
% power_zero_numeral
thf(fact_2107_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2108_power__Suc0__right,axiom,
! [A: real] :
( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2109_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2110_power__Suc0__right,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_2111_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_2112_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_2113_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_2114_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_2115_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_2116_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
= M ) ).
% div_by_Suc_0
thf(fact_2117_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_2118_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_2119_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_2120_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_2121_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_2122_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_2123_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_2124_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_2125_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_2126_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_2127_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_2128_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_2129_divide__le__0__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_2130_divide__le__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% divide_le_0_1_iff
thf(fact_2131_zero__le__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_2132_zero__le__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% zero_le_divide_1_iff
thf(fact_2133_zero__less__divide__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_2134_zero__less__divide__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% zero_less_divide_1_iff
thf(fact_2135_less__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_2136_less__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% less_divide_eq_1_pos
thf(fact_2137_less__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_real @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_2138_less__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% less_divide_eq_1_neg
thf(fact_2139_divide__less__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_2140_divide__less__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ B @ A ) ) ) ).
% divide_less_eq_1_pos
thf(fact_2141_divide__less__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_real @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_2142_divide__less__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_rat @ A @ B ) ) ) ).
% divide_less_eq_1_neg
thf(fact_2143_divide__less__0__1__iff,axiom,
! [A: real] :
( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% divide_less_0_1_iff
thf(fact_2144_divide__less__0__1__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% divide_less_0_1_iff
thf(fact_2145_divide__eq__eq__numeral1_I1_J,axiom,
! [B: complex,W2: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
= A )
= ( ( ( ( numera6690914467698888265omplex @ W2 )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
& ( ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2146_divide__eq__eq__numeral1_I1_J,axiom,
! [B: real,W2: num,A: real] :
( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
= A )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2147_divide__eq__eq__numeral1_I1_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
= A )
= ( ( ( ( numeral_numeral_rat @ W2 )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
& ( ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2148_eq__divide__eq__numeral1_I1_J,axiom,
! [A: complex,B: complex,W2: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( ( ( ( numera6690914467698888265omplex @ W2 )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
= B ) )
& ( ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2149_eq__divide__eq__numeral1_I1_J,axiom,
! [A: real,B: real,W2: num] :
( ( A
= ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ( numeral_numeral_real @ W2 )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
= B ) )
& ( ( ( numeral_numeral_real @ W2 )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2150_eq__divide__eq__numeral1_I1_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( A
= ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ( numeral_numeral_rat @ W2 )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
= B ) )
& ( ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2151_nonzero__divide__mult__cancel__left,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2152_nonzero__divide__mult__cancel__left,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2153_nonzero__divide__mult__cancel__left,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_2154_nonzero__divide__mult__cancel__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2155_nonzero__divide__mult__cancel__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2156_nonzero__divide__mult__cancel__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
= ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_2157_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_2158_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_2159_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_2160_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_2161_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_2162_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_2163_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_2164_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_2165_power__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( power_power_rat @ A @ N )
= zero_zero_rat )
= ( ( A = zero_zero_rat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2166_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2167_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2168_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2169_power__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( power_power_complex @ A @ N )
= zero_zero_complex )
= ( ( A = zero_zero_complex )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_2170_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_2171_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_2172_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_2173_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_2174_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_2175_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_2176_divide__le__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_2177_divide__le__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% divide_le_eq_1_neg
thf(fact_2178_divide__le__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_2179_divide__le__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% divide_le_eq_1_pos
thf(fact_2180_le__divide__eq__1__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_2181_le__divide__eq__1__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% le_divide_eq_1_neg
thf(fact_2182_le__divide__eq__1__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_2183_le__divide__eq__1__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% le_divide_eq_1_pos
thf(fact_2184_power__strict__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2185_power__strict__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2186_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2187_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_2188_zero__eq__power2,axiom,
! [A: rat] :
( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% zero_eq_power2
thf(fact_2189_zero__eq__power2,axiom,
! [A: nat] :
( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% zero_eq_power2
thf(fact_2190_zero__eq__power2,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% zero_eq_power2
thf(fact_2191_zero__eq__power2,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% zero_eq_power2
thf(fact_2192_zero__eq__power2,axiom,
! [A: complex] :
( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ).
% zero_eq_power2
thf(fact_2193_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2194_power__mono__iff,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2195_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2196_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_2197_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_2198_bits__1__div__2,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% bits_1_div_2
thf(fact_2199_bits__1__div__2,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% bits_1_div_2
thf(fact_2200_one__div__two__eq__zero,axiom,
( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% one_div_two_eq_zero
thf(fact_2201_one__div__two__eq__zero,axiom,
( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% one_div_two_eq_zero
thf(fact_2202_power__decreasing__iff,axiom,
! [B: real,M: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ B @ one_one_real )
=> ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_2203_power__decreasing__iff,axiom,
! [B: rat,M: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ B @ one_one_rat )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_2204_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_2205_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_2206_power2__less__eq__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% power2_less_eq_zero_iff
thf(fact_2207_power2__less__eq__zero__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% power2_less_eq_zero_iff
thf(fact_2208_power2__less__eq__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% power2_less_eq_zero_iff
thf(fact_2209_power2__eq__iff__nonneg,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y4 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_2210_power2__eq__iff__nonneg,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y4 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_2211_power2__eq__iff__nonneg,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y4 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_2212_power2__eq__iff__nonneg,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( X = Y4 ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_2213_zero__less__power2,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_real ) ) ).
% zero_less_power2
thf(fact_2214_zero__less__power2,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_rat ) ) ).
% zero_less_power2
thf(fact_2215_zero__less__power2,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( A != zero_zero_int ) ) ).
% zero_less_power2
thf(fact_2216_sum__power2__eq__zero__iff,axiom,
! [X: rat,Y4: rat] :
( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y4 = zero_zero_rat ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_2217_sum__power2__eq__zero__iff,axiom,
! [X: real,Y4: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y4 = zero_zero_real ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_2218_sum__power2__eq__zero__iff,axiom,
! [X: int,Y4: int] :
( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_2219_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_2220_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% power_0_left
thf(fact_2221_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_2222_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% power_0_left
thf(fact_2223_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_2224_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% power_0_left
thf(fact_2225_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ).
% zero_power
thf(fact_2226_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_2227_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_2228_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_2229_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ).
% zero_power
thf(fact_2230_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_2231_le__numeral__extra_I3_J,axiom,
ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% le_numeral_extra(3)
thf(fact_2232_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_2233_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_2234_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_2235_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_2236_field__lbound__gt__zero,axiom,
! [D1: rat,D22: rat] :
( ( ord_less_rat @ zero_zero_rat @ D1 )
=> ( ( ord_less_rat @ zero_zero_rat @ D22 )
=> ? [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
& ( ord_less_rat @ E2 @ D1 )
& ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_2237_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_2238_less__numeral__extra_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% less_numeral_extra(3)
thf(fact_2239_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_2240_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_2241_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_2242_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_2243_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_2244_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_2245_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( numera6690914467698888265omplex @ N ) ) ).
% zero_neq_numeral
thf(fact_2246_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_2247_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( numeral_numeral_rat @ N ) ) ).
% zero_neq_numeral
thf(fact_2248_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_2249_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_2250_mult__right__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= ( times_times_complex @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2251_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2252_mult__right__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= ( times_times_rat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2253_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2254_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_2255_mult__left__cancel,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A )
= ( times_times_complex @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2256_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2257_mult__left__cancel,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ C @ A )
= ( times_times_rat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2258_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2259_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_2260_no__zero__divisors,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_2261_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_2262_no__zero__divisors,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( times_times_rat @ A @ B )
!= zero_zero_rat ) ) ) ).
% no_zero_divisors
thf(fact_2263_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_2264_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_2265_divisors__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= zero_zero_complex )
=> ( ( A = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_2266_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_2267_divisors__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= zero_zero_rat )
=> ( ( A = zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divisors_zero
thf(fact_2268_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_2269_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_2270_mult__not__zero,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
!= zero_zero_complex )
=> ( ( A != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_2271_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_2272_mult__not__zero,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
!= zero_zero_rat )
=> ( ( A != zero_zero_rat )
& ( B != zero_zero_rat ) ) ) ).
% mult_not_zero
thf(fact_2273_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_2274_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_2275_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_2276_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_2277_zero__neq__one,axiom,
zero_zero_rat != one_one_rat ).
% zero_neq_one
thf(fact_2278_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_2279_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_2280_verit__sum__simplify,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_sum_simplify
thf(fact_2281_verit__sum__simplify,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% verit_sum_simplify
thf(fact_2282_verit__sum__simplify,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% verit_sum_simplify
thf(fact_2283_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_2284_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_2285_add_Ogroup__left__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% add.group_left_neutral
thf(fact_2286_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_2287_add_Ogroup__left__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% add.group_left_neutral
thf(fact_2288_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_2289_add_Ocomm__neutral,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ zero_zero_complex )
= A ) ).
% add.comm_neutral
thf(fact_2290_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_2291_add_Ocomm__neutral,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ zero_zero_rat )
= A ) ).
% add.comm_neutral
thf(fact_2292_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_2293_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_2294_comm__monoid__add__class_Oadd__0,axiom,
! [A: complex] :
( ( plus_plus_complex @ zero_zero_complex @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_2295_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_2296_comm__monoid__add__class_Oadd__0,axiom,
! [A: rat] :
( ( plus_plus_rat @ zero_zero_rat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_2297_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_2298_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_2299_power__not__zero,axiom,
! [A: rat,N: nat] :
( ( A != zero_zero_rat )
=> ( ( power_power_rat @ A @ N )
!= zero_zero_rat ) ) ).
% power_not_zero
thf(fact_2300_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_2301_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_2302_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_2303_power__not__zero,axiom,
! [A: complex,N: nat] :
( ( A != zero_zero_complex )
=> ( ( power_power_complex @ A @ N )
!= zero_zero_complex ) ) ).
% power_not_zero
thf(fact_2304_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_2305_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_2306_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_2307_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_2308_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_2309_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_2310_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_2311_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_2312_old_Onat_Oexhaust,axiom,
! [Y4: nat] :
( ( Y4 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y4
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_2313_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_2314_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_2315_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_2316_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_2317_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ( ( X
!= ( suc @ zero_zero_nat ) )
=> ~ ! [Va3: nat] :
( X
!= ( suc @ ( suc @ Va3 ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_2318_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_2319_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_2320_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_2321_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_2322_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_2323_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_2324_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_2325_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_2326_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_2327_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_2328_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_2329_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_2330_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_2331_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_2332_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_2333_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_2334_divide__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_2335_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_2336_divide__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_2337_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_2338_zero__le__divide__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_divide_iff
thf(fact_2339_divide__nonneg__nonneg,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_2340_divide__nonneg__nonneg,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_2341_divide__nonneg__nonpos,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_2342_divide__nonneg__nonpos,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ Y4 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_nonpos
thf(fact_2343_divide__nonpos__nonneg,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_2344_divide__nonpos__nonneg,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_nonneg
thf(fact_2345_divide__nonpos__nonpos,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_2346_divide__nonpos__nonpos,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y4 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_2347_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_2348_divide__right__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_2349_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_2350_divide__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_2351_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_2352_divide__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_2353_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_2354_zero__less__divide__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_less_divide_iff
thf(fact_2355_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_2356_divide__less__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) )
& ( C != zero_zero_rat ) ) ) ).
% divide_less_cancel
thf(fact_2357_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_2358_divide__less__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_2359_divide__pos__pos,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% divide_pos_pos
thf(fact_2360_divide__pos__pos,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ zero_zero_rat @ Y4 )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% divide_pos_pos
thf(fact_2361_divide__pos__neg,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_2362_divide__pos__neg,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ Y4 @ zero_zero_rat )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% divide_pos_neg
thf(fact_2363_divide__neg__pos,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_2364_divide__neg__pos,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y4 )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% divide_neg_pos
thf(fact_2365_divide__neg__neg,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% divide_neg_neg
thf(fact_2366_divide__neg__neg,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ Y4 @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% divide_neg_neg
thf(fact_2367_frac__eq__eq,axiom,
! [Y4: complex,Z2: complex,X: complex,W2: complex] :
( ( Y4 != zero_zero_complex )
=> ( ( Z2 != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ X @ Y4 )
= ( divide1717551699836669952omplex @ W2 @ Z2 ) )
= ( ( times_times_complex @ X @ Z2 )
= ( times_times_complex @ W2 @ Y4 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2368_frac__eq__eq,axiom,
! [Y4: real,Z2: real,X: real,W2: real] :
( ( Y4 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ( divide_divide_real @ X @ Y4 )
= ( divide_divide_real @ W2 @ Z2 ) )
= ( ( times_times_real @ X @ Z2 )
= ( times_times_real @ W2 @ Y4 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2369_frac__eq__eq,axiom,
! [Y4: rat,Z2: rat,X: rat,W2: rat] :
( ( Y4 != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( ( divide_divide_rat @ X @ Y4 )
= ( divide_divide_rat @ W2 @ Z2 ) )
= ( ( times_times_rat @ X @ Z2 )
= ( times_times_rat @ W2 @ Y4 ) ) ) ) ) ).
% frac_eq_eq
thf(fact_2370_divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq
thf(fact_2371_divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( divide_divide_real @ B @ C )
= A )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq
thf(fact_2372_divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ( divide_divide_rat @ B @ C )
= A )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq
thf(fact_2373_eq__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ A @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq
thf(fact_2374_eq__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq
thf(fact_2375_eq__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq
thf(fact_2376_divide__eq__imp,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( B
= ( times_times_complex @ A @ C ) )
=> ( ( divide1717551699836669952omplex @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_2377_divide__eq__imp,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( B
= ( times_times_real @ A @ C ) )
=> ( ( divide_divide_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_2378_divide__eq__imp,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( B
= ( times_times_rat @ A @ C ) )
=> ( ( divide_divide_rat @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_2379_eq__divide__imp,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A @ C )
= B )
=> ( A
= ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_2380_eq__divide__imp,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= B )
=> ( A
= ( divide_divide_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_2381_eq__divide__imp,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( ( times_times_rat @ A @ C )
= B )
=> ( A
= ( divide_divide_rat @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_2382_nonzero__divide__eq__eq,axiom,
! [C: complex,B: complex,A: complex] :
( ( C != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ B @ C )
= A )
= ( B
= ( times_times_complex @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2383_nonzero__divide__eq__eq,axiom,
! [C: real,B: real,A: real] :
( ( C != zero_zero_real )
=> ( ( ( divide_divide_real @ B @ C )
= A )
= ( B
= ( times_times_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2384_nonzero__divide__eq__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( C != zero_zero_rat )
=> ( ( ( divide_divide_rat @ B @ C )
= A )
= ( B
= ( times_times_rat @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_2385_nonzero__eq__divide__eq,axiom,
! [C: complex,A: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( A
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( times_times_complex @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2386_nonzero__eq__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( A
= ( divide_divide_real @ B @ C ) )
= ( ( times_times_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2387_nonzero__eq__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( C != zero_zero_rat )
=> ( ( A
= ( divide_divide_rat @ B @ C ) )
= ( ( times_times_rat @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_2388_right__inverse__eq,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= one_one_complex )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2389_right__inverse__eq,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( ( divide_divide_real @ A @ B )
= one_one_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2390_right__inverse__eq,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( ( divide_divide_rat @ A @ B )
= one_one_rat )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_2391_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2392_power__eq__imp__eq__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ( power_power_rat @ A @ N )
= ( power_power_rat @ B @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2393_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2394_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_2395_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2396_power__eq__iff__eq__base,axiom,
! [N: nat,A: rat,B: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ( power_power_rat @ A @ N )
= ( power_power_rat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2397_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2398_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_2399_lambda__zero,axiom,
( ( ^ [H: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_2400_lambda__zero,axiom,
( ( ^ [H: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_2401_lambda__zero,axiom,
( ( ^ [H: rat] : zero_zero_rat )
= ( times_times_rat @ zero_zero_rat ) ) ).
% lambda_zero
thf(fact_2402_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_2403_lambda__zero,axiom,
( ( ^ [H: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_2404_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2405_power__strict__mono,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2406_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2407_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_2408_field__le__epsilon,axiom,
! [X: real,Y4: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y4 @ E2 ) ) )
=> ( ord_less_eq_real @ X @ Y4 ) ) ).
% field_le_epsilon
thf(fact_2409_field__le__epsilon,axiom,
! [X: rat,Y4: rat] :
( ! [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
=> ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y4 @ E2 ) ) )
=> ( ord_less_eq_rat @ X @ Y4 ) ) ).
% field_le_epsilon
thf(fact_2410_frac__le,axiom,
! [Y4: real,X: real,W2: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ X @ Y4 )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y4 @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_2411_frac__le,axiom,
! [Y4: rat,X: rat,W2: rat,Z2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ( ord_less_eq_rat @ X @ Y4 )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z2 )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y4 @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_2412_frac__less,axiom,
! [X: real,Y4: real,W2: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y4 )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y4 @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_2413_frac__less,axiom,
! [X: rat,Y4: rat,W2: rat,Z2: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ X @ Y4 )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_eq_rat @ W2 @ Z2 )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y4 @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_2414_frac__less2,axiom,
! [X: real,Y4: real,W2: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y4 )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_real @ W2 @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y4 @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_2415_frac__less2,axiom,
! [X: rat,Y4: rat,W2: rat,Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ X @ Y4 )
=> ( ( ord_less_rat @ zero_zero_rat @ W2 )
=> ( ( ord_less_rat @ W2 @ Z2 )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y4 @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_2416_divide__le__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_2417_divide__le__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% divide_le_cancel
thf(fact_2418_divide__nonneg__neg,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_2419_divide__nonneg__neg,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ Y4 @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% divide_nonneg_neg
thf(fact_2420_divide__nonneg__pos,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_2421_divide__nonneg__pos,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_rat @ zero_zero_rat @ Y4 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% divide_nonneg_pos
thf(fact_2422_divide__nonpos__neg,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y4 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_2423_divide__nonpos__neg,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ Y4 @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% divide_nonpos_neg
thf(fact_2424_divide__nonpos__pos,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_2425_divide__nonpos__pos,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ Y4 )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ zero_zero_rat ) ) ) ).
% divide_nonpos_pos
thf(fact_2426_divide__strict__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_2427_divide__strict__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_2428_divide__strict__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_2429_divide__strict__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_2430_mult__imp__less__div__pos,axiom,
! [Y4: real,Z2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_real @ ( times_times_real @ Z2 @ Y4 ) @ X )
=> ( ord_less_real @ Z2 @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2431_mult__imp__less__div__pos,axiom,
! [Y4: rat,Z2: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y4 )
=> ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y4 ) @ X )
=> ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_2432_mult__imp__div__pos__less,axiom,
! [Y4: real,X: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_real @ X @ ( times_times_real @ Z2 @ Y4 ) )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ Z2 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2433_mult__imp__div__pos__less,axiom,
! [Y4: rat,X: rat,Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y4 )
=> ( ( ord_less_rat @ X @ ( times_times_rat @ Z2 @ Y4 ) )
=> ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ Z2 ) ) ) ).
% mult_imp_div_pos_less
thf(fact_2434_pos__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_2435_pos__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% pos_less_divide_eq
thf(fact_2436_pos__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_2437_pos__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_divide_less_eq
thf(fact_2438_neg__less__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_2439_neg__less__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_less_divide_eq
thf(fact_2440_neg__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_2441_neg__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% neg_divide_less_eq
thf(fact_2442_less__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_2443_less__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_2444_divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_2445_divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_2446_less__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_2447_less__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% less_divide_eq_1
thf(fact_2448_divide__less__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_less_eq_1
thf(fact_2449_divide__less__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ A ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ A @ B ) )
| ( A = zero_zero_rat ) ) ) ).
% divide_less_eq_1
thf(fact_2450_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2451_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2452_add__divide__eq__if__simps_I2_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= B ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_2453_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2454_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2455_add__divide__eq__if__simps_I1_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_2456_add__frac__eq,axiom,
! [Y4: complex,Z2: complex,X: complex,W2: complex] :
( ( Y4 != zero_zero_complex )
=> ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W2 @ Y4 ) ) @ ( times_times_complex @ Y4 @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_2457_add__frac__eq,axiom,
! [Y4: real,Z2: real,X: real,W2: real] :
( ( Y4 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y4 ) ) @ ( times_times_real @ Y4 @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_2458_add__frac__eq,axiom,
! [Y4: rat,Z2: rat,X: rat,W2: rat] :
( ( Y4 != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z2 ) ) ) ) ) ).
% add_frac_eq
thf(fact_2459_add__frac__num,axiom,
! [Y4: complex,X: complex,Z2: complex] :
( ( Y4 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ Z2 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y4 ) ) @ Y4 ) ) ) ).
% add_frac_num
thf(fact_2460_add__frac__num,axiom,
! [Y4: real,X: real,Z2: real] :
( ( Y4 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Y4 ) @ Z2 )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y4 ) ) @ Y4 ) ) ) ).
% add_frac_num
thf(fact_2461_add__frac__num,axiom,
! [Y4: rat,X: rat,Z2: rat] :
( ( Y4 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y4 ) @ Z2 )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y4 ) ) @ Y4 ) ) ) ).
% add_frac_num
thf(fact_2462_add__num__frac,axiom,
! [Y4: complex,Z2: complex,X: complex] :
( ( Y4 != zero_zero_complex )
=> ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X @ Y4 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y4 ) ) @ Y4 ) ) ) ).
% add_num_frac
thf(fact_2463_add__num__frac,axiom,
! [Y4: real,Z2: real,X: real] :
( ( Y4 != zero_zero_real )
=> ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X @ Y4 ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y4 ) ) @ Y4 ) ) ) ).
% add_num_frac
thf(fact_2464_add__num__frac,axiom,
! [Y4: rat,Z2: rat,X: rat] :
( ( Y4 != zero_zero_rat )
=> ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X @ Y4 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y4 ) ) @ Y4 ) ) ) ).
% add_num_frac
thf(fact_2465_add__divide__eq__iff,axiom,
! [Z2: complex,X: complex,Y4: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y4 @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ Y4 ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_2466_add__divide__eq__iff,axiom,
! [Z2: real,X: real,Y4: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ X @ ( divide_divide_real @ Y4 @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ Y4 ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_2467_add__divide__eq__iff,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y4 @ Z2 ) )
= ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ Y4 ) @ Z2 ) ) ) ).
% add_divide_eq_iff
thf(fact_2468_divide__add__eq__iff,axiom,
! [Z2: complex,X: complex,Y4: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y4 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_2469_divide__add__eq__iff,axiom,
! [Z2: real,X: real,Y4: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( divide_divide_real @ X @ Z2 ) @ Y4 )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_2470_divide__add__eq__iff,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y4 )
= ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_add_eq_iff
thf(fact_2471_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2472_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2473_add__divide__eq__if__simps_I4_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= A ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_2474_diff__frac__eq,axiom,
! [Y4: complex,Z2: complex,X: complex,W2: complex] :
( ( Y4 != zero_zero_complex )
=> ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W2 @ Y4 ) ) @ ( times_times_complex @ Y4 @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2475_diff__frac__eq,axiom,
! [Y4: real,Z2: real,X: real,W2: real] :
( ( Y4 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y4 ) ) @ ( times_times_real @ Y4 @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2476_diff__frac__eq,axiom,
! [Y4: rat,Z2: rat,X: rat,W2: rat] :
( ( Y4 != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z2 ) ) ) ) ) ).
% diff_frac_eq
thf(fact_2477_diff__divide__eq__iff,axiom,
! [Z2: complex,X: complex,Y4: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y4 @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ Y4 ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_2478_diff__divide__eq__iff,axiom,
! [Z2: real,X: real,Y4: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ X @ ( divide_divide_real @ Y4 @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y4 ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_2479_diff__divide__eq__iff,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y4 @ Z2 ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ Y4 ) @ Z2 ) ) ) ).
% diff_divide_eq_iff
thf(fact_2480_divide__diff__eq__iff,axiom,
! [Z2: complex,X: complex,Y4: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y4 )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_2481_divide__diff__eq__iff,axiom,
! [Z2: real,X: real,Y4: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y4 )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_2482_divide__diff__eq__iff,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y4 )
= ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% divide_diff_eq_iff
thf(fact_2483_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_2484_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_le_numeral
thf(fact_2485_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_2486_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_2487_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_2488_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_le_zero
thf(fact_2489_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_2490_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_2491_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2492_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2493_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2494_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2495_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2496_zero__le__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2497_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_2498_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2499_mult__nonneg__nonpos2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2500_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2501_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_2502_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2503_mult__nonpos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2504_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2505_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_2506_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2507_mult__nonneg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2508_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2509_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_2510_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2511_mult__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2512_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2513_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_2514_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_2515_split__mult__neg__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% split_mult_neg_le
thf(fact_2516_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_2517_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_2518_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2519_mult__le__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2520_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_2521_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_2522_mult__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_2523_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_2524_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_2525_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2526_mult__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2527_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_2528_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2529_mult__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2530_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2531_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_2532_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2533_mult__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2534_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_2535_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2536_mult__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2537_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_2538_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2539_split__mult__pos__le,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_eq_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2540_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_2541_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_2542_zero__le__square,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% zero_le_square
thf(fact_2543_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_2544_mult__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2545_mult__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2546_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2547_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_2548_mult__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2549_mult__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2550_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2551_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_2552_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_2553_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% not_numeral_less_zero
thf(fact_2554_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_2555_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_2556_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_2557_zero__less__numeral,axiom,
! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% zero_less_numeral
thf(fact_2558_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_2559_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_2560_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_2561_not__one__le__zero,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_le_zero
thf(fact_2562_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_2563_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_2564_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2565_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2566_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2567_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2568_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_2569_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one_class.zero_le_one
thf(fact_2570_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_2571_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_2572_add__nonpos__eq__0__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y4 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y4 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_2573_add__nonpos__eq__0__iff,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ Y4 @ zero_zero_rat )
=> ( ( ( plus_plus_rat @ X @ Y4 )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y4 = zero_zero_rat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_2574_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y4 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y4 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_2575_add__nonpos__eq__0__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y4 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y4 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_2576_add__nonneg__eq__0__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ( plus_plus_real @ X @ Y4 )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y4 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_2577_add__nonneg__eq__0__iff,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ( ( plus_plus_rat @ X @ Y4 )
= zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y4 = zero_zero_rat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_2578_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ( ( plus_plus_nat @ X @ Y4 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_2579_add__nonneg__eq__0__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( ( plus_plus_int @ X @ Y4 )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_2580_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_2581_add__nonpos__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_nonpos
thf(fact_2582_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_2583_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_2584_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_2585_add__nonneg__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_2586_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_2587_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_2588_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_2589_add__increasing2,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ B @ A )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_2590_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_2591_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_2592_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_2593_add__decreasing2,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_2594_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_2595_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_2596_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_2597_add__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_2598_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_2599_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_2600_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_2601_add__decreasing,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_2602_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_2603_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_2604_mult__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_2605_mult__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_2606_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_2607_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_2608_not__square__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% not_square_less_zero
thf(fact_2609_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_2610_mult__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2611_mult__less__0__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ B @ zero_zero_rat ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2612_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_2613_mult__neg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_2614_mult__neg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_neg_pos
thf(fact_2615_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_2616_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_2617_mult__pos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_2618_mult__pos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg
thf(fact_2619_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_2620_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_2621_mult__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2622_mult__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2623_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2624_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_2625_mult__pos__neg2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_2626_mult__pos__neg2,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% mult_pos_neg2
thf(fact_2627_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_2628_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_2629_zero__less__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2630_zero__less__mult__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_rat @ zero_zero_rat @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2631_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_2632_zero__less__mult__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2633_zero__less__mult__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2634_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2635_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_2636_zero__less__mult__pos2,axiom,
! [B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2637_zero__less__mult__pos2,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2638_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2639_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_2640_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2641_mult__less__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2642_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_2643_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2644_mult__less__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_rat @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2645_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_2646_mult__strict__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2647_mult__strict__left__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2648_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_2649_mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2650_mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2651_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2652_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_2653_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2654_mult__less__cancel__left__disj,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2655_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_2656_mult__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2657_mult__strict__right__mono__neg,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2658_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_2659_mult__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2660_mult__strict__right__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2661_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2662_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_2663_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2664_mult__less__cancel__right__disj,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
& ( ord_less_rat @ A @ B ) )
| ( ( ord_less_rat @ C @ zero_zero_rat )
& ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2665_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_2666_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2667_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2668_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2669_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2670_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A4: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_2671_le__iff__diff__le__0,axiom,
( ord_less_eq_rat
= ( ^ [A4: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B3 ) @ zero_zero_rat ) ) ) ).
% le_iff_diff_le_0
thf(fact_2672_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_2673_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_2674_less__numeral__extra_I1_J,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% less_numeral_extra(1)
thf(fact_2675_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_2676_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_2677_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_2678_zero__less__one,axiom,
ord_less_rat @ zero_zero_rat @ one_one_rat ).
% zero_less_one
thf(fact_2679_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_2680_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_2681_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_2682_not__one__less__zero,axiom,
~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% not_one_less_zero
thf(fact_2683_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_2684_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_2685_add__less__zeroD,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y4 ) @ zero_zero_real )
=> ( ( ord_less_real @ X @ zero_zero_real )
| ( ord_less_real @ Y4 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_2686_add__less__zeroD,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ ( plus_plus_rat @ X @ Y4 ) @ zero_zero_rat )
=> ( ( ord_less_rat @ X @ zero_zero_rat )
| ( ord_less_rat @ Y4 @ zero_zero_rat ) ) ) ).
% add_less_zeroD
thf(fact_2687_add__less__zeroD,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y4 ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y4 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_2688_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_2689_add__neg__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_neg
thf(fact_2690_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_2691_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_2692_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_2693_add__pos__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_2694_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_2695_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_2696_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_2697_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_2698_pos__add__strict,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_2699_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_2700_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_2701_less__iff__diff__less__0,axiom,
( ord_less_real
= ( ^ [A4: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B3 ) @ zero_zero_real ) ) ) ).
% less_iff_diff_less_0
thf(fact_2702_less__iff__diff__less__0,axiom,
( ord_less_rat
= ( ^ [A4: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B3 ) @ zero_zero_rat ) ) ) ).
% less_iff_diff_less_0
thf(fact_2703_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_2704_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2705_power__mono,axiom,
! [A: rat,B: rat,N: nat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2706_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2707_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_2708_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2709_zero__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2710_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2711_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_2712_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2713_zero__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2714_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2715_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_2716_power__0,axiom,
! [A: rat] :
( ( power_power_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% power_0
thf(fact_2717_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_2718_power__0,axiom,
! [A: real] :
( ( power_power_real @ A @ zero_zero_nat )
= one_one_real ) ).
% power_0
thf(fact_2719_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_2720_power__0,axiom,
! [A: complex] :
( ( power_power_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% power_0
thf(fact_2721_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_2722_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_2723_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_2724_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_2725_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_2726_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_2727_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_2728_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_2729_option_Osize_I4_J,axiom,
! [X22: nat] :
( ( size_size_option_nat @ ( some_nat @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_2730_option_Osize_I4_J,axiom,
! [X22: product_prod_nat_nat] :
( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_2731_option_Osize_I4_J,axiom,
! [X22: num] :
( ( size_size_option_num @ ( some_num @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_2732_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_2733_option_Osize_I3_J,axiom,
( ( size_size_option_nat @ none_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_2734_option_Osize_I3_J,axiom,
( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_2735_option_Osize_I3_J,axiom,
( ( size_size_option_num @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_2736_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_2737_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_2738_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_2739_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_2740_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_2741_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_2742_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_2743_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_2744_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_2745_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_2746_vebt__insert_Osimps_I2_J,axiom,
! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S2 ) @ X )
= ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S2 ) ) ).
% vebt_insert.simps(2)
thf(fact_2747_field__le__mult__one__interval,axiom,
! [X: real,Y4: real] :
( ! [Z: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ Z @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ Y4 ) ) )
=> ( ord_less_eq_real @ X @ Y4 ) ) ).
% field_le_mult_one_interval
thf(fact_2748_field__le__mult__one__interval,axiom,
! [X: rat,Y4: rat] :
( ! [Z: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z )
=> ( ( ord_less_rat @ Z @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ Y4 ) ) )
=> ( ord_less_eq_rat @ X @ Y4 ) ) ).
% field_le_mult_one_interval
thf(fact_2749_divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_2750_divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_2751_le__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_2752_le__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_2753_divide__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_2754_divide__left__mono,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_left_mono
thf(fact_2755_neg__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_2756_neg__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% neg_divide_le_eq
thf(fact_2757_neg__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_2758_neg__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_le_divide_eq
thf(fact_2759_pos__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_2760_pos__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_divide_le_eq
thf(fact_2761_pos__le__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_2762_pos__le__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% pos_le_divide_eq
thf(fact_2763_mult__imp__div__pos__le,axiom,
! [Y4: real,X: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ X @ ( times_times_real @ Z2 @ Y4 ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ Z2 ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2764_mult__imp__div__pos__le,axiom,
! [Y4: rat,X: rat,Z2: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y4 )
=> ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z2 @ Y4 ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ Z2 ) ) ) ).
% mult_imp_div_pos_le
thf(fact_2765_mult__imp__le__div__pos,axiom,
! [Y4: real,Z2: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y4 ) @ X )
=> ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2766_mult__imp__le__div__pos,axiom,
! [Y4: rat,Z2: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y4 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y4 ) @ X )
=> ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_2767_divide__left__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_2768_divide__left__mono__neg,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_2769_divide__le__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ A ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ A @ B ) )
| ( A = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_2770_divide__le__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ B @ A ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ A @ B ) )
| ( A = zero_zero_rat ) ) ) ).
% divide_le_eq_1
thf(fact_2771_le__divide__eq__1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ A @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_2772_le__divide__eq__1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ A )
& ( ord_less_eq_rat @ A @ B ) )
| ( ( ord_less_rat @ A @ zero_zero_rat )
& ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% le_divide_eq_1
thf(fact_2773_frac__le__eq,axiom,
! [Y4: real,Z2: real,X: real,W2: real] :
( ( Y4 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y4 ) ) @ ( times_times_real @ Y4 @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% frac_le_eq
thf(fact_2774_frac__le__eq,axiom,
! [Y4: rat,Z2: rat,X: rat,W2: rat] :
( ( Y4 != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% frac_le_eq
thf(fact_2775_frac__less__eq,axiom,
! [Y4: real,Z2: real,X: real,W2: real] :
( ( Y4 != zero_zero_real )
=> ( ( Z2 != zero_zero_real )
=> ( ( ord_less_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ W2 @ Z2 ) )
= ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W2 @ Y4 ) ) @ ( times_times_real @ Y4 @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% frac_less_eq
thf(fact_2776_frac__less__eq,axiom,
! [Y4: rat,Z2: rat,X: rat,W2: rat] :
( ( Y4 != zero_zero_rat )
=> ( ( Z2 != zero_zero_rat )
=> ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
= ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W2 @ Y4 ) ) @ ( times_times_rat @ Y4 @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% frac_less_eq
thf(fact_2777_linordered__field__no__ub,axiom,
! [X5: real] :
? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_2778_linordered__field__no__ub,axiom,
! [X5: rat] :
? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_2779_linordered__field__no__lb,axiom,
! [X5: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_2780_linordered__field__no__lb,axiom,
! [X5: rat] :
? [Y3: rat] : ( ord_less_rat @ Y3 @ X5 ) ).
% linordered_field_no_lb
thf(fact_2781_mult__less__le__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2782_mult__less__le__imp__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_eq_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2783_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2784_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_2785_mult__le__less__imp__less,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2786_mult__le__less__imp__less,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2787_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2788_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_2789_mult__right__le__imp__le,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2790_mult__right__le__imp__le,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2791_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2792_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_2793_mult__left__le__imp__le,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2794_mult__left__le__imp__le,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2795_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2796_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_2797_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_2798_mult__le__cancel__left__pos,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_eq_rat @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_2799_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_2800_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_2801_mult__le__cancel__left__neg,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_2802_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_2803_mult__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_2804_mult__less__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_2805_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_2806_mult__strict__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2807_mult__strict__mono_H,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2808_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2809_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_2810_mult__right__less__imp__less,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2811_mult__right__less__imp__less,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2812_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2813_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_2814_mult__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_2815_mult__less__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_2816_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_2817_mult__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2818_mult__strict__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ C @ D )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2819_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2820_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_2821_mult__left__less__imp__less,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2822_mult__left__less__imp__less,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2823_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2824_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_2825_mult__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_2826_mult__le__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_2827_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_2828_mult__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_2829_mult__le__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_2830_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_2831_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_2832_add__strict__increasing2,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_2833_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_2834_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_2835_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_2836_add__strict__increasing,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ B @ C )
=> ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_2837_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_2838_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_2839_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_2840_add__pos__nonneg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_2841_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_2842_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_2843_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_2844_add__nonpos__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_nonpos_neg
thf(fact_2845_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_2846_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_2847_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_2848_add__nonneg__pos,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_2849_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_2850_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_2851_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_2852_add__neg__nonpos,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% add_neg_nonpos
thf(fact_2853_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_2854_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_2855_mult__left__le,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_2856_mult__left__le,axiom,
! [C: rat,A: rat] :
( ( ord_less_eq_rat @ C @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_2857_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_2858_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_2859_mult__le__one,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ B @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_2860_mult__le__one,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ B @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% mult_le_one
thf(fact_2861_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_2862_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_2863_mult__right__le__one__le,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y4 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_2864_mult__right__le__one__le,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ( ord_less_eq_rat @ Y4 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ X @ Y4 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_2865_mult__right__le__one__le,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y4 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_2866_mult__left__le__one__le,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y4 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_2867_mult__left__le__one__le,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ( ord_less_eq_rat @ Y4 @ one_one_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ Y4 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_2868_mult__left__le__one__le,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y4 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_2869_sum__squares__le__zero__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y4 = zero_zero_real ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_2870_sum__squares__le__zero__iff,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y4 = zero_zero_rat ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_2871_sum__squares__le__zero__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_2872_sum__squares__ge__zero,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) ) ).
% sum_squares_ge_zero
thf(fact_2873_sum__squares__ge__zero,axiom,
! [X: rat,Y4: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) ) ).
% sum_squares_ge_zero
thf(fact_2874_sum__squares__ge__zero,axiom,
! [X: int,Y4: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) ) ).
% sum_squares_ge_zero
thf(fact_2875_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2876_power__less__imp__less__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_rat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2877_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2878_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_2879_sum__squares__gt__zero__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) )
= ( ( X != zero_zero_real )
| ( Y4 != zero_zero_real ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_2880_sum__squares__gt__zero__iff,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) )
= ( ( X != zero_zero_rat )
| ( Y4 != zero_zero_rat ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_2881_sum__squares__gt__zero__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) )
= ( ( X != zero_zero_int )
| ( Y4 != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_2882_not__sum__squares__lt__zero,axiom,
! [X: real,Y4: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) @ zero_zero_real ) ).
% not_sum_squares_lt_zero
thf(fact_2883_not__sum__squares__lt__zero,axiom,
! [X: rat,Y4: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y4 @ Y4 ) ) @ zero_zero_rat ) ).
% not_sum_squares_lt_zero
thf(fact_2884_not__sum__squares__lt__zero,axiom,
! [X: int,Y4: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y4 @ Y4 ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_2885_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_2886_zero__less__two,axiom,
ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% zero_less_two
thf(fact_2887_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_2888_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_2889_power__le__one,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% power_le_one
thf(fact_2890_power__le__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% power_le_one
thf(fact_2891_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_2892_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_2893_divide__eq__eq__numeral_I1_J,axiom,
! [B: complex,C: complex,W2: num] :
( ( ( divide1717551699836669952omplex @ B @ C )
= ( numera6690914467698888265omplex @ W2 ) )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2894_divide__eq__eq__numeral_I1_J,axiom,
! [B: real,C: real,W2: num] :
( ( ( divide_divide_real @ B @ C )
= ( numeral_numeral_real @ W2 ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2895_divide__eq__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ( divide_divide_rat @ B @ C )
= ( numeral_numeral_rat @ W2 ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2896_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: complex,C: complex] :
( ( ( numera6690914467698888265omplex @ W2 )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( ( numera6690914467698888265omplex @ W2 )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2897_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C: real] :
( ( ( numeral_numeral_real @ W2 )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( numeral_numeral_real @ W2 )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2898_eq__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ( numeral_numeral_rat @ W2 )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( numeral_numeral_rat @ W2 )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2899_power__inject__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ ( suc @ N ) )
= ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2900_power__inject__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ( power_power_rat @ A @ ( suc @ N ) )
= ( power_power_rat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2901_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2902_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_2903_power__le__imp__le__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2904_power__le__imp__le__base,axiom,
! [A: rat,N: nat,B: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2905_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2906_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_2907_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_2908_div__add__self1,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self1
thf(fact_2909_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_2910_div__add__self2,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
= ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% div_add_self2
thf(fact_2911_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_2912_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size_num @ ( bit0 @ X22 ) )
= ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_2913_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K2 )
=> ~ ( P @ I ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_2914_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_2915_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_2916_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_2917_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_2918_length__pos__if__in__set,axiom,
! [X: real,Xs2: list_real] :
( ( member_real @ X @ ( set_real2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2919_length__pos__if__in__set,axiom,
! [X: complex,Xs2: list_complex] :
( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2920_length__pos__if__in__set,axiom,
! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2921_length__pos__if__in__set,axiom,
! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2922_length__pos__if__in__set,axiom,
! [X: $o,Xs2: list_o] :
( ( member_o @ X @ ( set_o2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2923_length__pos__if__in__set,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2924_length__pos__if__in__set,axiom,
! [X: int,Xs2: list_int] :
( ( member_int @ X @ ( set_int2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2925_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_2926_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_2927_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_2928_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_2929_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_2930_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_2931_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_2932_nat__one__le__power,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).
% nat_one_le_power
thf(fact_2933_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
= ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_2934_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_2935_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_2936_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_2937_vebt__insert_Osimps_I3_J,axiom,
! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ X )
= ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) ) ).
% vebt_insert.simps(3)
thf(fact_2938_scaling__mono,axiom,
! [U2: real,V: real,R2: real,S2: real] :
( ( ord_less_eq_real @ U2 @ V )
=> ( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_eq_real @ R2 @ S2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ U2 @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U2 ) ) @ S2 ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_2939_scaling__mono,axiom,
! [U2: rat,V: rat,R2: rat,S2: rat] :
( ( ord_less_eq_rat @ U2 @ V )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
=> ( ( ord_less_eq_rat @ R2 @ S2 )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ U2 @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U2 ) ) @ S2 ) ) @ V ) ) ) ) ).
% scaling_mono
thf(fact_2940_vebt__member_Osimps_I3_J,axiom,
! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).
% vebt_member.simps(3)
thf(fact_2941_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_2942_mult__le__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_2943_mult__le__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_2944_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_2945_mult__le__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_2946_mult__le__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ one_one_rat ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_2947_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_2948_mult__le__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_2949_mult__le__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ one_one_rat @ B ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_2950_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_2951_mult__le__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_2952_mult__le__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ A @ one_one_rat ) )
& ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_2953_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_2954_mult__less__cancel__left1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_2955_mult__less__cancel__left1,axiom,
! [C: rat,B: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_2956_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_2957_mult__less__cancel__left2,axiom,
! [C: real,A: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_2958_mult__less__cancel__left2,axiom,
! [C: rat,A: rat] :
( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_2959_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_2960_mult__less__cancel__right1,axiom,
! [C: real,B: real] :
( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_2961_mult__less__cancel__right1,axiom,
! [C: rat,B: rat] :
( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ one_one_rat @ B ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_2962_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_2963_mult__less__cancel__right2,axiom,
! [A: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_2964_mult__less__cancel__right2,axiom,
! [A: rat,C: rat] :
( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
= ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ A @ one_one_rat ) )
& ( ( ord_less_eq_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_2965_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_2966_convex__bound__le,axiom,
! [X: real,A: real,Y4: real,U2: real,V: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ Y4 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U2 @ V )
= one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U2 @ X ) @ ( times_times_real @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_2967_convex__bound__le,axiom,
! [X: rat,A: rat,Y4: rat,U2: rat,V: rat] :
( ( ord_less_eq_rat @ X @ A )
=> ( ( ord_less_eq_rat @ Y4 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U2 @ V )
= one_one_rat )
=> ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U2 @ X ) @ ( times_times_rat @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_2968_convex__bound__le,axiom,
! [X: int,A: int,Y4: int,U2: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y4 @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U2 @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U2 @ X ) @ ( times_times_int @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_2969_divide__less__eq__numeral_I1_J,axiom,
! [B: real,C: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2970_divide__less__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2971_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C: real] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2972_less__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2973_power__Suc__less,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_2974_power__Suc__less,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_2975_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_2976_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_2977_power__Suc__le__self,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2978_power__Suc__le__self,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2979_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2980_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_2981_power__Suc__less__one,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% power_Suc_less_one
thf(fact_2982_power__Suc__less__one,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% power_Suc_less_one
thf(fact_2983_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_2984_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_2985_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2986_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: rat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2987_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2988_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_2989_power__decreasing,axiom,
! [N: nat,N4: nat,A: real] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2990_power__decreasing,axiom,
! [N: nat,N4: nat,A: rat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2991_power__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2992_power__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_2993_zero__power2,axiom,
( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_rat ) ).
% zero_power2
thf(fact_2994_zero__power2,axiom,
( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% zero_power2
thf(fact_2995_zero__power2,axiom,
( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_real ) ).
% zero_power2
thf(fact_2996_zero__power2,axiom,
( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% zero_power2
thf(fact_2997_zero__power2,axiom,
( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_complex ) ).
% zero_power2
thf(fact_2998_self__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_2999_self__le__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_3000_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_3001_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_3002_one__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_3003_one__less__power,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ one_one_rat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_3004_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_3005_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_3006_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_3007_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_3008_power__diff,axiom,
! [A: complex,N: nat,M: nat] :
( ( A != zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_3009_power__diff,axiom,
! [A: real,N: nat,M: nat] :
( ( A != zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_3010_power__diff,axiom,
! [A: rat,N: nat,M: nat] :
( ( A != zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_3011_power__diff,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_3012_power__diff,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
= ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_diff
thf(fact_3013_div__if,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N2: nat] :
( if_nat
@ ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_3014_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_3015_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_3016_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_3017_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_3018_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_3019_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_3020_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
=> ( P @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_3021_mult__commute__abs,axiom,
! [C: real] :
( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
= ( times_times_real @ C ) ) ).
% mult_commute_abs
thf(fact_3022_mult__commute__abs,axiom,
! [C: rat] :
( ( ^ [X2: rat] : ( times_times_rat @ X2 @ C ) )
= ( times_times_rat @ C ) ) ).
% mult_commute_abs
thf(fact_3023_mult__commute__abs,axiom,
! [C: nat] :
( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
= ( times_times_nat @ C ) ) ).
% mult_commute_abs
thf(fact_3024_mult__commute__abs,axiom,
! [C: int] :
( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
= ( times_times_int @ C ) ) ).
% mult_commute_abs
thf(fact_3025_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_3026_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).
% vebt_member.simps(4)
thf(fact_3027_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_3028_vebt__pred_Osimps_I5_J,axiom,
! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
= none_nat ) ).
% vebt_pred.simps(5)
thf(fact_3029_convex__bound__lt,axiom,
! [X: real,A: real,Y4: real,U2: real,V: real] :
( ( ord_less_real @ X @ A )
=> ( ( ord_less_real @ Y4 @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ U2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ V )
=> ( ( ( plus_plus_real @ U2 @ V )
= one_one_real )
=> ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U2 @ X ) @ ( times_times_real @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_3030_convex__bound__lt,axiom,
! [X: rat,A: rat,Y4: rat,U2: rat,V: rat] :
( ( ord_less_rat @ X @ A )
=> ( ( ord_less_rat @ Y4 @ A )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ U2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ V )
=> ( ( ( plus_plus_rat @ U2 @ V )
= one_one_rat )
=> ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U2 @ X ) @ ( times_times_rat @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_3031_convex__bound__lt,axiom,
! [X: int,A: int,Y4: int,U2: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y4 @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U2 @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U2 @ X ) @ ( times_times_int @ V @ Y4 ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_3032_divide__le__eq__numeral_I1_J,axiom,
! [B: real,C: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_3033_divide__le__eq__numeral_I1_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_3034_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: real,C: real] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_3035_le__divide__eq__numeral_I1_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_3036_half__gt__zero,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_3037_half__gt__zero,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% half_gt_zero
thf(fact_3038_half__gt__zero__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% half_gt_zero_iff
thf(fact_3039_half__gt__zero__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% half_gt_zero_iff
thf(fact_3040_zero__le__power2,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_3041_zero__le__power2,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_3042_zero__le__power2,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% zero_le_power2
thf(fact_3043_power2__eq__imp__eq,axiom,
! [X: real,Y4: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( X = Y4 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_3044_power2__eq__imp__eq,axiom,
! [X: rat,Y4: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( X = Y4 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_3045_power2__eq__imp__eq,axiom,
! [X: nat,Y4: nat] :
( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( X = Y4 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_3046_power2__eq__imp__eq,axiom,
! [X: int,Y4: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( X = Y4 ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_3047_power2__le__imp__le,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% power2_le_imp_le
thf(fact_3048_power2__le__imp__le,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ord_less_eq_rat @ X @ Y4 ) ) ) ).
% power2_le_imp_le
thf(fact_3049_power2__le__imp__le,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ord_less_eq_nat @ X @ Y4 ) ) ) ).
% power2_le_imp_le
thf(fact_3050_power2__le__imp__le,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ord_less_eq_int @ X @ Y4 ) ) ) ).
% power2_le_imp_le
thf(fact_3051_power2__less__0,axiom,
! [A: real] :
~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% power2_less_0
thf(fact_3052_power2__less__0,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% power2_less_0
thf(fact_3053_power2__less__0,axiom,
! [A: int] :
~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% power2_less_0
thf(fact_3054_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_left
thf(fact_3055_exp__add__not__zero__imp__left,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_left
thf(fact_3056_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat ) ) ).
% exp_add_not_zero_imp_right
thf(fact_3057_exp__add__not__zero__imp__right,axiom,
! [M: nat,N: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int ) ) ).
% exp_add_not_zero_imp_right
thf(fact_3058_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
!= zero_zero_nat )
=> ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_nat ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3059_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [N: nat,M: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
!= zero_zero_int )
=> ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
!= zero_zero_int ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3060_power__diff__power__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( A != zero_zero_nat )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
= ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_3061_power__diff__power__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( A != zero_zero_int )
=> ( ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
= ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_3062_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_3063_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_3064_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_3065_power__eq__if,axiom,
( power_power_complex
= ( ^ [P4: complex,M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P4 @ ( power_power_complex @ P4 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_3066_power__eq__if,axiom,
( power_power_real
= ( ^ [P4: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P4 @ ( power_power_real @ P4 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_3067_power__eq__if,axiom,
( power_power_rat
= ( ^ [P4: rat,M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P4 @ ( power_power_rat @ P4 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_3068_power__eq__if,axiom,
( power_power_nat
= ( ^ [P4: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_3069_power__eq__if,axiom,
( power_power_int
= ( ^ [P4: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P4 @ ( power_power_int @ P4 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_3070_power__minus__mult,axiom,
! [N: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_complex @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_3071_power__minus__mult,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_real @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_3072_power__minus__mult,axiom,
! [N: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_rat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_3073_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_3074_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_3075_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_3076_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
& ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_3077_vebt__pred_Osimps_I6_J,axiom,
! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
= none_nat ) ).
% vebt_pred.simps(6)
thf(fact_3078_power2__less__imp__less,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_real @ X @ Y4 ) ) ) ).
% power2_less_imp_less
thf(fact_3079_power2__less__imp__less,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ord_less_rat @ X @ Y4 ) ) ) ).
% power2_less_imp_less
thf(fact_3080_power2__less__imp__less,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ord_less_nat @ X @ Y4 ) ) ) ).
% power2_less_imp_less
thf(fact_3081_power2__less__imp__less,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ord_less_int @ X @ Y4 ) ) ) ).
% power2_less_imp_less
thf(fact_3082_sum__power2__ge__zero,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_3083_sum__power2__ge__zero,axiom,
! [X: rat,Y4: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_3084_sum__power2__ge__zero,axiom,
! [X: int,Y4: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_3085_sum__power2__le__zero__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y4 = zero_zero_real ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_3086_sum__power2__le__zero__iff,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
= ( ( X = zero_zero_rat )
& ( Y4 = zero_zero_rat ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_3087_sum__power2__le__zero__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_3088_not__sum__power2__lt__zero,axiom,
! [X: real,Y4: real] :
~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% not_sum_power2_lt_zero
thf(fact_3089_not__sum__power2__lt__zero,axiom,
! [X: rat,Y4: rat] :
~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% not_sum_power2_lt_zero
thf(fact_3090_not__sum__power2__lt__zero,axiom,
! [X: int,Y4: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% not_sum_power2_lt_zero
thf(fact_3091_sum__power2__gt__zero__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_real )
| ( Y4 != zero_zero_real ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_3092_sum__power2__gt__zero__iff,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_rat )
| ( Y4 != zero_zero_rat ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_3093_sum__power2__gt__zero__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( ( X != zero_zero_int )
| ( Y4 != zero_zero_int ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_3094_zero__le__even__power_H,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_3095_zero__le__even__power_H,axiom,
! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_3096_zero__le__even__power_H,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% zero_le_even_power'
thf(fact_3097_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_3098_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_3099_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_3100_times__divide__times__eq,axiom,
! [X: complex,Y4: complex,Z2: complex,W2: complex] :
( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ Z2 @ W2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ Y4 @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_3101_times__divide__times__eq,axiom,
! [X: real,Y4: real,Z2: real,W2: real] :
( ( times_times_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ Z2 @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y4 @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_3102_times__divide__times__eq,axiom,
! [X: rat,Y4: rat,Z2: rat,W2: rat] :
( ( times_times_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ Z2 @ W2 ) )
= ( divide_divide_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y4 @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_3103_divide__divide__times__eq,axiom,
! [X: complex,Y4: complex,Z2: complex,W2: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y4 ) @ ( divide1717551699836669952omplex @ Z2 @ W2 ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W2 ) @ ( times_times_complex @ Y4 @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_3104_divide__divide__times__eq,axiom,
! [X: real,Y4: real,Z2: real,W2: real] :
( ( divide_divide_real @ ( divide_divide_real @ X @ Y4 ) @ ( divide_divide_real @ Z2 @ W2 ) )
= ( divide_divide_real @ ( times_times_real @ X @ W2 ) @ ( times_times_real @ Y4 @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_3105_divide__divide__times__eq,axiom,
! [X: rat,Y4: rat,Z2: rat,W2: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y4 ) @ ( divide_divide_rat @ Z2 @ W2 ) )
= ( divide_divide_rat @ ( times_times_rat @ X @ W2 ) @ ( times_times_rat @ Y4 @ Z2 ) ) ) ).
% divide_divide_times_eq
thf(fact_3106_divide__divide__eq__left_H,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
= ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_3107_divide__divide__eq__left_H,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
= ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_3108_divide__divide__eq__left_H,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
= ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_3109_add__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
= ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_3110_add__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_3111_add__divide__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
= ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_3112_diff__divide__distrib,axiom,
! [A: complex,B: complex,C: complex] :
( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
= ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_3113_diff__divide__distrib,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_3114_diff__divide__distrib,axiom,
! [A: rat,B: rat,C: rat] :
( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
= ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_3115_odd__0__le__power__imp__0__le,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_3116_odd__0__le__power__imp__0__le,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_3117_odd__0__le__power__imp__0__le,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_3118_odd__power__less__zero,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% odd_power_less_zero
thf(fact_3119_odd__power__less__zero,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% odd_power_less_zero
thf(fact_3120_odd__power__less__zero,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% odd_power_less_zero
thf(fact_3121_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_3122_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_3123_arith__geo__mean,axiom,
! [U2: real,X: real,Y4: real] :
( ( ( power_power_real @ U2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ X @ Y4 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ U2 @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_3124_arith__geo__mean,axiom,
! [U2: rat,X: rat,Y4: rat] :
( ( ( power_power_rat @ U2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_rat @ X @ Y4 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ord_less_eq_rat @ U2 @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_3125_less__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% less_half_sum
thf(fact_3126_less__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% less_half_sum
thf(fact_3127_gt__half__sum,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% gt_half_sum
thf(fact_3128_gt__half__sum,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% gt_half_sum
thf(fact_3129_vebt__pred_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: option_nat] :
( ( ( vEBT_vebt_pred @ X @ Xa2 )
= Y4 )
=> ( ( ? [Uu3: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ( Xa2 = zero_zero_nat )
=> ( Y4 != none_nat ) ) )
=> ( ! [A3: $o] :
( ? [Uw2: $o] :
( X
= ( vEBT_Leaf @ A3 @ Uw2 ) )
=> ( ( Xa2
= ( suc @ zero_zero_nat ) )
=> ~ ( ( A3
=> ( Y4
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y4 = none_nat ) ) ) ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ? [Va3: nat] :
( Xa2
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( B2
=> ( Y4
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A3
=> ( Y4
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y4 = none_nat ) ) ) ) ) ) )
=> ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
=> ( Y4 != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
=> ( Y4 != none_nat ) )
=> ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
=> ( Y4 != none_nat ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
=> ( Y4
= ( some_nat @ Ma2 ) ) )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( Y4
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
@ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.elims
thf(fact_3130_inrange,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% inrange
thf(fact_3131_set__bit__0,axiom,
! [A: int] :
( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_3132_set__bit__0,axiom,
! [A: nat] :
( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_bit_0
thf(fact_3133_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A1: vEBT_VEBT,A22: nat] :
( ( ? [A4: $o,B3: $o] :
( A1
= ( vEBT_Leaf @ A4 @ B3 ) )
& ( A22
= ( suc @ zero_zero_nat ) ) )
| ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
& ( A22
= ( plus_plus_nat @ N2 @ N2 ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
| ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
& ( A22
= ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
| ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
& ( A22
= ( plus_plus_nat @ N2 @ N2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A1
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ( vEBT_invar_vebt @ X2 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
& ( A22
= ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_3134_invar__vebt_Ocases,axiom,
! [A12: vEBT_VEBT,A23: nat] :
( ( vEBT_invar_vebt @ A12 @ A23 )
=> ( ( ? [A3: $o,B2: $o] :
( A12
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( A23
!= ( suc @ zero_zero_nat ) ) )
=> ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
=> ( ( M4 = N3 )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M4 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M4 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
=> ( ( M4 = N3 )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M4 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N3 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X5 @ N3 ) ) )
=> ( ( ord_less_nat @ Mi2 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A12
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus_nat @ N3 @ M4 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
=> ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I: nat] :
( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N3 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X5 @ N3 ) ) )
=> ( ( ord_less_nat @ Mi2 @ X5 )
& ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_3135_div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_3136_empty__subsetI,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% empty_subsetI
thf(fact_3137_empty__subsetI,axiom,
! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% empty_subsetI
thf(fact_3138_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_3139_subset__empty,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
= ( A2 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_3140_subset__empty,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
= ( A2 = bot_bot_set_real ) ) ).
% subset_empty
thf(fact_3141_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_3142_Leaf__0__not,axiom,
! [A: $o,B: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% Leaf_0_not
thf(fact_3143_deg1Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
= ( ? [A4: $o,B3: $o] :
( T
= ( vEBT_Leaf @ A4 @ B3 ) ) ) ) ).
% deg1Leaf
thf(fact_3144_deg__1__Leaf,axiom,
! [T: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T @ one_one_nat )
=> ? [A3: $o,B2: $o] :
( T
= ( vEBT_Leaf @ A3 @ B2 ) ) ) ).
% deg_1_Leaf
thf(fact_3145_deg__1__Leafy,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( ( N = one_one_nat )
=> ? [A3: $o,B2: $o] :
( T
= ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ).
% deg_1_Leafy
thf(fact_3146_zdiv__numeral__Bit0,axiom,
! [V: num,W2: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_3147_Diff__empty,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
= A2 ) ).
% Diff_empty
thf(fact_3148_Diff__empty,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
= A2 ) ).
% Diff_empty
thf(fact_3149_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_3150_empty__Diff,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
= bot_bot_set_int ) ).
% empty_Diff
thf(fact_3151_empty__Diff,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
= bot_bot_set_real ) ).
% empty_Diff
thf(fact_3152_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_3153_Diff__cancel,axiom,
! [A2: set_int] :
( ( minus_minus_set_int @ A2 @ A2 )
= bot_bot_set_int ) ).
% Diff_cancel
thf(fact_3154_Diff__cancel,axiom,
! [A2: set_real] :
( ( minus_minus_set_real @ A2 @ A2 )
= bot_bot_set_real ) ).
% Diff_cancel
thf(fact_3155_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_3156_empty__iff,axiom,
! [C: complex] :
~ ( member_complex @ C @ bot_bot_set_complex ) ).
% empty_iff
thf(fact_3157_empty__iff,axiom,
! [C: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% empty_iff
thf(fact_3158_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_3159_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_3160_empty__iff,axiom,
! [C: real] :
~ ( member_real @ C @ bot_bot_set_real ) ).
% empty_iff
thf(fact_3161_all__not__in__conv,axiom,
! [A2: set_complex] :
( ( ! [X2: complex] :
~ ( member_complex @ X2 @ A2 ) )
= ( A2 = bot_bot_set_complex ) ) ).
% all_not_in_conv
thf(fact_3162_all__not__in__conv,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( ! [X2: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X2 @ A2 ) )
= ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
thf(fact_3163_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_3164_all__not__in__conv,axiom,
! [A2: set_int] :
( ( ! [X2: int] :
~ ( member_int @ X2 @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_3165_all__not__in__conv,axiom,
! [A2: set_real] :
( ( ! [X2: real] :
~ ( member_real @ X2 @ A2 ) )
= ( A2 = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_3166_Collect__empty__eq,axiom,
! [P: complex > $o] :
( ( ( collect_complex @ P )
= bot_bot_set_complex )
= ( ! [X2: complex] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_3167_Collect__empty__eq,axiom,
! [P: product_prod_nat_nat > $o] :
( ( ( collec3392354462482085612at_nat @ P )
= bot_bo2099793752762293965at_nat )
= ( ! [X2: product_prod_nat_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_3168_Collect__empty__eq,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( ! [X2: list_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_3169_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_3170_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X2: int] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_3171_Collect__empty__eq,axiom,
! [P: real > $o] :
( ( ( collect_real @ P )
= bot_bot_set_real )
= ( ! [X2: real] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_3172_empty__Collect__eq,axiom,
! [P: complex > $o] :
( ( bot_bot_set_complex
= ( collect_complex @ P ) )
= ( ! [X2: complex] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_3173_empty__Collect__eq,axiom,
! [P: product_prod_nat_nat > $o] :
( ( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat @ P ) )
= ( ! [X2: product_prod_nat_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_3174_empty__Collect__eq,axiom,
! [P: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P ) )
= ( ! [X2: list_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_3175_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_3176_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X2: int] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_3177_empty__Collect__eq,axiom,
! [P: real > $o] :
( ( bot_bot_set_real
= ( collect_real @ P ) )
= ( ! [X2: real] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_3178_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X222 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_3179_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_3180_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
= N ) ).
% idiff_0_right
thf(fact_3181_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
= zero_z5237406670263579293d_enat ) ).
% idiff_0
thf(fact_3182_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_3183_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% half_negative_int_iff
thf(fact_3184_Diff__eq__empty__iff,axiom,
! [A2: set_int,B5: set_int] :
( ( ( minus_minus_set_int @ A2 @ B5 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_3185_Diff__eq__empty__iff,axiom,
! [A2: set_real,B5: set_real] :
( ( ( minus_minus_set_real @ A2 @ B5 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ A2 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_3186_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B5: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B5 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_3187_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_3188_VEBT_Osize_I4_J,axiom,
! [X21: $o,X222: $o] :
( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= zero_zero_nat ) ).
% VEBT.size(4)
thf(fact_3189_VEBT_Odistinct_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X222 ) ) ).
% VEBT.distinct(1)
thf(fact_3190_VEBT_Oexhaust,axiom,
! [Y4: vEBT_VEBT] :
( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
( Y4
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y4
!= ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% VEBT.exhaust
thf(fact_3191_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,Uw: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_3192_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_3193_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_3194_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_3195_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_3196_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_3197_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_3198_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
= ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
& ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_3199_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_p3455044024723400733d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
& ( N = zero_z5237406670263579293d_enat ) ) ) ).
% iadd_is_0
thf(fact_3200_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_3201_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_3202_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_3203_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_3204_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ zero_zero_nat )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_3205_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_3206_invar__vebt_Ointros_I1_J,axiom,
! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% invar_vebt.intros(1)
thf(fact_3207_int__power__div__base,axiom,
! [M: nat,K: int] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
= ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_3208_vebt__member_Osimps_I1_J,axiom,
! [A: $o,B: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
= ( ( ( X = zero_zero_nat )
=> A )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> B )
& ( X = one_one_nat ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_3209_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_3210_VEBT__internal_OminNull_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu3: $o] :
( X
!= ( vEBT_Leaf @ Uu3 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_3211_vebt__insert_Osimps_I1_J,axiom,
! [X: nat,A: $o,B: $o] :
( ( ( X = zero_zero_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
= ( vEBT_Leaf @ $true @ B ) ) )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
= ( vEBT_Leaf @ A @ $true ) ) )
& ( ( X != one_one_nat )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
= ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_3212_vebt__pred_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o] :
( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
= none_nat ) ).
% vebt_pred.simps(1)
thf(fact_3213_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A: $o,B: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
= ( ( ( X = zero_zero_nat )
=> A )
& ( ( X != zero_zero_nat )
=> ( ( ( X = one_one_nat )
=> B )
& ( X = one_one_nat ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_3214_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu3: $o] :
( X
!= ( vEBT_Leaf @ Uu3 @ $true ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_3215_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_3216_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_3217_not__psubset__empty,axiom,
! [A2: set_int] :
~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% not_psubset_empty
thf(fact_3218_not__psubset__empty,axiom,
! [A2: set_real] :
~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% not_psubset_empty
thf(fact_3219_emptyE,axiom,
! [A: complex] :
~ ( member_complex @ A @ bot_bot_set_complex ) ).
% emptyE
thf(fact_3220_emptyE,axiom,
! [A: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_3221_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_3222_emptyE,axiom,
! [A: int] :
~ ( member_int @ A @ bot_bot_set_int ) ).
% emptyE
thf(fact_3223_emptyE,axiom,
! [A: real] :
~ ( member_real @ A @ bot_bot_set_real ) ).
% emptyE
thf(fact_3224_equals0D,axiom,
! [A2: set_complex,A: complex] :
( ( A2 = bot_bot_set_complex )
=> ~ ( member_complex @ A @ A2 ) ) ).
% equals0D
thf(fact_3225_equals0D,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( A2 = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_3226_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_3227_equals0D,axiom,
! [A2: set_int,A: int] :
( ( A2 = bot_bot_set_int )
=> ~ ( member_int @ A @ A2 ) ) ).
% equals0D
thf(fact_3228_equals0D,axiom,
! [A2: set_real,A: real] :
( ( A2 = bot_bot_set_real )
=> ~ ( member_real @ A @ A2 ) ) ).
% equals0D
thf(fact_3229_equals0I,axiom,
! [A2: set_complex] :
( ! [Y3: complex] :
~ ( member_complex @ Y3 @ A2 )
=> ( A2 = bot_bot_set_complex ) ) ).
% equals0I
thf(fact_3230_equals0I,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ! [Y3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y3 @ A2 )
=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_3231_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_3232_equals0I,axiom,
! [A2: set_int] :
( ! [Y3: int] :
~ ( member_int @ Y3 @ A2 )
=> ( A2 = bot_bot_set_int ) ) ).
% equals0I
thf(fact_3233_equals0I,axiom,
! [A2: set_real] :
( ! [Y3: real] :
~ ( member_real @ Y3 @ A2 )
=> ( A2 = bot_bot_set_real ) ) ).
% equals0I
thf(fact_3234_ex__in__conv,axiom,
! [A2: set_complex] :
( ( ? [X2: complex] : ( member_complex @ X2 @ A2 ) )
= ( A2 != bot_bot_set_complex ) ) ).
% ex_in_conv
thf(fact_3235_ex__in__conv,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( ? [X2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X2 @ A2 ) )
= ( A2 != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
thf(fact_3236_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_3237_ex__in__conv,axiom,
! [A2: set_int] :
( ( ? [X2: int] : ( member_int @ X2 @ A2 ) )
= ( A2 != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_3238_ex__in__conv,axiom,
! [A2: set_real] :
( ( ? [X2: real] : ( member_real @ X2 @ A2 ) )
= ( A2 != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_3239_bot__set__def,axiom,
( bot_bot_set_complex
= ( collect_complex @ bot_bot_complex_o ) ) ).
% bot_set_def
thf(fact_3240_bot__set__def,axiom,
( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).
% bot_set_def
thf(fact_3241_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_3242_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_3243_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_3244_bot__set__def,axiom,
( bot_bot_set_real
= ( collect_real @ bot_bot_real_o ) ) ).
% bot_set_def
thf(fact_3245_realpow__pos__nth2,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ ( suc @ N ) )
= A ) ) ) ).
% realpow_pos_nth2
thf(fact_3246_real__arch__pow__inv,axiom,
! [Y4: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y4 ) ) ) ).
% real_arch_pow_inv
thf(fact_3247_vebt__mint_Ocases,axiom,
! [X: vEBT_VEBT] :
( ! [A3: $o,B2: $o] :
( X
!= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% vebt_mint.cases
thf(fact_3248_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y4: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y4 )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y4 )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y4 )
=> ( ( ? [Uu3: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ $true ) )
=> Y4 )
=> ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y4 )
=> ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> Y4 ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_3249_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_3250_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y5: real] :
( ( ( ord_less_real @ zero_zero_real @ Y5 )
& ( ( power_power_real @ Y5 @ N )
= A ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_3251_empty__def,axiom,
( bot_bot_set_complex
= ( collect_complex
@ ^ [X2: complex] : $false ) ) ).
% empty_def
thf(fact_3252_empty__def,axiom,
( bot_bo2099793752762293965at_nat
= ( collec3392354462482085612at_nat
@ ^ [X2: product_prod_nat_nat] : $false ) ) ).
% empty_def
thf(fact_3253_empty__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat
@ ^ [X2: list_nat] : $false ) ) ).
% empty_def
thf(fact_3254_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_3255_empty__def,axiom,
( bot_bot_set_int
= ( collect_int
@ ^ [X2: int] : $false ) ) ).
% empty_def
thf(fact_3256_empty__def,axiom,
( bot_bot_set_real
= ( collect_real
@ ^ [X2: real] : $false ) ) ).
% empty_def
thf(fact_3257_vebt__pred_Osimps_I2_J,axiom,
! [A: $o,Uw: $o] :
( ( A
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
= none_nat ) ) ) ).
% vebt_pred.simps(2)
thf(fact_3258_vebt__mint_Osimps_I1_J,axiom,
! [A: $o,B: $o] :
( ( A
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A
=> ( ( B
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
= ( some_nat @ one_one_nat ) ) )
& ( ~ B
=> ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
= none_nat ) ) ) ) ) ).
% vebt_mint.simps(1)
thf(fact_3259_vebt__maxt_Osimps_I1_J,axiom,
! [B: $o,A: $o] :
( ( B
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
= ( some_nat @ one_one_nat ) ) )
& ( ~ B
=> ( ( A
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A
=> ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
= none_nat ) ) ) ) ) ).
% vebt_maxt.simps(1)
thf(fact_3260_vebt__pred_Osimps_I3_J,axiom,
! [B: $o,A: $o,Va: nat] :
( ( B
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
= ( some_nat @ one_one_nat ) ) )
& ( ~ B
=> ( ( A
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A
=> ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
= none_nat ) ) ) ) ) ).
% vebt_pred.simps(3)
thf(fact_3261_vebt__mint_Oelims,axiom,
! [X: vEBT_VEBT,Y4: option_nat] :
( ( ( vEBT_vebt_mint @ X )
= Y4 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ~ ( ( A3
=> ( Y4
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( ( B2
=> ( Y4
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( Y4 = none_nat ) ) ) ) ) )
=> ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( Y4 != none_nat ) )
=> ~ ! [Mi2: nat] :
( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y4
!= ( some_nat @ Mi2 ) ) ) ) ) ) ).
% vebt_mint.elims
thf(fact_3262_vebt__maxt_Oelims,axiom,
! [X: vEBT_VEBT,Y4: option_nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y4 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ~ ( ( B2
=> ( Y4
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A3
=> ( Y4
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y4 = none_nat ) ) ) ) ) )
=> ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( Y4 != none_nat ) )
=> ~ ! [Mi2: nat,Ma2: nat] :
( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( Y4
!= ( some_nat @ Ma2 ) ) ) ) ) ) ).
% vebt_maxt.elims
thf(fact_3263_div__positive,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_positive
thf(fact_3264_div__positive,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_positive
thf(fact_3265_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ B )
=> ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3266_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( divide_divide_int @ A @ B )
= zero_zero_int ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3267_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3268_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3269_discrete,axiom,
( ord_less_nat
= ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).
% discrete
thf(fact_3270_discrete,axiom,
( ord_less_int
= ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).
% discrete
thf(fact_3271_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_3272_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_3273_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y4 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( Y4
= ( ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> Y4 )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [S: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ( Y4
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_3274_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_3275_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Uu3: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_3276_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y4 )
=> ( ( ? [Uu3: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> Y4 )
=> ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> Y4 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ( Y4
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( Y4
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
( ? [Vd2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ( Y4
= ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_3277_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_3278_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y4 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( Y4
= ( ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) ) )
=> ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> Y4 )
=> ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> Y4 )
=> ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> Y4 )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( Y4
= ( ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_3279_divmod__step__eq,axiom,
! [L2: num,R2: nat,Q2: nat] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L2 ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R2 )
=> ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_3280_divmod__step__eq,axiom,
! [L2: num,R2: int,Q2: int] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L2 ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R2 )
=> ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_3281_divmod__step__eq,axiom,
! [L2: num,R2: code_integer,Q2: code_integer] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
=> ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
= ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R2 )
=> ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
= ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% divmod_step_eq
thf(fact_3282_atLeastatMost__empty,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat ) ) ).
% atLeastatMost_empty
thf(fact_3283_atLeastatMost__empty,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num ) ) ).
% atLeastatMost_empty
thf(fact_3284_atLeastatMost__empty,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_3285_atLeastatMost__empty,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int ) ) ).
% atLeastatMost_empty
thf(fact_3286_atLeastatMost__empty,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real ) ) ).
% atLeastatMost_empty
thf(fact_3287_atLeastatMost__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3288_atLeastatMost__subset__iff,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ~ ( ord_less_eq_rat @ A @ B )
| ( ( ord_less_eq_rat @ C @ A )
& ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3289_atLeastatMost__subset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( ( ord_less_eq_num @ C @ A )
& ( ord_less_eq_num @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3290_atLeastatMost__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3291_atLeastatMost__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3292_atLeastatMost__subset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_3293_atLeastatMost__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( set_or4548717258645045905et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3294_atLeastatMost__empty__iff,axiom,
! [A: rat,B: rat] :
( ( ( set_or633870826150836451st_rat @ A @ B )
= bot_bot_set_rat )
= ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3295_atLeastatMost__empty__iff,axiom,
! [A: num,B: num] :
( ( ( set_or7049704709247886629st_num @ A @ B )
= bot_bot_set_num )
= ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3296_atLeastatMost__empty__iff,axiom,
! [A: nat,B: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3297_atLeastatMost__empty__iff,axiom,
! [A: int,B: int] :
( ( ( set_or1266510415728281911st_int @ A @ B )
= bot_bot_set_int )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3298_atLeastatMost__empty__iff,axiom,
! [A: real,B: real] :
( ( ( set_or1222579329274155063t_real @ A @ B )
= bot_bot_set_real )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3299_atLeastatMost__empty__iff2,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat
= ( set_or4548717258645045905et_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3300_atLeastatMost__empty__iff2,axiom,
! [A: rat,B: rat] :
( ( bot_bot_set_rat
= ( set_or633870826150836451st_rat @ A @ B ) )
= ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3301_atLeastatMost__empty__iff2,axiom,
! [A: num,B: num] :
( ( bot_bot_set_num
= ( set_or7049704709247886629st_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3302_atLeastatMost__empty__iff2,axiom,
! [A: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3303_atLeastatMost__empty__iff2,axiom,
! [A: int,B: int] :
( ( bot_bot_set_int
= ( set_or1266510415728281911st_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3304_atLeastatMost__empty__iff2,axiom,
! [A: real,B: real] :
( ( bot_bot_set_real
= ( set_or1222579329274155063t_real @ A @ B ) )
= ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3305_atLeastAtMost__iff,axiom,
! [I2: set_nat,L2: set_nat,U2: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4548717258645045905et_nat @ L2 @ U2 ) )
= ( ( ord_less_eq_set_nat @ L2 @ I2 )
& ( ord_less_eq_set_nat @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_3306_atLeastAtMost__iff,axiom,
! [I2: rat,L2: rat,U2: rat] :
( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L2 @ U2 ) )
= ( ( ord_less_eq_rat @ L2 @ I2 )
& ( ord_less_eq_rat @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_3307_atLeastAtMost__iff,axiom,
! [I2: num,L2: num,U2: num] :
( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L2 @ U2 ) )
= ( ( ord_less_eq_num @ L2 @ I2 )
& ( ord_less_eq_num @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_3308_atLeastAtMost__iff,axiom,
! [I2: nat,L2: nat,U2: nat] :
( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L2 @ U2 ) )
= ( ( ord_less_eq_nat @ L2 @ I2 )
& ( ord_less_eq_nat @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_3309_atLeastAtMost__iff,axiom,
! [I2: int,L2: int,U2: int] :
( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L2 @ U2 ) )
= ( ( ord_less_eq_int @ L2 @ I2 )
& ( ord_less_eq_int @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_3310_atLeastAtMost__iff,axiom,
! [I2: real,L2: real,U2: real] :
( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L2 @ U2 ) )
= ( ( ord_less_eq_real @ L2 @ I2 )
& ( ord_less_eq_real @ I2 @ U2 ) ) ) ).
% atLeastAtMost_iff
thf(fact_3311_Icc__eq__Icc,axiom,
! [L2: set_nat,H2: set_nat,L3: set_nat,H3: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L2 @ H2 )
= ( set_or4548717258645045905et_nat @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
& ~ ( ord_less_eq_set_nat @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3312_Icc__eq__Icc,axiom,
! [L2: rat,H2: rat,L3: rat,H3: rat] :
( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
= ( set_or633870826150836451st_rat @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_rat @ L2 @ H2 )
& ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3313_Icc__eq__Icc,axiom,
! [L2: num,H2: num,L3: num,H3: num] :
( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
= ( set_or7049704709247886629st_num @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_num @ L2 @ H2 )
& ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3314_Icc__eq__Icc,axiom,
! [L2: nat,H2: nat,L3: nat,H3: nat] :
( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
= ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_nat @ L2 @ H2 )
& ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3315_Icc__eq__Icc,axiom,
! [L2: int,H2: int,L3: int,H3: int] :
( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
= ( set_or1266510415728281911st_int @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_int @ L2 @ H2 )
& ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3316_Icc__eq__Icc,axiom,
! [L2: real,H2: real,L3: real,H3: real] :
( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
= ( set_or1222579329274155063t_real @ L3 @ H3 ) )
= ( ( ( L2 = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq_real @ L2 @ H2 )
& ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_3317_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_3318_double__eq__0__iff,axiom,
! [A: rat] :
( ( ( plus_plus_rat @ A @ A )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% double_eq_0_iff
thf(fact_3319_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_3320_unset__bit__0,axiom,
! [A: nat] :
( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_3321_unset__bit__0,axiom,
! [A: int] :
( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% unset_bit_0
thf(fact_3322_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_3323_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_3324_div__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L2 )
=> ( ( divide_divide_int @ K @ L2 )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_3325_div__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L2 @ K )
=> ( ( divide_divide_int @ K @ L2 )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_3326_add1__zle__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% add1_zle_eq
thf(fact_3327_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_3328_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I4: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_3329_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_3330_div__pos__geq,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ( ord_less_eq_int @ L2 @ K )
=> ( ( divide_divide_int @ K @ L2 )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_3331_q__pos__lemma,axiom,
! [B4: int,Q5: int,R4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
=> ( ( ord_less_int @ R4 @ B4 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% q_pos_lemma
thf(fact_3332_int__div__neg__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( divide_divide_int @ A @ B )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_3333_int__div__pos__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( divide_divide_int @ A @ B )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_3334_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_3335_zdiv__mono2__lemma,axiom,
! [B: int,Q2: int,R2: int,B4: int,Q5: int,R4: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
=> ( ( ord_less_int @ R4 @ B4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_3336_zdiv__mono2__neg__lemma,axiom,
! [B: int,Q2: int,R2: int,B4: int,Q5: int,R4: int] :
( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
= ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
=> ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) @ zero_zero_int )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_3337_unique__quotient__lemma,axiom,
! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R4 )
=> ( ( ord_less_int @ R4 @ B )
=> ( ( ord_less_int @ R2 @ B )
=> ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_3338_unique__quotient__lemma__neg,axiom,
! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( ord_less_int @ B @ R4 )
=> ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_3339_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_3340_int__ge__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_3341_int__induct,axiom,
! [P: int > $o,K: int,I2: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_induct
thf(fact_3342_int__le__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_3343_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_3344_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_7803423173614009249d_enat @ M @ N )
= zero_z5237406670263579293d_enat )
= ( ( M = zero_z5237406670263579293d_enat )
| ( N = zero_z5237406670263579293d_enat ) ) ) ).
% imult_is_0
thf(fact_3345_zero__one__enat__neq_I1_J,axiom,
zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% zero_one_enat_neq(1)
thf(fact_3346_times__int__code_I2_J,axiom,
! [L2: int] :
( ( times_times_int @ zero_zero_int @ L2 )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_3347_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_3348_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_3349_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_3350_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_3351_zmult__zless__mono2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_3352_plus__int__code_I2_J,axiom,
! [L2: int] :
( ( plus_plus_int @ zero_zero_int @ L2 )
= L2 ) ).
% plus_int_code(2)
thf(fact_3353_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_3354_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
thf(fact_3355_int__less__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_3356_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_3357_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_3358_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_3359_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_3360_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z2 )
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_3361_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_3362_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc8306885398267862888on_nat] :
( ! [Uu3: nat > nat > nat,Uv2: option_nat] :
( X
!= ( produc8929957630744042906on_nat @ Uu3 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
=> ( ! [Uw2: nat > nat > nat,V2: nat] :
( X
!= ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
=> ~ ! [F2: nat > nat > nat,A3: nat,B2: nat] :
( X
!= ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B2 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_3363_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc5542196010084753463at_nat] :
( ! [Uu3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
( X
!= ( produc2899441246263362727at_nat @ Uu3 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
=> ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
( X
!= ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
=> ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
( X
!= ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B2 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_3364_VEBT__internal_Ooption__shift_Ocases,axiom,
! [X: produc1193250871479095198on_num] :
( ! [Uu3: num > num > num,Uv2: option_num] :
( X
!= ( produc5778274026573060048on_num @ Uu3 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
=> ( ! [Uw2: num > num > num,V2: num] :
( X
!= ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
=> ~ ! [F2: num > num > num,A3: num,B2: num] :
( X
!= ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B2 ) ) ) ) ) ) ).
% VEBT_internal.option_shift.cases
thf(fact_3365_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc2233624965454879586on_nat] :
( ! [Uu3: nat > nat > $o,Uv2: option_nat] :
( X
!= ( produc4035269172776083154on_nat @ Uu3 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
=> ( ! [Uw2: nat > nat > $o,V2: nat] :
( X
!= ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
=> ~ ! [F2: nat > nat > $o,X3: nat,Y3: nat] :
( X
!= ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_3366_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc5491161045314408544at_nat] :
( ! [Uu3: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
( X
!= ( produc3994169339658061776at_nat @ Uu3 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
=> ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
( X
!= ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
=> ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( X
!= ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_3367_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
! [X: produc7036089656553540234on_num] :
( ! [Uu3: num > num > $o,Uv2: option_num] :
( X
!= ( produc3576312749637752826on_num @ Uu3 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
=> ( ! [Uw2: num > num > $o,V2: num] :
( X
!= ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
=> ~ ! [F2: num > num > $o,X3: num,Y3: num] :
( X
!= ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y3 ) ) ) ) ) ) ).
% VEBT_internal.option_comp_shift.cases
thf(fact_3368_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu3: $o,Uv2: $o,D3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ D3 ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_3369_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_3370_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A3: $o,B2: $o,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
=> ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_3371_vebt__insert_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A3: $o,B2: $o,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) @ X3 ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ X3 ) )
=> ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% vebt_insert.cases
thf(fact_3372_vebt__pred_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu3: $o,Uv2: $o] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ zero_zero_nat ) )
=> ( ! [A3: $o,Uw2: $o] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
=> ( ! [A3: $o,B2: $o,Va3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) )
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
=> ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).
% vebt_pred.cases
thf(fact_3373_VEBT__internal_Omembermima_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [Uu3: $o,Uv2: $o,Uw2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ X3 ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X3 ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_3374_vebt__member_Ocases,axiom,
! [X: produc9072475918466114483BT_nat] :
( ! [A3: $o,B2: $o,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X3 ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ X3 ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X3 ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_3375_atLeastatMost__psubset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ B @ D )
& ( ( ord_less_set_nat @ C @ A )
| ( ord_less_set_nat @ B @ D ) ) ) )
& ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3376_atLeastatMost__psubset__iff,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
= ( ( ~ ( ord_less_eq_rat @ A @ B )
| ( ( ord_less_eq_rat @ C @ A )
& ( ord_less_eq_rat @ B @ D )
& ( ( ord_less_rat @ C @ A )
| ( ord_less_rat @ B @ D ) ) ) )
& ( ord_less_eq_rat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3377_atLeastatMost__psubset__iff,axiom,
! [A: num,B: num,C: num,D: num] :
( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
= ( ( ~ ( ord_less_eq_num @ A @ B )
| ( ( ord_less_eq_num @ C @ A )
& ( ord_less_eq_num @ B @ D )
& ( ( ord_less_num @ C @ A )
| ( ord_less_num @ B @ D ) ) ) )
& ( ord_less_eq_num @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3378_atLeastatMost__psubset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3379_atLeastatMost__psubset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
= ( ( ~ ( ord_less_eq_int @ A @ B )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D )
& ( ( ord_less_int @ C @ A )
| ( ord_less_int @ B @ D ) ) ) )
& ( ord_less_eq_int @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3380_atLeastatMost__psubset__iff,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
= ( ( ~ ( ord_less_eq_real @ A @ B )
| ( ( ord_less_eq_real @ C @ A )
& ( ord_less_eq_real @ B @ D )
& ( ( ord_less_real @ C @ A )
| ( ord_less_real @ B @ D ) ) ) )
& ( ord_less_eq_real @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_3381_cpmi,axiom,
! [D4: int,P: int > $o,P5: int > $o,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P5 @ X3 )
= ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ( P5 @ X2 ) )
| ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ? [Y: int] :
( ( member_int @ Y @ B5 )
& ( P @ ( plus_plus_int @ Y @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_3382_cppi,axiom,
! [D4: int,P: int > $o,P5: int > $o,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P5 @ X3 )
= ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ( P5 @ X2 ) )
| ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
& ? [Y: int] :
( ( member_int @ Y @ A2 )
& ( P @ ( minus_minus_int @ Y @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_3383_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_3384_aset_I8_J,axiom,
! [D4: int,A2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_3385_aset_I6_J,axiom,
! [D4: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% aset(6)
thf(fact_3386_bset_I8_J,axiom,
! [D4: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ T @ X5 )
=> ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_3387_bset_I6_J,axiom,
! [D4: int,B5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq_int @ X5 @ T )
=> ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% bset(6)
thf(fact_3388_vebt__pred_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: option_nat] :
( ( ( vEBT_vebt_pred @ X @ Xa2 )
= Y4 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ( Xa2 = zero_zero_nat )
=> ( ( Y4 = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
=> ( ! [A3: $o,Uw2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ Uw2 ) )
=> ( ( Xa2
= ( suc @ zero_zero_nat ) )
=> ( ( ( A3
=> ( Y4
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y4 = none_nat ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ! [Va3: nat] :
( ( Xa2
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( B2
=> ( Y4
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A3
=> ( Y4
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y4 = none_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
=> ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
=> ( ( Y4 = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
=> ( ( Y4 = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
=> ( ( Y4 = none_nat )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
=> ( Y4
= ( some_nat @ Ma2 ) ) )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( Y4
= ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
@ ( if_option_nat
@ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
!= none_nat )
& ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( if_option_nat
@ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= none_nat )
@ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
@ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
@ none_nat ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_pred.pelims
thf(fact_3389_minf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z )
=> ~ ( ord_less_real @ T @ X5 ) ) ).
% minf(7)
thf(fact_3390_minf_I7_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z )
=> ~ ( ord_less_rat @ T @ X5 ) ) ).
% minf(7)
thf(fact_3391_minf_I7_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z )
=> ~ ( ord_less_num @ T @ X5 ) ) ).
% minf(7)
thf(fact_3392_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_3393_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_3394_minf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z )
=> ( ord_less_real @ X5 @ T ) ) ).
% minf(5)
thf(fact_3395_minf_I5_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z )
=> ( ord_less_rat @ X5 @ T ) ) ).
% minf(5)
thf(fact_3396_minf_I5_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z )
=> ( ord_less_num @ X5 @ T ) ) ).
% minf(5)
thf(fact_3397_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_3398_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_3399_minf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_3400_minf_I4_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_3401_minf_I4_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_3402_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_3403_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_3404_minf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_3405_minf_I3_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_3406_minf_I3_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_3407_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_3408_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_3409_minf_I2_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_3410_minf_I2_J,axiom,
! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
( ? [Z5: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_3411_minf_I2_J,axiom,
! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
( ? [Z5: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_3412_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_3413_minf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_3414_minf_I1_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_3415_minf_I1_J,axiom,
! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
( ? [Z5: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: rat] :
! [X3: rat] :
( ( ord_less_rat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_3416_minf_I1_J,axiom,
! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
( ? [Z5: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_3417_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_3418_minf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_3419_pinf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ Z @ X5 )
=> ( ord_less_real @ T @ X5 ) ) ).
% pinf(7)
thf(fact_3420_pinf_I7_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z @ X5 )
=> ( ord_less_rat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_3421_pinf_I7_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ Z @ X5 )
=> ( ord_less_num @ T @ X5 ) ) ).
% pinf(7)
thf(fact_3422_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_3423_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_3424_pinf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ Z @ X5 )
=> ~ ( ord_less_real @ X5 @ T ) ) ).
% pinf(5)
thf(fact_3425_pinf_I5_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z @ X5 )
=> ~ ( ord_less_rat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_3426_pinf_I5_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ Z @ X5 )
=> ~ ( ord_less_num @ X5 @ T ) ) ).
% pinf(5)
thf(fact_3427_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_3428_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_3429_pinf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_3430_pinf_I4_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_3431_pinf_I4_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_3432_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_3433_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_3434_pinf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_3435_pinf_I3_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_3436_pinf_I3_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_3437_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_3438_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_3439_pinf_I2_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: real] :
! [X5: real] :
( ( ord_less_real @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_3440_pinf_I2_J,axiom,
! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
( ? [Z5: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_3441_pinf_I2_J,axiom,
! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
( ? [Z5: num] :
! [X3: num] :
( ( ord_less_num @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: num] :
! [X3: num] :
( ( ord_less_num @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: num] :
! [X5: num] :
( ( ord_less_num @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_3442_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_3443_pinf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_3444_pinf_I1_J,axiom,
! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: real] :
! [X3: real] :
( ( ord_less_real @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: real] :
! [X5: real] :
( ( ord_less_real @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_3445_pinf_I1_J,axiom,
! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
( ? [Z5: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: rat] :
! [X3: rat] :
( ( ord_less_rat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_3446_pinf_I1_J,axiom,
! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
( ? [Z5: num] :
! [X3: num] :
( ( ord_less_num @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: num] :
! [X3: num] :
( ( ord_less_num @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: num] :
! [X5: num] :
( ( ord_less_num @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_3447_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_3448_pinf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q6 @ X3 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q6 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_3449_pinf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ Z @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% pinf(6)
thf(fact_3450_pinf_I6_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z @ X5 )
=> ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_3451_pinf_I6_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ Z @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% pinf(6)
thf(fact_3452_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_3453_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_3454_pinf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ Z @ X5 )
=> ( ord_less_eq_real @ T @ X5 ) ) ).
% pinf(8)
thf(fact_3455_pinf_I8_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z @ X5 )
=> ( ord_less_eq_rat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_3456_pinf_I8_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ Z @ X5 )
=> ( ord_less_eq_num @ T @ X5 ) ) ).
% pinf(8)
thf(fact_3457_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_3458_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_3459_minf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z )
=> ( ord_less_eq_real @ X5 @ T ) ) ).
% minf(6)
thf(fact_3460_minf_I6_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z )
=> ( ord_less_eq_rat @ X5 @ T ) ) ).
% minf(6)
thf(fact_3461_minf_I6_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z )
=> ( ord_less_eq_num @ X5 @ T ) ) ).
% minf(6)
thf(fact_3462_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_3463_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_3464_minf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z )
=> ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% minf(8)
thf(fact_3465_minf_I8_J,axiom,
! [T: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z )
=> ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% minf(8)
thf(fact_3466_minf_I8_J,axiom,
! [T: num] :
? [Z: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z )
=> ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% minf(8)
thf(fact_3467_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_3468_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_3469_inf__period_I2_J,axiom,
! [P: real > $o,D4: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
=> ! [X5: real,K4: real] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
| ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_3470_inf__period_I2_J,axiom,
! [P: rat > $o,D4: rat,Q: rat > $o] :
( ! [X3: rat,K2: rat] :
( ( P @ X3 )
= ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
=> ( ! [X3: rat,K2: rat] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
=> ! [X5: rat,K4: rat] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
| ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_3471_inf__period_I2_J,axiom,
! [P: int > $o,D4: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
| ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_3472_inf__period_I1_J,axiom,
! [P: real > $o,D4: real,Q: real > $o] :
( ! [X3: real,K2: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
=> ( ! [X3: real,K2: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K2 @ D4 ) ) ) )
=> ! [X5: real,K4: real] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
& ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_3473_inf__period_I1_J,axiom,
! [P: rat > $o,D4: rat,Q: rat > $o] :
( ! [X3: rat,K2: rat] :
( ( P @ X3 )
= ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
=> ( ! [X3: rat,K2: rat] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K2 @ D4 ) ) ) )
=> ! [X5: rat,K4: rat] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
& ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_3474_inf__period_I1_J,axiom,
! [P: int > $o,D4: int,Q: int > $o] :
( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
=> ! [X5: int,K4: int] :
( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
& ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_3475_aset_I2_J,axiom,
! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
| ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_3476_aset_I1_J,axiom,
! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ A2 )
=> ( X3
!= ( minus_minus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
& ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_3477_bset_I2_J,axiom,
! [D4: int,B5: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
| ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_3478_bset_I1_J,axiom,
! [D4: int,B5: set_int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb: int] :
( ( member_int @ Xb @ B5 )
=> ( X3
!= ( plus_plus_int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
=> ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
& ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_3479_plusinfinity,axiom,
! [D: int,P5: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P5 @ X3 )
= ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [X_12: int] : ( P5 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_3480_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_3481_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X5: int] :
( ( P @ X5 )
=> ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_3482_periodic__finite__ex,axiom,
! [D: int,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
=> ( ( ? [X6: int] : ( P @ X6 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_3483_bset_I3_J,axiom,
! [D4: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T )
=> ( ( minus_minus_int @ X5 @ D4 )
= T ) ) ) ) ) ).
% bset(3)
thf(fact_3484_bset_I4_J,axiom,
! [D4: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ B5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T )
=> ( ( minus_minus_int @ X5 @ D4 )
!= T ) ) ) ) ) ).
% bset(4)
thf(fact_3485_bset_I5_J,axiom,
! [D4: int,B5: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% bset(5)
thf(fact_3486_bset_I7_J,axiom,
! [D4: int,T: int,B5: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ B5 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_3487_aset_I3_J,axiom,
! [D4: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 = T )
=> ( ( plus_plus_int @ X5 @ D4 )
= T ) ) ) ) ) ).
% aset(3)
thf(fact_3488_aset_I4_J,axiom,
! [D4: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X5 != T )
=> ( ( plus_plus_int @ X5 @ D4 )
!= T ) ) ) ) ) ).
% aset(4)
thf(fact_3489_aset_I5_J,axiom,
! [D4: int,T: int,A2: set_int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ( ( member_int @ T @ A2 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ X5 @ T )
=> ( ord_less_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% aset(5)
thf(fact_3490_aset_I7_J,axiom,
! [D4: int,A2: set_int,T: int] :
( ( ord_less_int @ zero_zero_int @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_int @ T @ X5 )
=> ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_3491_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_3492_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y4 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( Y4
= ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( ~ Y4
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
=> ( ~ Y4
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
=> ( ~ Y4
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y4
= ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_3493_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
=> ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) @ Xa2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_3494_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) @ Xa2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_3495_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y4 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( Y4
= ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
=> ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
=> ( ~ Y4
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) )
=> ( ( Y4
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_3496_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> A3 )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> B2 )
& ( Xa2 = one_one_nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_3497_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y4 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ~ Y4
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa2 ) ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ( ~ Y4
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ( ( Y4
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( Y4
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ( ( Y4
= ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_3498_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa2 ) ) )
=> ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
=> ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_3499_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
=> ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
=> ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
=> ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_3500_mult__le__cancel__iff2,axiom,
! [Z2: real,X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y4 ) )
= ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3501_mult__le__cancel__iff2,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ ( times_times_rat @ Z2 @ Y4 ) )
= ( ord_less_eq_rat @ X @ Y4 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3502_mult__le__cancel__iff2,axiom,
! [Z2: int,X: int,Y4: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y4 ) )
= ( ord_less_eq_int @ X @ Y4 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_3503_mult__le__cancel__iff1,axiom,
! [Z2: real,X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y4 @ Z2 ) )
= ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3504_mult__le__cancel__iff1,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y4 @ Z2 ) )
= ( ord_less_eq_rat @ X @ Y4 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3505_mult__le__cancel__iff1,axiom,
! [Z2: int,X: int,Y4: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y4 @ Z2 ) )
= ( ord_less_eq_int @ X @ Y4 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_3506_divides__aux__eq,axiom,
! [Q2: nat,R2: nat] :
( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_3507_divides__aux__eq,axiom,
! [Q2: int,R2: int] :
( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( R2 = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_3508_neg__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q2: int,R2: int] :
( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_3509_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X2: nat,N2: nat] : ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% low_def
thf(fact_3510_obtain__set__pred,axiom,
! [Z2: nat,X: nat,A2: set_nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( vEBT_VEBT_min_in_set @ A2 @ Z2 )
=> ( ( finite_finite_nat @ A2 )
=> ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_1 ) ) ) ) ).
% obtain_set_pred
thf(fact_3511_set__vebt__finite,axiom,
! [T: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T @ N )
=> ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% set_vebt_finite
thf(fact_3512_pred__none__empty,axiom,
! [Xs2: set_nat,A: nat] :
( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_1 )
=> ( ( finite_finite_nat @ Xs2 )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ Xs2 )
& ( ord_less_nat @ X5 @ A ) ) ) ) ).
% pred_none_empty
thf(fact_3513_mod__mod__trivial,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_3514_mod__mod__trivial,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_3515_mod__mod__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mod_trivial
thf(fact_3516_mod__add__self2,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self2
thf(fact_3517_mod__add__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self2
thf(fact_3518_mod__add__self2,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_add_self2
thf(fact_3519_mod__add__self1,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self1
thf(fact_3520_mod__add__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_add_self1
thf(fact_3521_mod__add__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_add_self1
thf(fact_3522_minus__mod__self2,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_mod_self2
thf(fact_3523_minus__mod__self2,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% minus_mod_self2
thf(fact_3524_List_Ofinite__set,axiom,
! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_3525_List_Ofinite__set,axiom,
! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_3526_List_Ofinite__set,axiom,
! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_3527_List_Ofinite__set,axiom,
! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_3528_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_3529_mod__mult__self2__is__0,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_3530_mod__mult__self2__is__0,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_3531_mod__mult__self2__is__0,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_mult_self2_is_0
thf(fact_3532_mod__mult__self1__is__0,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_3533_mod__mult__self1__is__0,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_3534_mod__mult__self1__is__0,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_mult_self1_is_0
thf(fact_3535_mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_3536_mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_3537_mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% mod_by_1
thf(fact_3538_bits__mod__by__1,axiom,
! [A: nat] :
( ( modulo_modulo_nat @ A @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_3539_bits__mod__by__1,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_3540_bits__mod__by__1,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% bits_mod_by_1
thf(fact_3541_mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% mod_div_trivial
thf(fact_3542_mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% mod_div_trivial
thf(fact_3543_mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% mod_div_trivial
thf(fact_3544_bits__mod__div__trivial,axiom,
! [A: nat,B: nat] :
( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
= zero_zero_nat ) ).
% bits_mod_div_trivial
thf(fact_3545_bits__mod__div__trivial,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
= zero_zero_int ) ).
% bits_mod_div_trivial
thf(fact_3546_bits__mod__div__trivial,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
= zero_z3403309356797280102nteger ) ).
% bits_mod_div_trivial
thf(fact_3547_mod__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self4
thf(fact_3548_mod__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self4
thf(fact_3549_mod__mult__self4,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self4
thf(fact_3550_mod__mult__self3,axiom,
! [C: nat,B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self3
thf(fact_3551_mod__mult__self3,axiom,
! [C: int,B: int,A: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self3
thf(fact_3552_mod__mult__self3,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self3
thf(fact_3553_mod__mult__self2,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self2
thf(fact_3554_mod__mult__self2,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self2
thf(fact_3555_mod__mult__self2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self2
thf(fact_3556_mod__mult__self1,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_mult_self1
thf(fact_3557_mod__mult__self1,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ).
% mod_mult_self1
thf(fact_3558_mod__mult__self1,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% mod_mult_self1
thf(fact_3559_infinite__Icc__iff,axiom,
! [A: rat,B: rat] :
( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
= ( ord_less_rat @ A @ B ) ) ).
% infinite_Icc_iff
thf(fact_3560_infinite__Icc__iff,axiom,
! [A: real,B: real] :
( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
= ( ord_less_real @ A @ B ) ) ).
% infinite_Icc_iff
thf(fact_3561_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_3562_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_3563_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_3564_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_3565_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_3566_one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% one_mod_two_eq_one
thf(fact_3567_one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% one_mod_two_eq_one
thf(fact_3568_one__mod__two__eq__one,axiom,
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% one_mod_two_eq_one
thf(fact_3569_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ).
% bits_one_mod_two_eq_one
thf(fact_3570_bits__one__mod__two__eq__one,axiom,
( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% bits_one_mod_two_eq_one
thf(fact_3571_bits__one__mod__two__eq__one,axiom,
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% bits_one_mod_two_eq_one
thf(fact_3572_mod2__Suc__Suc,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_3573_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
!= one_one_nat )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
= one_one_nat ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_3574_not__mod__2__eq__1__eq__0,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_3575_not__mod__2__eq__1__eq__0,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_3576_not__mod__2__eq__1__eq__0,axiom,
! [A: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= one_one_Code_integer )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_3577_not__mod__2__eq__0__eq__1,axiom,
! [A: nat] :
( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_3578_not__mod__2__eq__0__eq__1,axiom,
! [A: int] :
( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_3579_not__mod__2__eq__0__eq__1,axiom,
! [A: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_3580_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= ( suc @ zero_zero_nat ) )
= ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_3581_add__self__mod__2,axiom,
! [M: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% add_self_mod_2
thf(fact_3582_mod2__gr__0,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% mod2_gr_0
thf(fact_3583_unique__quotient,axiom,
! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
=> ( Q2 = Q5 ) ) ) ).
% unique_quotient
thf(fact_3584_unique__remainder,axiom,
! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
=> ( R2 = R4 ) ) ) ).
% unique_remainder
thf(fact_3585_mod__mult__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_3586_mod__mult__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_3587_mod__mult__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% mod_mult_right_eq
thf(fact_3588_mod__mult__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_3589_mod__mult__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_3590_mod__mult__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% mod_mult_left_eq
thf(fact_3591_mult__mod__right,axiom,
! [C: nat,A: nat,B: nat] :
( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_3592_mult__mod__right,axiom,
! [C: int,A: int,B: int] :
( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
= ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_3593_mult__mod__right,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% mult_mod_right
thf(fact_3594_mod__mult__mult2,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_3595_mod__mult__mult2,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_3596_mod__mult__mult2,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% mod_mult_mult2
thf(fact_3597_mod__mult__cong,axiom,
! [A: nat,C: nat,A5: nat,B: nat,B4: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A5 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B4 @ C ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B4 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_3598_mod__mult__cong,axiom,
! [A: int,C: int,A5: int,B: int,B4: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B4 @ C ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A5 @ B4 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_3599_mod__mult__cong,axiom,
! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B4: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A5 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B4 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B4 ) @ C ) ) ) ) ).
% mod_mult_cong
thf(fact_3600_mod__mult__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_3601_mod__mult__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_3602_mod__mult__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% mod_mult_eq
thf(fact_3603_mod__add__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_3604_mod__add__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_3605_mod__add__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_3606_mod__add__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_3607_mod__add__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_3608_mod__add__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_3609_mod__add__cong,axiom,
! [A: nat,C: nat,A5: nat,B: nat,B4: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A5 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B4 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B4 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_3610_mod__add__cong,axiom,
! [A: int,C: int,A5: int,B: int,B4: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B4 @ C ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B4 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_3611_mod__add__cong,axiom,
! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B4: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A5 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B4 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B4 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_3612_mod__add__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_3613_mod__add__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_3614_mod__add__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_3615_mod__diff__right__eq,axiom,
! [A: int,B: int,C: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_3616_mod__diff__right__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% mod_diff_right_eq
thf(fact_3617_mod__diff__left__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_3618_mod__diff__left__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% mod_diff_left_eq
thf(fact_3619_mod__diff__cong,axiom,
! [A: int,C: int,A5: int,B: int,B4: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ A5 @ C ) )
=> ( ( ( modulo_modulo_int @ B @ C )
= ( modulo_modulo_int @ B4 @ C ) )
=> ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B4 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_3620_mod__diff__cong,axiom,
! [A: code_integer,C: code_integer,A5: code_integer,B: code_integer,B4: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ A5 @ C ) )
=> ( ( ( modulo364778990260209775nteger @ B @ C )
= ( modulo364778990260209775nteger @ B4 @ C ) )
=> ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A5 @ B4 ) @ C ) ) ) ) ).
% mod_diff_cong
thf(fact_3621_mod__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
= ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% mod_diff_eq
thf(fact_3622_mod__diff__eq,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
= ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% mod_diff_eq
thf(fact_3623_power__mod,axiom,
! [A: nat,B: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
= ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% power_mod
thf(fact_3624_power__mod,axiom,
! [A: int,B: int,N: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
= ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% power_mod
thf(fact_3625_power__mod,axiom,
! [A: code_integer,B: code_integer,N: nat] :
( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
= ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% power_mod
thf(fact_3626_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_3627_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
= ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_3628_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_3629_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N6 )
=> ( ord_less_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_3630_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_3631_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N6 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_3632_finite__list,axiom,
! [A2: set_VEBT_VEBT] :
( ( finite5795047828879050333T_VEBT @ A2 )
=> ? [Xs3: list_VEBT_VEBT] :
( ( set_VEBT_VEBT2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_3633_finite__list,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ? [Xs3: list_int] :
( ( set_int2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_3634_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_3635_finite__list,axiom,
! [A2: set_complex] :
( ( finite3207457112153483333omplex @ A2 )
=> ? [Xs3: list_complex] :
( ( set_complex2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_3636_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_3637_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_3638_finite__lists__length__eq,axiom,
! [A2: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
& ( ( size_s3451745648224563538omplex @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3639_finite__lists__length__eq,axiom,
! [A2: set_VEBT_VEBT,N: nat] :
( ( finite5795047828879050333T_VEBT @ A2 )
=> ( finite3004134309566078307T_VEBT
@ ( collec5608196760682091941T_VEBT
@ ^ [Xs: list_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
& ( ( size_s6755466524823107622T_VEBT @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3640_finite__lists__length__eq,axiom,
! [A2: set_o,N: nat] :
( ( finite_finite_o @ A2 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
& ( ( size_size_list_o @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3641_finite__lists__length__eq,axiom,
! [A2: set_int,N: nat] :
( ( finite_finite_int @ A2 )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
& ( ( size_size_list_int @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3642_finite__lists__length__eq,axiom,
! [A2: set_nat,N: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
& ( ( size_size_list_nat @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3643_eucl__rel__int__by0,axiom,
! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% eucl_rel_int_by0
thf(fact_3644_div__int__unique,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( divide_divide_int @ K @ L2 )
= Q2 ) ) ).
% div_int_unique
thf(fact_3645_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3646_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3647_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3648_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3649_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3650_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3651_mod__eq__self__iff__div__eq__0,axiom,
! [A: nat,B: nat] :
( ( ( modulo_modulo_nat @ A @ B )
= A )
= ( ( divide_divide_nat @ A @ B )
= zero_zero_nat ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_3652_mod__eq__self__iff__div__eq__0,axiom,
! [A: int,B: int] :
( ( ( modulo_modulo_int @ A @ B )
= A )
= ( ( divide_divide_int @ A @ B )
= zero_zero_int ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_3653_mod__eq__self__iff__div__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= A )
= ( ( divide6298287555418463151nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_3654_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_3655_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_3656_cong__exp__iff__simps_I9_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_3657_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_3658_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_3659_cong__exp__iff__simps_I4_J,axiom,
! [M: num,N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_3660_mod__eqE,axiom,
! [A: int,C: int,B: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B @ C ) )
=> ~ ! [D3: int] :
( B
!= ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% mod_eqE
thf(fact_3661_mod__eqE,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ B @ C ) )
=> ~ ! [D3: code_integer] :
( B
!= ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).
% mod_eqE
thf(fact_3662_div__add1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_3663_div__add1__eq,axiom,
! [A: int,B: int,C: int] :
( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_3664_div__add1__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% div_add1_eq
thf(fact_3665_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_3666_mod__induct,axiom,
! [P: nat > $o,N: nat,P6: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less_nat @ N @ P6 )
=> ( ( ord_less_nat @ M @ P6 )
=> ( ! [N3: nat] :
( ( ord_less_nat @ N3 @ P6 )
=> ( ( P @ N3 )
=> ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P6 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_3667_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_3668_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_3669_mod__eq__0D,axiom,
! [M: nat,D: nat] :
( ( ( modulo_modulo_nat @ M @ D )
= zero_zero_nat )
=> ? [Q3: nat] :
( M
= ( times_times_nat @ D @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_3670_mod__geq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( modulo_modulo_nat @ M @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_3671_mod__if,axiom,
( modulo_modulo_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N2 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_3672_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( modulo_modulo_nat @ M @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_3673_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y4: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y4 @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y4 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_3674_infinite__Icc,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% infinite_Icc
thf(fact_3675_infinite__Icc,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% infinite_Icc
thf(fact_3676_finite__lists__length__le,axiom,
! [A2: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
& ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3677_finite__lists__length__le,axiom,
! [A2: set_VEBT_VEBT,N: nat] :
( ( finite5795047828879050333T_VEBT @ A2 )
=> ( finite3004134309566078307T_VEBT
@ ( collec5608196760682091941T_VEBT
@ ^ [Xs: list_VEBT_VEBT] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
& ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3678_finite__lists__length__le,axiom,
! [A2: set_o,N: nat] :
( ( finite_finite_o @ A2 )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
& ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3679_finite__lists__length__le,axiom,
! [A2: set_int,N: nat] :
( ( finite_finite_int @ A2 )
=> ( finite3922522038869484883st_int
@ ( collect_list_int
@ ^ [Xs: list_int] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
& ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3680_finite__lists__length__le,axiom,
! [A2: set_nat,N: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3681_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ( modulo364778990260209775nteger @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3682_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ B )
=> ( ( modulo_modulo_nat @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3683_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ B )
=> ( ( modulo_modulo_int @ A @ B )
= A ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3684_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3685_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3686_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3687_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= zero_zero_nat )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(2)
thf(fact_3688_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= zero_zero_int )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(2)
thf(fact_3689_cong__exp__iff__simps_I2_J,axiom,
! [N: num,Q2: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
= zero_z3403309356797280102nteger )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(2)
thf(fact_3690_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
= zero_zero_nat ) ).
% cong_exp_iff_simps(1)
thf(fact_3691_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
= zero_zero_int ) ).
% cong_exp_iff_simps(1)
thf(fact_3692_cong__exp__iff__simps_I1_J,axiom,
! [N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(1)
thf(fact_3693_cong__exp__iff__simps_I6_J,axiom,
! [Q2: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_3694_cong__exp__iff__simps_I6_J,axiom,
! [Q2: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_3695_cong__exp__iff__simps_I6_J,axiom,
! [Q2: num,N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_3696_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q2: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_3697_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_3698_cong__exp__iff__simps_I8_J,axiom,
! [M: num,Q2: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_3699_mult__div__mod__eq,axiom,
! [B: nat,A: nat] :
( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_3700_mult__div__mod__eq,axiom,
! [B: int,A: int] :
( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_3701_mult__div__mod__eq,axiom,
! [B: code_integer,A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
= A ) ).
% mult_div_mod_eq
thf(fact_3702_mod__mult__div__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_3703_mod__mult__div__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_3704_mod__mult__div__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
= A ) ).
% mod_mult_div_eq
thf(fact_3705_mod__div__mult__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_3706_mod__div__mult__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_3707_mod__div__mult__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
= A ) ).
% mod_div_mult_eq
thf(fact_3708_div__mult__mod__eq,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_3709_div__mult__mod__eq,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_3710_div__mult__mod__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
= A ) ).
% div_mult_mod_eq
thf(fact_3711_mod__div__decomp,axiom,
! [A: nat,B: nat] :
( A
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_3712_mod__div__decomp,axiom,
! [A: int,B: int] :
( A
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_3713_mod__div__decomp,axiom,
! [A: code_integer,B: code_integer] :
( A
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% mod_div_decomp
thf(fact_3714_cancel__div__mod__rules_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_3715_cancel__div__mod__rules_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
= ( plus_plus_int @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_3716_cancel__div__mod__rules_I1_J,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
= ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% cancel_div_mod_rules(1)
thf(fact_3717_cancel__div__mod__rules_I2_J,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
= ( plus_plus_nat @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_3718_cancel__div__mod__rules_I2_J,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
= ( plus_plus_int @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_3719_cancel__div__mod__rules_I2_J,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
= ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% cancel_div_mod_rules(2)
thf(fact_3720_div__mult1__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_3721_div__mult1__eq,axiom,
! [A: int,B: int,C: int] :
( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_3722_div__mult1__eq,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% div_mult1_eq
thf(fact_3723_minus__mult__div__eq__mod,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
= ( modulo_modulo_nat @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_3724_minus__mult__div__eq__mod,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_3725_minus__mult__div__eq__mod,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% minus_mult_div_eq_mod
thf(fact_3726_minus__mod__eq__mult__div,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
= ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_3727_minus__mod__eq__mult__div,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
= ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_3728_minus__mod__eq__mult__div,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_3729_minus__mod__eq__div__mult,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_3730_minus__mod__eq__div__mult,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
= ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_3731_minus__mod__eq__div__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% minus_mod_eq_div_mult
thf(fact_3732_minus__div__mult__eq__mod,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
= ( modulo_modulo_nat @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_3733_minus__div__mult__eq__mod,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
= ( modulo_modulo_int @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_3734_minus__div__mult__eq__mod,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
= ( modulo364778990260209775nteger @ A @ B ) ) ).
% minus_div_mult_eq_mod
thf(fact_3735_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_3736_div__less__mono,axiom,
! [A2: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A2 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_nat @ A2 @ N )
= zero_zero_nat )
=> ( ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
=> ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_3737_mod__eq__nat1E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q2 )
= ( modulo_modulo_nat @ N @ Q2 ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ~ ! [S: nat] :
( M
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_3738_mod__eq__nat2E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo_nat @ M @ Q2 )
= ( modulo_modulo_nat @ N @ Q2 ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ~ ! [S: nat] :
( N
!= ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_3739_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y4: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y4 @ N ) )
=> ( ( ord_less_eq_nat @ Y4 @ X )
=> ? [Q3: nat] :
( X
= ( plus_plus_nat @ Y4 @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_3740_eucl__rel__int__dividesI,axiom,
! [L2: int,K: int,Q2: int] :
( ( L2 != zero_zero_int )
=> ( ( K
= ( times_times_int @ Q2 @ L2 ) )
=> ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_3741_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_3742_div__mod__decomp,axiom,
! [A2: nat,N: nat] :
( A2
= ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% div_mod_decomp
thf(fact_3743_modulo__nat__def,axiom,
( modulo_modulo_nat
= ( ^ [M2: nat,N2: nat] : ( minus_minus_nat @ M2 @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% modulo_nat_def
thf(fact_3744_split__mod,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( modulo_modulo_nat @ M @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ M ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( M
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_3745_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
=> ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3746_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3747_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_3748_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
= one_one_nat ) ) ).
% Suc_times_mod_eq
thf(fact_3749_divmod__digit__0_I2_J,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
= ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_3750_divmod__digit__0_I2_J,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
= ( modulo_modulo_int @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_3751_divmod__digit__0_I2_J,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
= ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_3752_bits__stable__imp__add__self,axiom,
! [A: nat] :
( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_nat ) ) ).
% bits_stable_imp_add_self
thf(fact_3753_bits__stable__imp__add__self,axiom,
! [A: int] :
( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= zero_zero_int ) ) ).
% bits_stable_imp_add_self
thf(fact_3754_bits__stable__imp__add__self,axiom,
! [A: code_integer] :
( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A )
=> ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
= zero_z3403309356797280102nteger ) ) ).
% bits_stable_imp_add_self
thf(fact_3755_div__exp__mod__exp__eq,axiom,
! [A: nat,N: nat,M: nat] :
( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_3756_div__exp__mod__exp__eq,axiom,
! [A: int,N: nat,M: nat] :
( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_3757_div__exp__mod__exp__eq,axiom,
! [A: code_integer,N: nat,M: nat] :
( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
= ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_3758_verit__le__mono__div,axiom,
! [A2: nat,B5: nat,N: nat] :
( ( ord_less_nat @ A2 @ B5 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat
@ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
@ ( if_nat
@ ( ( modulo_modulo_nat @ B5 @ N )
= zero_zero_nat )
@ one_one_nat
@ zero_zero_nat ) )
@ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_3759_divmod__digit__0_I1_J,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_3760_divmod__digit__0_I1_J,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_3761_divmod__digit__0_I1_J,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_3762_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_3763_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_3764_mult__exp__mod__exp__eq,axiom,
! [M: nat,N: nat,A: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_3765_eucl__rel__int__iff,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( ( K
= ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R2 ) )
& ( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
& ( ord_less_int @ R2 @ L2 ) ) )
& ( ~ ( ord_less_int @ zero_zero_int @ L2 )
=> ( ( ( ord_less_int @ L2 @ zero_zero_int )
=> ( ( ord_less_int @ L2 @ R2 )
& ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
& ( ~ ( ord_less_int @ L2 @ zero_zero_int )
=> ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_3766_mod__double__modulus,axiom,
! [M: code_integer,X: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
=> ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
= ( modulo364778990260209775nteger @ X @ M ) )
| ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_3767_mod__double__modulus,axiom,
! [M: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( modulo_modulo_nat @ X @ M ) )
| ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
= ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_3768_mod__double__modulus,axiom,
! [M: int,X: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( modulo_modulo_int @ X @ M ) )
| ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
= ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% mod_double_modulus
thf(fact_3769_divmod__digit__1_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_3770_divmod__digit__1_I2_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_3771_divmod__digit__1_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
= ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_3772_unset__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3773_unset__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3774_unset__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_3775_set__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3776_set__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3777_set__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_3778_divmod__digit__1_I1_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
=> ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_3779_divmod__digit__1_I1_J,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
= ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_3780_divmod__digit__1_I1_J,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
= ( divide_divide_int @ A @ B ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_3781_mult__less__iff1,axiom,
! [Z2: real,X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y4 @ Z2 ) )
= ( ord_less_real @ X @ Y4 ) ) ) ).
% mult_less_iff1
thf(fact_3782_mult__less__iff1,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( ord_less_rat @ zero_zero_rat @ Z2 )
=> ( ( ord_less_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y4 @ Z2 ) )
= ( ord_less_rat @ X @ Y4 ) ) ) ).
% mult_less_iff1
thf(fact_3783_mult__less__iff1,axiom,
! [Z2: int,X: int,Y4: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y4 @ Z2 ) )
= ( ord_less_int @ X @ Y4 ) ) ) ).
% mult_less_iff1
thf(fact_3784_pos__eucl__rel__int__mult__2,axiom,
! [B: int,A: int,Q2: int,R2: int] :
( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_3785_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_3786_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_3787_finite__Collect__bounded__ex,axiom,
! [P: real > $o,Q: real > real > $o] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] :
? [Y: real] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: real] :
( ( P @ Y )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_3788_finite__Collect__bounded__ex,axiom,
! [P: real > $o,Q: nat > real > $o] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y: real] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: real] :
( ( P @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_3789_finite__Collect__bounded__ex,axiom,
! [P: real > $o,Q: complex > real > $o] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
? [Y: real] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: real] :
( ( P @ Y )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_3790_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: real > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_3791_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: nat > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_3792_finite__Collect__bounded__ex,axiom,
! [P: nat > $o,Q: complex > nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
? [Y: nat] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: nat] :
( ( P @ Y )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_3793_finite__Collect__bounded__ex,axiom,
! [P: complex > $o,Q: real > complex > $o] :
( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] :
? [Y: complex] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: complex] :
( ( P @ Y )
=> ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_3794_finite__Collect__bounded__ex,axiom,
! [P: complex > $o,Q: nat > complex > $o] :
( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
? [Y: complex] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: complex] :
( ( P @ Y )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_3795_finite__Collect__bounded__ex,axiom,
! [P: complex > $o,Q: complex > complex > $o] :
( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
? [Y: complex] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: complex] :
( ( P @ Y )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_3796_finite__Collect__bounded__ex,axiom,
! [P: real > $o,Q: list_nat > real > $o] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
? [Y: real] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: real] :
( ( P @ Y )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_3797_finite__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Z4: real] :
( ( power_power_real @ Z4 @ N )
= one_one_real ) ) ) ) ).
% finite_roots_unity
thf(fact_3798_finite__roots__unity,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= one_one_complex ) ) ) ) ).
% finite_roots_unity
thf(fact_3799_flip__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3800_flip__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3801_flip__bit__Suc,axiom,
! [N: nat,A: nat] :
( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
= ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_3802_product__nth,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N )
= ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3803_product__nth,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N )
= ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3804_product__nth,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N )
= ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3805_product__nth,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
=> ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) @ N )
= ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3806_product__nth,axiom,
! [N: nat,Xs2: list_o,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) @ N )
= ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3807_product__nth,axiom,
! [N: nat,Xs2: list_o,Ys: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys ) @ N )
= ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3808_product__nth,axiom,
! [N: nat,Xs2: list_o,Ys: list_nat] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
=> ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys ) @ N )
= ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3809_product__nth,axiom,
! [N: nat,Xs2: list_o,Ys: list_int] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
=> ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys ) @ N )
= ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3810_product__nth,axiom,
! [N: nat,Xs2: list_nat,Ys: list_VEBT_VEBT] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
=> ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) @ N )
= ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3811_product__nth,axiom,
! [N: nat,Xs2: list_nat,Ys: list_o] :
( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
=> ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs2 @ Ys ) @ N )
= ( product_Pair_nat_o @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_3812_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > complex,Y4: int > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= one_one_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( times_times_complex @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3813_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > complex,Y4: real > complex] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= one_one_complex ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( times_times_complex @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3814_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > complex,Y4: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= one_one_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( times_times_complex @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3815_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > complex,Y4: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= one_one_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( times_times_complex @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3816_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > real,Y4: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= one_one_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( times_times_real @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3817_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > real,Y4: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= one_one_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( times_times_real @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3818_prod_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > real,Y4: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= one_one_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( times_times_real @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3819_prod_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > real,Y4: complex > real] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_real ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= one_one_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( times_times_real @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= one_one_real ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3820_prod_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > rat,Y4: int > rat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_rat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= one_one_rat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( times_times_rat @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3821_prod_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > rat,Y4: real > rat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( X @ I4 )
!= one_one_rat ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= one_one_rat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( times_times_rat @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= one_one_rat ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3822_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > complex,Y4: int > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3823_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > complex,Y4: real > complex] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3824_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > complex,Y4: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= zero_zero_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3825_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > complex,Y4: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= zero_zero_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= zero_zero_complex ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3826_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > real,Y4: int > real] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( plus_plus_real @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3827_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > real,Y4: real > real] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( plus_plus_real @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3828_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > real,Y4: nat > real] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_real ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= zero_zero_real ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( plus_plus_real @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3829_sum_Ofinite__Collect__op,axiom,
! [I5: set_complex,X: complex > real,Y4: complex > real] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_real ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= zero_zero_real ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I4: complex] :
( ( member_complex @ I4 @ I5 )
& ( ( plus_plus_real @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= zero_zero_real ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3830_sum_Ofinite__Collect__op,axiom,
! [I5: set_int,X: int > rat,Y4: int > rat] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I4: int] :
( ( member_int @ I4 @ I5 )
& ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3831_sum_Ofinite__Collect__op,axiom,
! [I5: set_real,X: real > rat,Y4: real > rat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( X @ I4 )
!= zero_zero_rat ) ) ) )
=> ( ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( Y4 @ I4 )
!= zero_zero_rat ) ) ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [I4: real] :
( ( member_real @ I4 @ I5 )
& ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y4 @ I4 ) )
!= zero_zero_rat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3832_mod__pos__pos__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L2 )
=> ( ( modulo_modulo_int @ K @ L2 )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_3833_mod__neg__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L2 @ K )
=> ( ( modulo_modulo_int @ K @ L2 )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_3834_length__product,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_3835_length__product,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_o] :
( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_3836_length__product,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_3837_length__product,axiom,
! [Xs2: list_VEBT_VEBT,Ys: list_int] :
( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_product
thf(fact_3838_length__product,axiom,
! [Xs2: list_o,Ys: list_VEBT_VEBT] :
( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_3839_length__product,axiom,
! [Xs2: list_o,Ys: list_o] :
( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_3840_length__product,axiom,
! [Xs2: list_o,Ys: list_nat] :
( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_3841_length__product,axiom,
! [Xs2: list_o,Ys: list_int] :
( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_product
thf(fact_3842_length__product,axiom,
! [Xs2: list_nat,Ys: list_VEBT_VEBT] :
( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% length_product
thf(fact_3843_length__product,axiom,
! [Xs2: list_nat,Ys: list_o] :
( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_product
thf(fact_3844_zmod__numeral__Bit0,axiom,
! [V: num,W2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_3845_neg__mod__bound,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ L2 @ zero_zero_int )
=> ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% neg_mod_bound
thf(fact_3846_Euclidean__Division_Opos__mod__bound,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_3847_zmod__eq__0D,axiom,
! [M: int,D: int] :
( ( ( modulo_modulo_int @ M @ D )
= zero_zero_int )
=> ? [Q3: int] :
( M
= ( times_times_int @ D @ Q3 ) ) ) ).
% zmod_eq_0D
thf(fact_3848_zmod__eq__0__iff,axiom,
! [M: int,D: int] :
( ( ( modulo_modulo_int @ M @ D )
= zero_zero_int )
= ( ? [Q4: int] :
( M
= ( times_times_int @ D @ Q4 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_3849_finite__maxlen,axiom,
! [M7: set_list_VEBT_VEBT] :
( ( finite3004134309566078307T_VEBT @ M7 )
=> ? [N3: nat] :
! [X5: list_VEBT_VEBT] :
( ( member2936631157270082147T_VEBT @ X5 @ M7 )
=> ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_3850_finite__maxlen,axiom,
! [M7: set_list_o] :
( ( finite_finite_list_o @ M7 )
=> ? [N3: nat] :
! [X5: list_o] :
( ( member_list_o @ X5 @ M7 )
=> ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_3851_finite__maxlen,axiom,
! [M7: set_list_nat] :
( ( finite8100373058378681591st_nat @ M7 )
=> ? [N3: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ M7 )
=> ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_3852_finite__maxlen,axiom,
! [M7: set_list_int] :
( ( finite3922522038869484883st_int @ M7 )
=> ? [N3: nat] :
! [X5: list_int] :
( ( member_list_int @ X5 @ M7 )
=> ( ord_less_nat @ ( size_size_list_int @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_3853_mod__int__unique,axiom,
! [K: int,L2: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( modulo_modulo_int @ K @ L2 )
= R2 ) ) ).
% mod_int_unique
thf(fact_3854_Euclidean__Division_Opos__mod__sign,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_3855_neg__mod__sign,axiom,
! [L2: int,K: int] :
( ( ord_less_int @ L2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_3856_zdiv__mono__strict,axiom,
! [A2: int,B5: int,N: int] :
( ( ord_less_int @ A2 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A2 @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B5 @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_3857_div__mod__decomp__int,axiom,
! [A2: int,N: int] :
( A2
= ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_3858_eucl__rel__int,axiom,
! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% eucl_rel_int
thf(fact_3859_mod__pos__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L2 )
= ( plus_plus_int @ K @ L2 ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_3860_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I4: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_3861_int__mod__neg__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ R2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ R2 )
=> ( ( modulo_modulo_int @ A @ B )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_3862_int__mod__pos__eq,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R2 )
=> ( ( ord_less_int @ R2 @ B )
=> ( ( modulo_modulo_int @ A @ B )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_3863_zmod__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_3864_verit__le__mono__div__int,axiom,
! [A2: int,B5: int,N: int] :
( ( ord_less_int @ A2 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B5 @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_3865_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_3866_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_3867_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ A @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3868_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ord_less_eq_set_nat @ A @ X3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3869_finite__has__maximal2,axiom,
! [A2: set_rat,A: rat] :
( ( finite_finite_rat @ A2 )
=> ( ( member_rat @ A @ A2 )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ( ord_less_eq_rat @ A @ X3 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A2 )
=> ( ( ord_less_eq_rat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3870_finite__has__maximal2,axiom,
! [A2: set_num,A: num] :
( ( finite_finite_num @ A2 )
=> ( ( member_num @ A @ A2 )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ( ord_less_eq_num @ A @ X3 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3871_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3872_finite__has__maximal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ A @ X3 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3873_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3874_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ord_less_eq_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3875_finite__has__minimal2,axiom,
! [A2: set_rat,A: rat] :
( ( finite_finite_rat @ A2 )
=> ( ( member_rat @ A @ A2 )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ( ord_less_eq_rat @ X3 @ A )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A2 )
=> ( ( ord_less_eq_rat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3876_finite__has__minimal2,axiom,
! [A2: set_num,A: num] :
( ( finite_finite_num @ A2 )
=> ( ( member_num @ A @ A2 )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ( ord_less_eq_num @ X3 @ A )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3877_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3878_finite__has__minimal2,axiom,
! [A2: set_int,A: int] :
( ( finite_finite_int @ A2 )
=> ( ( member_int @ A @ A2 )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( ord_less_eq_int @ X3 @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3879_finite_OemptyI,axiom,
finite3207457112153483333omplex @ bot_bot_set_complex ).
% finite.emptyI
thf(fact_3880_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_3881_finite_OemptyI,axiom,
finite_finite_int @ bot_bot_set_int ).
% finite.emptyI
thf(fact_3882_finite_OemptyI,axiom,
finite_finite_real @ bot_bot_set_real ).
% finite.emptyI
thf(fact_3883_infinite__imp__nonempty,axiom,
! [S3: set_complex] :
( ~ ( finite3207457112153483333omplex @ S3 )
=> ( S3 != bot_bot_set_complex ) ) ).
% infinite_imp_nonempty
thf(fact_3884_infinite__imp__nonempty,axiom,
! [S3: set_nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( S3 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_3885_infinite__imp__nonempty,axiom,
! [S3: set_int] :
( ~ ( finite_finite_int @ S3 )
=> ( S3 != bot_bot_set_int ) ) ).
% infinite_imp_nonempty
thf(fact_3886_infinite__imp__nonempty,axiom,
! [S3: set_real] :
( ~ ( finite_finite_real @ S3 )
=> ( S3 != bot_bot_set_real ) ) ).
% infinite_imp_nonempty
thf(fact_3887_pos__zmod__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_3888_finite__image__set,axiom,
! [P: real > $o,F: real > real] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: real] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3889_finite__image__set,axiom,
! [P: real > $o,F: real > nat] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: real] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3890_finite__image__set,axiom,
! [P: real > $o,F: real > complex] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Uu2: complex] :
? [X2: real] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3891_finite__image__set,axiom,
! [P: nat > $o,F: nat > real] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: nat] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3892_finite__image__set,axiom,
! [P: nat > $o,F: nat > nat] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: nat] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3893_finite__image__set,axiom,
! [P: nat > $o,F: nat > complex] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Uu2: complex] :
? [X2: nat] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3894_finite__image__set,axiom,
! [P: complex > $o,F: complex > real] :
( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: complex] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3895_finite__image__set,axiom,
! [P: complex > $o,F: complex > nat] :
( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: complex] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3896_finite__image__set,axiom,
! [P: complex > $o,F: complex > complex] :
( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Uu2: complex] :
? [X2: complex] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3897_finite__image__set,axiom,
! [P: real > $o,F: real > list_nat] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Uu2: list_nat] :
? [X2: real] :
( ( Uu2
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_3898_finite__image__set2,axiom,
! [P: real > $o,Q: real > $o,F: real > real > real] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_real @ ( collect_real @ Q ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: real,Y: real] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3899_finite__image__set2,axiom,
! [P: real > $o,Q: real > $o,F: real > real > nat] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_real @ ( collect_real @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: real,Y: real] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3900_finite__image__set2,axiom,
! [P: real > $o,Q: real > $o,F: real > real > complex] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_real @ ( collect_real @ Q ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Uu2: complex] :
? [X2: real,Y: real] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3901_finite__image__set2,axiom,
! [P: real > $o,Q: nat > $o,F: real > nat > real] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: real,Y: nat] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3902_finite__image__set2,axiom,
! [P: real > $o,Q: nat > $o,F: real > nat > nat] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: real,Y: nat] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3903_finite__image__set2,axiom,
! [P: real > $o,Q: nat > $o,F: real > nat > complex] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite_finite_nat @ ( collect_nat @ Q ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Uu2: complex] :
? [X2: real,Y: nat] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3904_finite__image__set2,axiom,
! [P: real > $o,Q: complex > $o,F: real > complex > real] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite3207457112153483333omplex @ ( collect_complex @ Q ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: real,Y: complex] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3905_finite__image__set2,axiom,
! [P: real > $o,Q: complex > $o,F: real > complex > nat] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite3207457112153483333omplex @ ( collect_complex @ Q ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [Uu2: nat] :
? [X2: real,Y: complex] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3906_finite__image__set2,axiom,
! [P: real > $o,Q: complex > $o,F: real > complex > complex] :
( ( finite_finite_real @ ( collect_real @ P ) )
=> ( ( finite3207457112153483333omplex @ ( collect_complex @ Q ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Uu2: complex] :
? [X2: real,Y: complex] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3907_finite__image__set2,axiom,
! [P: nat > $o,Q: real > $o,F: nat > real > real] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_real @ ( collect_real @ Q ) )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Uu2: real] :
? [X2: nat,Y: real] :
( ( Uu2
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_3908_neg__zmod__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% neg_zmod_mult_2
thf(fact_3909_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3910_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3911_finite__has__minimal,axiom,
! [A2: set_rat] :
( ( finite_finite_rat @ A2 )
=> ( ( A2 != bot_bot_set_rat )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A2 )
=> ( ( ord_less_eq_rat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3912_finite__has__minimal,axiom,
! [A2: set_num] :
( ( finite_finite_num @ A2 )
=> ( ( A2 != bot_bot_set_num )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3913_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3914_finite__has__minimal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3915_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3916_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3917_finite__has__maximal,axiom,
! [A2: set_rat] :
( ( finite_finite_rat @ A2 )
=> ( ( A2 != bot_bot_set_rat )
=> ? [X3: rat] :
( ( member_rat @ X3 @ A2 )
& ! [Xa: rat] :
( ( member_rat @ Xa @ A2 )
=> ( ( ord_less_eq_rat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3918_finite__has__maximal,axiom,
! [A2: set_num] :
( ( finite_finite_num @ A2 )
=> ( ( A2 != bot_bot_set_num )
=> ? [X3: num] :
( ( member_num @ X3 @ A2 )
& ! [Xa: num] :
( ( member_num @ Xa @ A2 )
=> ( ( ord_less_eq_num @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3919_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3920_finite__has__maximal,axiom,
! [A2: set_int] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A2 )
=> ( ( ord_less_eq_int @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3921_arcosh__1,axiom,
( ( arcosh_real @ one_one_real )
= zero_zero_real ) ).
% arcosh_1
thf(fact_3922_finite__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_3923_prod_Oinject,axiom,
! [X1: int,X22: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X22 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_3924_prod_Oinject,axiom,
! [X1: code_integer > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
( ( ( produc6137756002093451184nteger @ X1 @ X22 )
= ( produc6137756002093451184nteger @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_3925_prod_Oinject,axiom,
! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ X1 @ X22 )
= ( produc8603105652947943368nteger @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_3926_prod_Oinject,axiom,
! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
= ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_3927_prod_Oinject,axiom,
! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
= ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_3928_old_Oprod_Oinject,axiom,
! [A: int,B: int,A5: int,B4: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_3929_old_Oprod_Oinject,axiom,
! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
( ( ( produc6137756002093451184nteger @ A @ B )
= ( produc6137756002093451184nteger @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_3930_old_Oprod_Oinject,axiom,
! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ A @ B )
= ( produc8603105652947943368nteger @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_3931_old_Oprod_Oinject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A5: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_3932_old_Oprod_Oinject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A5: int > option6357759511663192854e_term,B4: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_3933_old_Oprod_Oexhaust,axiom,
! [Y4: product_prod_int_int] :
~ ! [A3: int,B2: int] :
( Y4
!= ( product_Pair_int_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_3934_old_Oprod_Oexhaust,axiom,
! [Y4: produc8763457246119570046nteger] :
~ ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( Y4
!= ( produc6137756002093451184nteger @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_3935_old_Oprod_Oexhaust,axiom,
! [Y4: produc1908205239877642774nteger] :
~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( Y4
!= ( produc8603105652947943368nteger @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_3936_old_Oprod_Oexhaust,axiom,
! [Y4: produc2285326912895808259nt_int] :
~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( Y4
!= ( produc5700946648718959541nt_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_3937_old_Oprod_Oexhaust,axiom,
! [Y4: produc7773217078559923341nt_int] :
~ ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( Y4
!= ( produc4305682042979456191nt_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_3938_surj__pair,axiom,
! [P6: product_prod_int_int] :
? [X3: int,Y3: int] :
( P6
= ( product_Pair_int_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3939_surj__pair,axiom,
! [P6: produc8763457246119570046nteger] :
? [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( P6
= ( produc6137756002093451184nteger @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3940_surj__pair,axiom,
! [P6: produc1908205239877642774nteger] :
? [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( P6
= ( produc8603105652947943368nteger @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3941_surj__pair,axiom,
! [P6: produc2285326912895808259nt_int] :
? [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P6
= ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3942_surj__pair,axiom,
! [P6: produc7773217078559923341nt_int] :
? [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P6
= ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_3943_prod__cases,axiom,
! [P: product_prod_int_int > $o,P6: product_prod_int_int] :
( ! [A3: int,B2: int] : ( P @ ( product_Pair_int_int @ A3 @ B2 ) )
=> ( P @ P6 ) ) ).
% prod_cases
thf(fact_3944_prod__cases,axiom,
! [P: produc8763457246119570046nteger > $o,P6: produc8763457246119570046nteger] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( P @ P6 ) ) ).
% prod_cases
thf(fact_3945_prod__cases,axiom,
! [P: produc1908205239877642774nteger > $o,P6: produc1908205239877642774nteger] :
( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A3 @ B2 ) )
=> ( P @ P6 ) ) ).
% prod_cases
thf(fact_3946_prod__cases,axiom,
! [P: produc2285326912895808259nt_int > $o,P6: produc2285326912895808259nt_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B2 ) )
=> ( P @ P6 ) ) ).
% prod_cases
thf(fact_3947_prod__cases,axiom,
! [P: produc7773217078559923341nt_int > $o,P6: produc7773217078559923341nt_int] :
( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B2 ) )
=> ( P @ P6 ) ) ).
% prod_cases
thf(fact_3948_Pair__inject,axiom,
! [A: int,B: int,A5: int,B4: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_3949_Pair__inject,axiom,
! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
( ( ( produc6137756002093451184nteger @ A @ B )
= ( produc6137756002093451184nteger @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_3950_Pair__inject,axiom,
! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A5: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
( ( ( produc8603105652947943368nteger @ A @ B )
= ( produc8603105652947943368nteger @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_3951_Pair__inject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A5: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_3952_Pair__inject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A5: int > option6357759511663192854e_term,B4: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A5 @ B4 ) )
=> ~ ( ( A = A5 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_3953_prod__cases3,axiom,
! [Y4: produc8763457246119570046nteger] :
~ ! [A3: code_integer > option6357759511663192854e_term,B2: code_integer,C3: code_integer] :
( Y4
!= ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_3954_prod__cases3,axiom,
! [Y4: produc1908205239877642774nteger] :
~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: code_integer,C3: code_integer] :
( Y4
!= ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_3955_prod__cases3,axiom,
! [Y4: produc2285326912895808259nt_int] :
~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C3: int] :
( Y4
!= ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_3956_prod__cases3,axiom,
! [Y4: produc7773217078559923341nt_int] :
~ ! [A3: int > option6357759511663192854e_term,B2: int,C3: int] :
( Y4
!= ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) ) ).
% prod_cases3
thf(fact_3957_prod__induct3,axiom,
! [P: produc8763457246119570046nteger > $o,X: produc8763457246119570046nteger] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: code_integer,C3: code_integer] : ( P @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_3958_prod__induct3,axiom,
! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: code_integer,C3: code_integer] : ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B2 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_3959_prod__induct3,axiom,
! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C3: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_3960_prod__induct3,axiom,
! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
( ! [A3: int > option6357759511663192854e_term,B2: int,C3: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_3961_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( ( ord_less_nat @ Mi @ X )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_3962_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
=> ( ( ord_less_nat @ X @ Mi )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_3963_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
=> ( P @ M4 @ N3 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_3964_concat__bit__Suc,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L2 )
= ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).
% concat_bit_Suc
thf(fact_3965_dbl__simps_I3_J,axiom,
( ( neg_nu7009210354673126013omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_3966_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_3967_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_3968_dbl__simps_I3_J,axiom,
( ( neg_numeral_dbl_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% dbl_simps(3)
thf(fact_3969_max__bot,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% max_bot
thf(fact_3970_max__bot,axiom,
! [X: set_int] :
( ( ord_max_set_int @ bot_bot_set_int @ X )
= X ) ).
% max_bot
thf(fact_3971_max__bot,axiom,
! [X: set_real] :
( ( ord_max_set_real @ bot_bot_set_real @ X )
= X ) ).
% max_bot
thf(fact_3972_max__bot,axiom,
! [X: nat] :
( ( ord_max_nat @ bot_bot_nat @ X )
= X ) ).
% max_bot
thf(fact_3973_max__bot,axiom,
! [X: extended_enat] :
( ( ord_ma741700101516333627d_enat @ bot_bo4199563552545308370d_enat @ X )
= X ) ).
% max_bot
thf(fact_3974_max__bot2,axiom,
! [X: set_nat] :
( ( ord_max_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% max_bot2
thf(fact_3975_max__bot2,axiom,
! [X: set_int] :
( ( ord_max_set_int @ X @ bot_bot_set_int )
= X ) ).
% max_bot2
thf(fact_3976_max__bot2,axiom,
! [X: set_real] :
( ( ord_max_set_real @ X @ bot_bot_set_real )
= X ) ).
% max_bot2
thf(fact_3977_max__bot2,axiom,
! [X: nat] :
( ( ord_max_nat @ X @ bot_bot_nat )
= X ) ).
% max_bot2
thf(fact_3978_max__bot2,axiom,
! [X: extended_enat] :
( ( ord_ma741700101516333627d_enat @ X @ bot_bo4199563552545308370d_enat )
= X ) ).
% max_bot2
thf(fact_3979_length__list__update,axiom,
! [Xs2: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) )
= ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).
% length_list_update
thf(fact_3980_length__list__update,axiom,
! [Xs2: list_o,I2: nat,X: $o] :
( ( size_size_list_o @ ( list_update_o @ Xs2 @ I2 @ X ) )
= ( size_size_list_o @ Xs2 ) ) ).
% length_list_update
thf(fact_3981_length__list__update,axiom,
! [Xs2: list_nat,I2: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_list_update
thf(fact_3982_length__list__update,axiom,
! [Xs2: list_int,I2: nat,X: int] :
( ( size_size_list_int @ ( list_update_int @ Xs2 @ I2 @ X ) )
= ( size_size_list_int @ Xs2 ) ) ).
% length_list_update
thf(fact_3983_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_3984_nth__list__update__neq,axiom,
! [I2: nat,J: nat,Xs2: list_nat,X: nat] :
( ( I2 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_3985_nth__list__update__neq,axiom,
! [I2: nat,J: nat,Xs2: list_int,X: int] :
( ( I2 != J )
=> ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ J )
= ( nth_int @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_3986_nth__list__update__neq,axiom,
! [I2: nat,J: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( I2 != J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ J )
= ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_3987_list__update__id,axiom,
! [Xs2: list_nat,I2: nat] :
( ( list_update_nat @ Xs2 @ I2 @ ( nth_nat @ Xs2 @ I2 ) )
= Xs2 ) ).
% list_update_id
thf(fact_3988_list__update__id,axiom,
! [Xs2: list_int,I2: nat] :
( ( list_update_int @ Xs2 @ I2 @ ( nth_int @ Xs2 @ I2 ) )
= Xs2 ) ).
% list_update_id
thf(fact_3989_list__update__id,axiom,
! [Xs2: list_VEBT_VEBT,I2: nat] :
( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) )
= Xs2 ) ).
% list_update_id
thf(fact_3990_dbl__simps_I2_J,axiom,
( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
= zero_zero_complex ) ).
% dbl_simps(2)
thf(fact_3991_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_real @ zero_zero_real )
= zero_zero_real ) ).
% dbl_simps(2)
thf(fact_3992_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_rat @ zero_zero_rat )
= zero_zero_rat ) ).
% dbl_simps(2)
thf(fact_3993_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_3994_max__number__of_I1_J,axiom,
! [U2: num,V: num] :
( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U2 ) @ ( numera1916890842035813515d_enat @ V ) )
=> ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U2 ) @ ( numera1916890842035813515d_enat @ V ) )
= ( numera1916890842035813515d_enat @ V ) ) )
& ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U2 ) @ ( numera1916890842035813515d_enat @ V ) )
=> ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U2 ) @ ( numera1916890842035813515d_enat @ V ) )
= ( numera1916890842035813515d_enat @ U2 ) ) ) ) ).
% max_number_of(1)
thf(fact_3995_max__number__of_I1_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U2 ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U2 ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ V ) ) )
& ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U2 ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U2 ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ U2 ) ) ) ) ).
% max_number_of(1)
thf(fact_3996_max__number__of_I1_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U2 ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U2 ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ V ) ) )
& ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U2 ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U2 ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ U2 ) ) ) ) ).
% max_number_of(1)
thf(fact_3997_max__number__of_I1_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U2 ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U2 ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U2 ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U2 ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U2 ) ) ) ) ).
% max_number_of(1)
thf(fact_3998_max__number__of_I1_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U2 ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U2 ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U2 ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U2 ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ U2 ) ) ) ) ).
% max_number_of(1)
thf(fact_3999_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(3)
thf(fact_4000_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(3)
thf(fact_4001_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(3)
thf(fact_4002_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(3)
thf(fact_4003_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(3)
thf(fact_4004_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(4)
thf(fact_4005_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(4)
thf(fact_4006_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(4)
thf(fact_4007_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(4)
thf(fact_4008_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(4)
thf(fact_4009_max__0__1_I2_J,axiom,
( ( ord_max_real @ one_one_real @ zero_zero_real )
= one_one_real ) ).
% max_0_1(2)
thf(fact_4010_max__0__1_I2_J,axiom,
( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
= one_one_rat ) ).
% max_0_1(2)
thf(fact_4011_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_4012_max__0__1_I2_J,axiom,
( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
= one_on7984719198319812577d_enat ) ).
% max_0_1(2)
thf(fact_4013_max__0__1_I2_J,axiom,
( ( ord_max_int @ one_one_int @ zero_zero_int )
= one_one_int ) ).
% max_0_1(2)
thf(fact_4014_max__0__1_I1_J,axiom,
( ( ord_max_real @ zero_zero_real @ one_one_real )
= one_one_real ) ).
% max_0_1(1)
thf(fact_4015_max__0__1_I1_J,axiom,
( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
= one_one_rat ) ).
% max_0_1(1)
thf(fact_4016_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_4017_max__0__1_I1_J,axiom,
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
= one_on7984719198319812577d_enat ) ).
% max_0_1(1)
thf(fact_4018_max__0__1_I1_J,axiom,
( ( ord_max_int @ zero_zero_int @ one_one_int )
= one_one_int ) ).
% max_0_1(1)
thf(fact_4019_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(5)
thf(fact_4020_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(5)
thf(fact_4021_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(5)
thf(fact_4022_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(5)
thf(fact_4023_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(5)
thf(fact_4024_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
= ( numera1916890842035813515d_enat @ X ) ) ).
% max_0_1(6)
thf(fact_4025_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
= ( numeral_numeral_real @ X ) ) ).
% max_0_1(6)
thf(fact_4026_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
= ( numeral_numeral_rat @ X ) ) ).
% max_0_1(6)
thf(fact_4027_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(6)
thf(fact_4028_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
= ( numeral_numeral_int @ X ) ) ).
% max_0_1(6)
thf(fact_4029_list__update__beyond,axiom,
! [Xs2: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I2 )
=> ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_4030_list__update__beyond,axiom,
! [Xs2: list_o,I2: nat,X: $o] :
( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I2 )
=> ( ( list_update_o @ Xs2 @ I2 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_4031_list__update__beyond,axiom,
! [Xs2: list_nat,I2: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I2 )
=> ( ( list_update_nat @ Xs2 @ I2 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_4032_list__update__beyond,axiom,
! [Xs2: list_int,I2: nat,X: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I2 )
=> ( ( list_update_int @ Xs2 @ I2 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_4033_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_4034_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_4035_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
= ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_4036_dbl__simps_I5_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% dbl_simps(5)
thf(fact_4037_nth__list__update__eq,axiom,
! [I2: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_4038_nth__list__update__eq,axiom,
! [I2: nat,Xs2: list_o,X: $o] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
=> ( ( nth_o @ ( list_update_o @ Xs2 @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_4039_nth__list__update__eq,axiom,
! [I2: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_4040_nth__list__update__eq,axiom,
! [I2: nat,Xs2: list_int,X: int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_4041_set__swap,axiom,
! [I2: nat,Xs2: list_VEBT_VEBT,J: nat] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
= ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_4042_set__swap,axiom,
! [I2: nat,Xs2: list_o,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
=> ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I2 @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I2 ) ) )
= ( set_o2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_4043_set__swap,axiom,
! [I2: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I2 @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I2 ) ) )
= ( set_nat2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_4044_set__swap,axiom,
! [I2: nat,Xs2: list_int,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
=> ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I2 @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I2 ) ) )
= ( set_int2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_4045_max__def,axiom,
( ord_ma741700101516333627d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4046_max__def,axiom,
( ord_max_set_nat
= ( ^ [A4: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4047_max__def,axiom,
( ord_max_rat
= ( ^ [A4: rat,B3: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4048_max__def,axiom,
( ord_max_num
= ( ^ [A4: num,B3: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4049_max__def,axiom,
( ord_max_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4050_max__def,axiom,
( ord_max_int
= ( ^ [A4: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def
thf(fact_4051_max__absorb1,axiom,
! [Y4: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y4 @ X )
=> ( ( ord_ma741700101516333627d_enat @ X @ Y4 )
= X ) ) ).
% max_absorb1
thf(fact_4052_max__absorb1,axiom,
! [Y4: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X )
=> ( ( ord_max_set_nat @ X @ Y4 )
= X ) ) ).
% max_absorb1
thf(fact_4053_max__absorb1,axiom,
! [Y4: rat,X: rat] :
( ( ord_less_eq_rat @ Y4 @ X )
=> ( ( ord_max_rat @ X @ Y4 )
= X ) ) ).
% max_absorb1
thf(fact_4054_max__absorb1,axiom,
! [Y4: num,X: num] :
( ( ord_less_eq_num @ Y4 @ X )
=> ( ( ord_max_num @ X @ Y4 )
= X ) ) ).
% max_absorb1
thf(fact_4055_max__absorb1,axiom,
! [Y4: nat,X: nat] :
( ( ord_less_eq_nat @ Y4 @ X )
=> ( ( ord_max_nat @ X @ Y4 )
= X ) ) ).
% max_absorb1
thf(fact_4056_max__absorb1,axiom,
! [Y4: int,X: int] :
( ( ord_less_eq_int @ Y4 @ X )
=> ( ( ord_max_int @ X @ Y4 )
= X ) ) ).
% max_absorb1
thf(fact_4057_max__absorb2,axiom,
! [X: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y4 )
=> ( ( ord_ma741700101516333627d_enat @ X @ Y4 )
= Y4 ) ) ).
% max_absorb2
thf(fact_4058_max__absorb2,axiom,
! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ( ord_max_set_nat @ X @ Y4 )
= Y4 ) ) ).
% max_absorb2
thf(fact_4059_max__absorb2,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ Y4 )
=> ( ( ord_max_rat @ X @ Y4 )
= Y4 ) ) ).
% max_absorb2
thf(fact_4060_max__absorb2,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
=> ( ( ord_max_num @ X @ Y4 )
= Y4 ) ) ).
% max_absorb2
thf(fact_4061_max__absorb2,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_max_nat @ X @ Y4 )
= Y4 ) ) ).
% max_absorb2
thf(fact_4062_max__absorb2,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ( ord_max_int @ X @ Y4 )
= Y4 ) ) ).
% max_absorb2
thf(fact_4063_max__add__distrib__right,axiom,
! [X: real,Y4: real,Z2: real] :
( ( plus_plus_real @ X @ ( ord_max_real @ Y4 @ Z2 ) )
= ( ord_max_real @ ( plus_plus_real @ X @ Y4 ) @ ( plus_plus_real @ X @ Z2 ) ) ) ).
% max_add_distrib_right
thf(fact_4064_max__add__distrib__right,axiom,
! [X: rat,Y4: rat,Z2: rat] :
( ( plus_plus_rat @ X @ ( ord_max_rat @ Y4 @ Z2 ) )
= ( ord_max_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( plus_plus_rat @ X @ Z2 ) ) ) ).
% max_add_distrib_right
thf(fact_4065_max__add__distrib__right,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( plus_plus_nat @ X @ ( ord_max_nat @ Y4 @ Z2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Y4 ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).
% max_add_distrib_right
thf(fact_4066_max__add__distrib__right,axiom,
! [X: int,Y4: int,Z2: int] :
( ( plus_plus_int @ X @ ( ord_max_int @ Y4 @ Z2 ) )
= ( ord_max_int @ ( plus_plus_int @ X @ Y4 ) @ ( plus_plus_int @ X @ Z2 ) ) ) ).
% max_add_distrib_right
thf(fact_4067_max__add__distrib__left,axiom,
! [X: real,Y4: real,Z2: real] :
( ( plus_plus_real @ ( ord_max_real @ X @ Y4 ) @ Z2 )
= ( ord_max_real @ ( plus_plus_real @ X @ Z2 ) @ ( plus_plus_real @ Y4 @ Z2 ) ) ) ).
% max_add_distrib_left
thf(fact_4068_max__add__distrib__left,axiom,
! [X: rat,Y4: rat,Z2: rat] :
( ( plus_plus_rat @ ( ord_max_rat @ X @ Y4 ) @ Z2 )
= ( ord_max_rat @ ( plus_plus_rat @ X @ Z2 ) @ ( plus_plus_rat @ Y4 @ Z2 ) ) ) ).
% max_add_distrib_left
thf(fact_4069_max__add__distrib__left,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X @ Y4 ) @ Z2 )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y4 @ Z2 ) ) ) ).
% max_add_distrib_left
thf(fact_4070_max__add__distrib__left,axiom,
! [X: int,Y4: int,Z2: int] :
( ( plus_plus_int @ ( ord_max_int @ X @ Y4 ) @ Z2 )
= ( ord_max_int @ ( plus_plus_int @ X @ Z2 ) @ ( plus_plus_int @ Y4 @ Z2 ) ) ) ).
% max_add_distrib_left
thf(fact_4071_nat__add__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_4072_nat__add__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_4073_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_4074_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
= ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_4075_max__def__raw,axiom,
( ord_ma741700101516333627d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_4076_max__def__raw,axiom,
( ord_max_set_nat
= ( ^ [A4: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_4077_max__def__raw,axiom,
( ord_max_rat
= ( ^ [A4: rat,B3: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_4078_max__def__raw,axiom,
( ord_max_num
= ( ^ [A4: num,B3: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_4079_max__def__raw,axiom,
( ord_max_nat
= ( ^ [A4: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_4080_max__def__raw,axiom,
( ord_max_int
= ( ^ [A4: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B3 ) @ B3 @ A4 ) ) ) ).
% max_def_raw
thf(fact_4081_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L2: int,R2: int] :
( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L2 @ R2 ) )
= ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L2 ) @ R2 ) ) ).
% concat_bit_assoc
thf(fact_4082_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_4083_set__update__subsetI,axiom,
! [Xs2: list_real,A2: set_real,X: real,I2: nat] :
( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
=> ( ( member_real @ X @ A2 )
=> ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_4084_set__update__subsetI,axiom,
! [Xs2: list_complex,A2: set_complex,X: complex,I2: nat] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
=> ( ( member_complex @ X @ A2 )
=> ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_4085_set__update__subsetI,axiom,
! [Xs2: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I2: nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
=> ( ( member8440522571783428010at_nat @ X @ A2 )
=> ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_4086_set__update__subsetI,axiom,
! [Xs2: list_int,A2: set_int,X: int,I2: nat] :
( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
=> ( ( member_int @ X @ A2 )
=> ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_4087_set__update__subsetI,axiom,
! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I2: nat] :
( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
=> ( ( member_VEBT_VEBT @ X @ A2 )
=> ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_4088_set__update__subsetI,axiom,
! [Xs2: list_nat,A2: set_nat,X: nat,I2: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I2 @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_4089_dbl__def,axiom,
( neg_numeral_dbl_real
= ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_4090_dbl__def,axiom,
( neg_numeral_dbl_rat
= ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_4091_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).
% dbl_def
thf(fact_4092_set__update__memI,axiom,
! [N: nat,Xs2: list_real,X: real] :
( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
=> ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_4093_set__update__memI,axiom,
! [N: nat,Xs2: list_complex,X: complex] :
( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_4094_set__update__memI,axiom,
! [N: nat,Xs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_4095_set__update__memI,axiom,
! [N: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_4096_set__update__memI,axiom,
! [N: nat,Xs2: list_o,X: $o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
=> ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_4097_set__update__memI,axiom,
! [N: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_4098_set__update__memI,axiom,
! [N: nat,Xs2: list_int,X: int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
=> ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_4099_list__update__same__conv,axiom,
! [I2: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X )
= Xs2 )
= ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_4100_list__update__same__conv,axiom,
! [I2: nat,Xs2: list_o,X: $o] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
=> ( ( ( list_update_o @ Xs2 @ I2 @ X )
= Xs2 )
= ( ( nth_o @ Xs2 @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_4101_list__update__same__conv,axiom,
! [I2: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( list_update_nat @ Xs2 @ I2 @ X )
= Xs2 )
= ( ( nth_nat @ Xs2 @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_4102_list__update__same__conv,axiom,
! [I2: nat,Xs2: list_int,X: int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( ( list_update_int @ Xs2 @ I2 @ X )
= Xs2 )
= ( ( nth_int @ Xs2 @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_4103_nth__list__update,axiom,
! [I2: nat,Xs2: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
=> ( ( ( I2 = J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ J )
= ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_4104_nth__list__update,axiom,
! [I2: nat,Xs2: list_o,X: $o,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
=> ( ( nth_o @ ( list_update_o @ Xs2 @ I2 @ X ) @ J )
= ( ( ( I2 = J )
=> X )
& ( ( I2 != J )
=> ( nth_o @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_4105_nth__list__update,axiom,
! [I2: nat,Xs2: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( I2 = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_4106_nth__list__update,axiom,
! [I2: nat,Xs2: list_int,J: nat,X: int] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( ( I2 = J )
=> ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ J )
= ( nth_int @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_4107_small__lazy_H_Ocases,axiom,
! [X: product_prod_int_int] :
~ ! [D3: int,I3: int] :
( X
!= ( product_Pair_int_int @ D3 @ I3 ) ) ).
% small_lazy'.cases
thf(fact_4108_exhaustive__int_H_Ocases,axiom,
! [X: produc7773217078559923341nt_int] :
~ ! [F2: int > option6357759511663192854e_term,D3: int,I3: int] :
( X
!= ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I3 ) ) ) ).
% exhaustive_int'.cases
thf(fact_4109_full__exhaustive__int_H_Ocases,axiom,
! [X: produc2285326912895808259nt_int] :
~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D3: int,I3: int] :
( X
!= ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I3 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_4110_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
= ( P @ B2 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_4111_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_4112_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y4 )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ~ ( ( ( Xa2 = zero_zero_nat )
=> ( Y4
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y4
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y4
= ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
=> ( Y4
!= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
=> ( Y4
!= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) ) )
=> ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
=> ( Y4
!= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( Y4
!= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_4113_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y4 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y4
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( ( ( Xa2 = one_one_nat )
=> ( Y4
= ( vEBT_Leaf @ A3 @ $true ) ) )
& ( ( Xa2 != one_one_nat )
=> ( Y4
= ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
=> ( ( Y4
= ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S ) @ Xa2 ) ) ) )
=> ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
=> ( ( Y4
= ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S ) @ Xa2 ) ) ) )
=> ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y4
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
=> ( ( Y4
= ( if_VEBT_VEBT
@ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_4114_max__less__iff__conj,axiom,
! [X: extended_enat,Y4: extended_enat,Z2: extended_enat] :
( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y4 ) @ Z2 )
= ( ( ord_le72135733267957522d_enat @ X @ Z2 )
& ( ord_le72135733267957522d_enat @ Y4 @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_4115_max__less__iff__conj,axiom,
! [X: real,Y4: real,Z2: real] :
( ( ord_less_real @ ( ord_max_real @ X @ Y4 ) @ Z2 )
= ( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Y4 @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_4116_max__less__iff__conj,axiom,
! [X: rat,Y4: rat,Z2: rat] :
( ( ord_less_rat @ ( ord_max_rat @ X @ Y4 ) @ Z2 )
= ( ( ord_less_rat @ X @ Z2 )
& ( ord_less_rat @ Y4 @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_4117_max__less__iff__conj,axiom,
! [X: num,Y4: num,Z2: num] :
( ( ord_less_num @ ( ord_max_num @ X @ Y4 ) @ Z2 )
= ( ( ord_less_num @ X @ Z2 )
& ( ord_less_num @ Y4 @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_4118_max__less__iff__conj,axiom,
! [X: nat,Y4: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y4 ) @ Z2 )
= ( ( ord_less_nat @ X @ Z2 )
& ( ord_less_nat @ Y4 @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_4119_max__less__iff__conj,axiom,
! [X: int,Y4: int,Z2: int] :
( ( ord_less_int @ ( ord_max_int @ X @ Y4 ) @ Z2 )
= ( ( ord_less_int @ X @ Z2 )
& ( ord_less_int @ Y4 @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_4120_max_Oabsorb4,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_4121_max_Oabsorb4,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_max_real @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_4122_max_Oabsorb4,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_max_rat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_4123_max_Oabsorb4,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_4124_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_4125_max_Oabsorb4,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_4126_max_Oabsorb3,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_4127_max_Oabsorb3,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_max_real @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_4128_max_Oabsorb3,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ B @ A )
=> ( ( ord_max_rat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_4129_max_Oabsorb3,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_4130_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_4131_max_Oabsorb3,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_4132_max_Oabsorb1,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_4133_max_Oabsorb1,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_max_rat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_4134_max_Oabsorb1,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_4135_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_4136_max_Oabsorb1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_max_int @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_4137_max_Oabsorb2,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_ma741700101516333627d_enat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_4138_max_Oabsorb2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_max_rat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_4139_max_Oabsorb2,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_4140_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_4141_max_Oabsorb2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_max_int @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_4142_max_Obounded__iff,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
= ( ( ord_le2932123472753598470d_enat @ B @ A )
& ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_4143_max_Obounded__iff,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
= ( ( ord_less_eq_rat @ B @ A )
& ( ord_less_eq_rat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_4144_max_Obounded__iff,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
= ( ( ord_less_eq_num @ B @ A )
& ( ord_less_eq_num @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_4145_max_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_4146_max_Obounded__iff,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_4147_max__enat__simps_I2_J,axiom,
! [Q2: extended_enat] :
( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
= Q2 ) ).
% max_enat_simps(2)
thf(fact_4148_max__enat__simps_I3_J,axiom,
! [Q2: extended_enat] :
( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
= Q2 ) ).
% max_enat_simps(3)
thf(fact_4149_prod__decode__aux_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [K2: nat,M4: nat] :
( X
!= ( product_Pair_nat_nat @ K2 @ M4 ) ) ).
% prod_decode_aux.cases
thf(fact_4150_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_4151_max_Omono,axiom,
! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ A )
=> ( ( ord_le2932123472753598470d_enat @ D @ B )
=> ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_4152_max_Omono,axiom,
! [C: rat,A: rat,D: rat,B: rat] :
( ( ord_less_eq_rat @ C @ A )
=> ( ( ord_less_eq_rat @ D @ B )
=> ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_4153_max_Omono,axiom,
! [C: num,A: num,D: num,B: num] :
( ( ord_less_eq_num @ C @ A )
=> ( ( ord_less_eq_num @ D @ B )
=> ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% max.mono
thf(fact_4154_max_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_4155_max_Omono,axiom,
! [C: int,A: int,D: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ( ord_less_eq_int @ D @ B )
=> ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% max.mono
thf(fact_4156_max_OorderE,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( A
= ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.orderE
thf(fact_4157_max_OorderE,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( A
= ( ord_max_rat @ A @ B ) ) ) ).
% max.orderE
thf(fact_4158_max_OorderE,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( A
= ( ord_max_num @ A @ B ) ) ) ).
% max.orderE
thf(fact_4159_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_4160_max_OorderE,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( A
= ( ord_max_int @ A @ B ) ) ) ).
% max.orderE
thf(fact_4161_max_OorderI,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A
= ( ord_ma741700101516333627d_enat @ A @ B ) )
=> ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% max.orderI
thf(fact_4162_max_OorderI,axiom,
! [A: rat,B: rat] :
( ( A
= ( ord_max_rat @ A @ B ) )
=> ( ord_less_eq_rat @ B @ A ) ) ).
% max.orderI
thf(fact_4163_max_OorderI,axiom,
! [A: num,B: num] :
( ( A
= ( ord_max_num @ A @ B ) )
=> ( ord_less_eq_num @ B @ A ) ) ).
% max.orderI
thf(fact_4164_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_4165_max_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( ord_max_int @ A @ B ) )
=> ( ord_less_eq_int @ B @ A ) ) ).
% max.orderI
thf(fact_4166_max_OboundedE,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
=> ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_4167_max_OboundedE,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_rat @ B @ A )
=> ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_4168_max_OboundedE,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_num @ B @ A )
=> ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% max.boundedE
thf(fact_4169_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_4170_max_OboundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_int @ B @ A )
=> ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% max.boundedE
thf(fact_4171_max_OboundedI,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ A )
=> ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_4172_max_OboundedI,axiom,
! [B: rat,A: rat,C: rat] :
( ( ord_less_eq_rat @ B @ A )
=> ( ( ord_less_eq_rat @ C @ A )
=> ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_4173_max_OboundedI,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ A )
=> ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_4174_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_4175_max_OboundedI,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_4176_max_Oorder__iff,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( A4
= ( ord_ma741700101516333627d_enat @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_4177_max_Oorder__iff,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A4: rat] :
( A4
= ( ord_max_rat @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_4178_max_Oorder__iff,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A4: num] :
( A4
= ( ord_max_num @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_4179_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( A4
= ( ord_max_nat @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_4180_max_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( A4
= ( ord_max_int @ A4 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_4181_max_Ocobounded1,axiom,
! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% max.cobounded1
thf(fact_4182_max_Ocobounded1,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% max.cobounded1
thf(fact_4183_max_Ocobounded1,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded1
thf(fact_4184_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_4185_max_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded1
thf(fact_4186_max_Ocobounded2,axiom,
! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% max.cobounded2
thf(fact_4187_max_Ocobounded2,axiom,
! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% max.cobounded2
thf(fact_4188_max_Ocobounded2,axiom,
! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded2
thf(fact_4189_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_4190_max_Ocobounded2,axiom,
! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% max.cobounded2
thf(fact_4191_le__max__iff__disj,axiom,
! [Z2: extended_enat,X: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z2 @ ( ord_ma741700101516333627d_enat @ X @ Y4 ) )
= ( ( ord_le2932123472753598470d_enat @ Z2 @ X )
| ( ord_le2932123472753598470d_enat @ Z2 @ Y4 ) ) ) ).
% le_max_iff_disj
thf(fact_4192_le__max__iff__disj,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( ord_less_eq_rat @ Z2 @ ( ord_max_rat @ X @ Y4 ) )
= ( ( ord_less_eq_rat @ Z2 @ X )
| ( ord_less_eq_rat @ Z2 @ Y4 ) ) ) ).
% le_max_iff_disj
thf(fact_4193_le__max__iff__disj,axiom,
! [Z2: num,X: num,Y4: num] :
( ( ord_less_eq_num @ Z2 @ ( ord_max_num @ X @ Y4 ) )
= ( ( ord_less_eq_num @ Z2 @ X )
| ( ord_less_eq_num @ Z2 @ Y4 ) ) ) ).
% le_max_iff_disj
thf(fact_4194_le__max__iff__disj,axiom,
! [Z2: nat,X: nat,Y4: nat] :
( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X @ Y4 ) )
= ( ( ord_less_eq_nat @ Z2 @ X )
| ( ord_less_eq_nat @ Z2 @ Y4 ) ) ) ).
% le_max_iff_disj
thf(fact_4195_le__max__iff__disj,axiom,
! [Z2: int,X: int,Y4: int] :
( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X @ Y4 ) )
= ( ( ord_less_eq_int @ Z2 @ X )
| ( ord_less_eq_int @ Z2 @ Y4 ) ) ) ).
% le_max_iff_disj
thf(fact_4196_max_Oabsorb__iff1,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_ma741700101516333627d_enat @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_4197_max_Oabsorb__iff1,axiom,
( ord_less_eq_rat
= ( ^ [B3: rat,A4: rat] :
( ( ord_max_rat @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_4198_max_Oabsorb__iff1,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A4: num] :
( ( ord_max_num @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_4199_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_max_nat @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_4200_max_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_max_int @ A4 @ B3 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_4201_max_Oabsorb__iff2,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_ma741700101516333627d_enat @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_4202_max_Oabsorb__iff2,axiom,
( ord_less_eq_rat
= ( ^ [A4: rat,B3: rat] :
( ( ord_max_rat @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_4203_max_Oabsorb__iff2,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B3: num] :
( ( ord_max_num @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_4204_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_max_nat @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_4205_max_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_max_int @ A4 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_4206_max_OcoboundedI1,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ A )
=> ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_4207_max_OcoboundedI1,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_eq_rat @ C @ A )
=> ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_4208_max_OcoboundedI1,axiom,
! [C: num,A: num,B: num] :
( ( ord_less_eq_num @ C @ A )
=> ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_4209_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_4210_max_OcoboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ A )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_4211_max_OcoboundedI2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_4212_max_OcoboundedI2,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_eq_rat @ C @ B )
=> ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_4213_max_OcoboundedI2,axiom,
! [C: num,B: num,A: num] :
( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_4214_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_4215_max_OcoboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_4216_less__max__iff__disj,axiom,
! [Z2: extended_enat,X: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z2 @ ( ord_ma741700101516333627d_enat @ X @ Y4 ) )
= ( ( ord_le72135733267957522d_enat @ Z2 @ X )
| ( ord_le72135733267957522d_enat @ Z2 @ Y4 ) ) ) ).
% less_max_iff_disj
thf(fact_4217_less__max__iff__disj,axiom,
! [Z2: real,X: real,Y4: real] :
( ( ord_less_real @ Z2 @ ( ord_max_real @ X @ Y4 ) )
= ( ( ord_less_real @ Z2 @ X )
| ( ord_less_real @ Z2 @ Y4 ) ) ) ).
% less_max_iff_disj
thf(fact_4218_less__max__iff__disj,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( ord_less_rat @ Z2 @ ( ord_max_rat @ X @ Y4 ) )
= ( ( ord_less_rat @ Z2 @ X )
| ( ord_less_rat @ Z2 @ Y4 ) ) ) ).
% less_max_iff_disj
thf(fact_4219_less__max__iff__disj,axiom,
! [Z2: num,X: num,Y4: num] :
( ( ord_less_num @ Z2 @ ( ord_max_num @ X @ Y4 ) )
= ( ( ord_less_num @ Z2 @ X )
| ( ord_less_num @ Z2 @ Y4 ) ) ) ).
% less_max_iff_disj
thf(fact_4220_less__max__iff__disj,axiom,
! [Z2: nat,X: nat,Y4: nat] :
( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X @ Y4 ) )
= ( ( ord_less_nat @ Z2 @ X )
| ( ord_less_nat @ Z2 @ Y4 ) ) ) ).
% less_max_iff_disj
thf(fact_4221_less__max__iff__disj,axiom,
! [Z2: int,X: int,Y4: int] :
( ( ord_less_int @ Z2 @ ( ord_max_int @ X @ Y4 ) )
= ( ( ord_less_int @ Z2 @ X )
| ( ord_less_int @ Z2 @ Y4 ) ) ) ).
% less_max_iff_disj
thf(fact_4222_max_Ostrict__boundedE,axiom,
! [B: extended_enat,C: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
=> ~ ( ( ord_le72135733267957522d_enat @ B @ A )
=> ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_4223_max_Ostrict__boundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
=> ~ ( ( ord_less_real @ B @ A )
=> ~ ( ord_less_real @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_4224_max_Ostrict__boundedE,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
=> ~ ( ( ord_less_rat @ B @ A )
=> ~ ( ord_less_rat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_4225_max_Ostrict__boundedE,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
=> ~ ( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_4226_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_4227_max_Ostrict__boundedE,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_4228_max_Ostrict__order__iff,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( A4
= ( ord_ma741700101516333627d_enat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_4229_max_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B3: real,A4: real] :
( ( A4
= ( ord_max_real @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_4230_max_Ostrict__order__iff,axiom,
( ord_less_rat
= ( ^ [B3: rat,A4: rat] :
( ( A4
= ( ord_max_rat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_4231_max_Ostrict__order__iff,axiom,
( ord_less_num
= ( ^ [B3: num,A4: num] :
( ( A4
= ( ord_max_num @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_4232_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( A4
= ( ord_max_nat @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_4233_max_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( A4
= ( ord_max_int @ A4 @ B3 ) )
& ( A4 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_4234_max_Ostrict__coboundedI1,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ C @ A )
=> ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_4235_max_Ostrict__coboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ A )
=> ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_4236_max_Ostrict__coboundedI1,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ A )
=> ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_4237_max_Ostrict__coboundedI1,axiom,
! [C: num,A: num,B: num] :
( ( ord_less_num @ C @ A )
=> ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_4238_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_4239_max_Ostrict__coboundedI1,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ A )
=> ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_4240_max_Ostrict__coboundedI2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_4241_max_Ostrict__coboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_4242_max_Ostrict__coboundedI2,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ B )
=> ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_4243_max_Ostrict__coboundedI2,axiom,
! [C: num,B: num,A: num] :
( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_4244_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_4245_max_Ostrict__coboundedI2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_4246_triangle__def,axiom,
( nat_triangle
= ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% triangle_def
thf(fact_4247_even__succ__mod__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_4248_even__succ__mod__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_4249_even__succ__mod__exp,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_4250_option_Osize__gen_I2_J,axiom,
! [X: nat > nat,X22: nat] :
( ( size_option_nat @ X @ ( some_nat @ X22 ) )
= ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_4251_option_Osize__gen_I2_J,axiom,
! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
= ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_4252_option_Osize__gen_I2_J,axiom,
! [X: num > nat,X22: num] :
( ( size_option_num @ X @ ( some_num @ X22 ) )
= ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_4253_even__succ__div__exp,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_4254_even__succ__div__exp,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_4255_even__succ__div__exp,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_4256_signed__take__bit__Suc,axiom,
! [N: nat,A: code_integer] :
( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
= ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_4257_signed__take__bit__Suc,axiom,
! [N: nat,A: int] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
= ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_4258_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
= ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_4259_infinite__growing,axiom,
! [X7: set_real] :
( ( X7 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X7 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X7 )
& ( ord_less_real @ X3 @ Xa ) ) )
=> ~ ( finite_finite_real @ X7 ) ) ) ).
% infinite_growing
thf(fact_4260_infinite__growing,axiom,
! [X7: set_rat] :
( ( X7 != bot_bot_set_rat )
=> ( ! [X3: rat] :
( ( member_rat @ X3 @ X7 )
=> ? [Xa: rat] :
( ( member_rat @ Xa @ X7 )
& ( ord_less_rat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_rat @ X7 ) ) ) ).
% infinite_growing
thf(fact_4261_infinite__growing,axiom,
! [X7: set_num] :
( ( X7 != bot_bot_set_num )
=> ( ! [X3: num] :
( ( member_num @ X3 @ X7 )
=> ? [Xa: num] :
( ( member_num @ Xa @ X7 )
& ( ord_less_num @ X3 @ Xa ) ) )
=> ~ ( finite_finite_num @ X7 ) ) ) ).
% infinite_growing
thf(fact_4262_infinite__growing,axiom,
! [X7: set_nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X7 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X7 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_4263_infinite__growing,axiom,
! [X7: set_int] :
( ( X7 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ X7 )
=> ? [Xa: int] :
( ( member_int @ Xa @ X7 )
& ( ord_less_int @ X3 @ Xa ) ) )
=> ~ ( finite_finite_int @ X7 ) ) ) ).
% infinite_growing
thf(fact_4264_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_4265_dvd__add__triv__right__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_4266_dvd__add__triv__right__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_4267_dvd__add__triv__right__iff,axiom,
! [A: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_4268_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_4269_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_4270_dvd__add__triv__left__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_4271_dvd__add__triv__left__iff,axiom,
! [A: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_4272_dvd__add__triv__left__iff,axiom,
! [A: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_4273_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_4274_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_4275_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_4276_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_4277_div__dvd__div,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ A @ C )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_4278_div__dvd__div,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_4279_div__dvd__div,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ) ).
% div_dvd_div
thf(fact_4280_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_4281_dvd__times__right__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_4282_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_4283_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_4284_dvd__times__left__cancel__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_4285_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_4286_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_4287_dvd__mult__cancel__right,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4288_dvd__mult__cancel__right,axiom,
! [A: complex,C: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4289_dvd__mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4290_dvd__mult__cancel__right,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
= ( ( C = zero_zero_rat )
| ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4291_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_4292_dvd__mult__cancel__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
= ( ( C = zero_z3403309356797280102nteger )
| ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4293_dvd__mult__cancel__left,axiom,
! [C: complex,A: complex,B: complex] :
( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( dvd_dvd_complex @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4294_dvd__mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4295_dvd__mult__cancel__left,axiom,
! [C: rat,A: rat,B: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
= ( ( C = zero_zero_rat )
| ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4296_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_4297_unit__prod,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% unit_prod
thf(fact_4298_unit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_4299_unit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_4300_dvd__add__times__triv__left__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4301_dvd__add__times__triv__left__iff,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4302_dvd__add__times__triv__left__iff,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4303_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4304_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_4305_dvd__add__times__triv__right__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4306_dvd__add__times__triv__right__iff,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
= ( dvd_dvd_real @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4307_dvd__add__times__triv__right__iff,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
= ( dvd_dvd_rat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4308_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4309_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_4310_dvd__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_4311_dvd__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_4312_dvd__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% dvd_div_mult_self
thf(fact_4313_dvd__mult__div__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_4314_dvd__mult__div__cancel,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_4315_dvd__mult__div__cancel,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
= B ) ) ).
% dvd_mult_div_cancel
thf(fact_4316_unit__div,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% unit_div
thf(fact_4317_unit__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_div
thf(fact_4318_unit__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_div
thf(fact_4319_unit__div__1__unit,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% unit_div_1_unit
thf(fact_4320_unit__div__1__unit,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% unit_div_1_unit
thf(fact_4321_unit__div__1__unit,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% unit_div_1_unit
thf(fact_4322_unit__div__1__div__1,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_4323_unit__div__1__div__1,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_4324_unit__div__1__div__1,axiom,
! [A: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
= A ) ) ).
% unit_div_1_div_1
thf(fact_4325_div__add,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_4326_div__add,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_4327_div__add,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_add
thf(fact_4328_div__diff,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
= ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% div_diff
thf(fact_4329_div__diff,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% div_diff
thf(fact_4330_signed__take__bit__Suc__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_Suc_1
thf(fact_4331_signed__take__bit__numeral__of__1,axiom,
! [K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
= one_one_int ) ).
% signed_take_bit_numeral_of_1
thf(fact_4332_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_4333_set__decode__zero,axiom,
( ( nat_set_decode @ zero_zero_nat )
= bot_bot_set_nat ) ).
% set_decode_zero
thf(fact_4334_unit__mult__div__div,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
= ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_4335_unit__mult__div__div,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
= ( divide_divide_nat @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_4336_unit__mult__div__div,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% unit_mult_div_div
thf(fact_4337_unit__div__mult__self,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_4338_unit__div__mult__self,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_4339_unit__div__mult__self,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
= B ) ) ).
% unit_div_mult_self
thf(fact_4340_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% even_Suc
thf(fact_4341_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_4342_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_4343_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_4344_even__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_4345_even__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_4346_even__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_mult_iff
thf(fact_4347_odd__add,axiom,
! [A: code_integer,B: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_4348_odd__add,axiom,
! [A: nat,B: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_4349_odd__add,axiom,
! [A: int,B: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
= ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% odd_add
thf(fact_4350_even__add,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_4351_even__add,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_4352_even__add,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% even_add
thf(fact_4353_even__mod__2__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% even_mod_2_iff
thf(fact_4354_even__mod__2__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_mod_2_iff
thf(fact_4355_even__mod__2__iff,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% even_mod_2_iff
thf(fact_4356_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_4357_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_4358_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_4359_set__decode__0,axiom,
! [X: nat] :
( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_4360_zero__le__power__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_4361_zero__le__power__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_4362_zero__le__power__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_4363_power__less__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq
thf(fact_4364_power__less__zero__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq
thf(fact_4365_power__less__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq
thf(fact_4366_power__less__zero__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_4367_power__less__zero__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_4368_power__less__zero__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_4369_even__plus__one__iff,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
= ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_4370_even__plus__one__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_4371_even__plus__one__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% even_plus_one_iff
thf(fact_4372_even__diff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% even_diff
thf(fact_4373_even__diff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% even_diff
thf(fact_4374_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_4375_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_4376_zero__less__power__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_4377_zero__less__power__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_4378_zero__less__power__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
= ( ( ( numeral_numeral_nat @ W2 )
= zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_4379_even__succ__div__2,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_4380_even__succ__div__2,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_4381_even__succ__div__2,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_2
thf(fact_4382_odd__succ__div__two,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% odd_succ_div_two
thf(fact_4383_odd__succ__div__two,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% odd_succ_div_two
thf(fact_4384_odd__succ__div__two,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% odd_succ_div_two
thf(fact_4385_even__succ__div__two,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_4386_even__succ__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_4387_even__succ__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% even_succ_div_two
thf(fact_4388_even__power,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_4389_even__power,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_4390_even__power,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% even_power
thf(fact_4391_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_4392_odd__two__times__div__two__succ,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_4393_odd__two__times__div__two__succ,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_4394_odd__two__times__div__two__succ,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
= A ) ) ).
% odd_two_times_div_two_succ
thf(fact_4395_power__le__zero__eq__numeral,axiom,
! [A: real,W2: num] :
( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_4396_power__le__zero__eq__numeral,axiom,
! [A: rat,W2: num] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_4397_power__le__zero__eq__numeral,axiom,
! [A: int,W2: num] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_4398_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4399_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4400_semiring__parity__class_Oeven__mask__iff,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
= ( N = zero_zero_nat ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_4401_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
=> ? [B6: nat,C4: nat] :
( ( A
= ( times_times_nat @ B6 @ C4 ) )
& ( dvd_dvd_nat @ B6 @ B )
& ( dvd_dvd_nat @ C4 @ C ) ) ) ).
% division_decomp
thf(fact_4402_division__decomp,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
=> ? [B6: int,C4: int] :
( ( A
= ( times_times_int @ B6 @ C4 ) )
& ( dvd_dvd_int @ B6 @ B )
& ( dvd_dvd_int @ C4 @ C ) ) ) ).
% division_decomp
thf(fact_4403_dvd__productE,axiom,
! [P6: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P6 @ ( times_times_nat @ A @ B ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( P6
= ( times_times_nat @ X3 @ Y3 ) )
=> ( ( dvd_dvd_nat @ X3 @ A )
=> ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_4404_dvd__productE,axiom,
! [P6: int,A: int,B: int] :
( ( dvd_dvd_int @ P6 @ ( times_times_int @ A @ B ) )
=> ~ ! [X3: int,Y3: int] :
( ( P6
= ( times_times_int @ X3 @ Y3 ) )
=> ( ( dvd_dvd_int @ X3 @ A )
=> ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_4405_dvd__triv__right,axiom,
! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% dvd_triv_right
thf(fact_4406_dvd__triv__right,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% dvd_triv_right
thf(fact_4407_dvd__triv__right,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_4408_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_4409_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_4410_dvd__mult__right,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
=> ( dvd_dvd_Code_integer @ B @ C ) ) ).
% dvd_mult_right
thf(fact_4411_dvd__mult__right,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
=> ( dvd_dvd_real @ B @ C ) ) ).
% dvd_mult_right
thf(fact_4412_dvd__mult__right,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
=> ( dvd_dvd_rat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_4413_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_4414_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_4415_mult__dvd__mono,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ C @ D )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4416_mult__dvd__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ C @ D )
=> ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4417_mult__dvd__mono,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ C @ D )
=> ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4418_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4419_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_4420_dvd__triv__left,axiom,
! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% dvd_triv_left
thf(fact_4421_dvd__triv__left,axiom,
! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% dvd_triv_left
thf(fact_4422_dvd__triv__left,axiom,
! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_4423_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_4424_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_4425_dvd__mult__left,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
=> ( dvd_dvd_Code_integer @ A @ C ) ) ).
% dvd_mult_left
thf(fact_4426_dvd__mult__left,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
=> ( dvd_dvd_real @ A @ C ) ) ).
% dvd_mult_left
thf(fact_4427_dvd__mult__left,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
=> ( dvd_dvd_rat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_4428_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_4429_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_4430_dvd__mult2,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_4431_dvd__mult2,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_4432_dvd__mult2,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_4433_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_4434_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_4435_dvd__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ C )
=> ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% dvd_mult
thf(fact_4436_dvd__mult,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ C )
=> ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% dvd_mult
thf(fact_4437_dvd__mult,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ A @ C )
=> ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_4438_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_4439_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_4440_dvd__def,axiom,
( dvd_dvd_Code_integer
= ( ^ [B3: code_integer,A4: code_integer] :
? [K3: code_integer] :
( A4
= ( times_3573771949741848930nteger @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4441_dvd__def,axiom,
( dvd_dvd_real
= ( ^ [B3: real,A4: real] :
? [K3: real] :
( A4
= ( times_times_real @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4442_dvd__def,axiom,
( dvd_dvd_rat
= ( ^ [B3: rat,A4: rat] :
? [K3: rat] :
( A4
= ( times_times_rat @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4443_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A4: nat] :
? [K3: nat] :
( A4
= ( times_times_nat @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4444_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B3: int,A4: int] :
? [K3: int] :
( A4
= ( times_times_int @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_4445_dvdI,axiom,
! [A: code_integer,B: code_integer,K: code_integer] :
( ( A
= ( times_3573771949741848930nteger @ B @ K ) )
=> ( dvd_dvd_Code_integer @ B @ A ) ) ).
% dvdI
thf(fact_4446_dvdI,axiom,
! [A: real,B: real,K: real] :
( ( A
= ( times_times_real @ B @ K ) )
=> ( dvd_dvd_real @ B @ A ) ) ).
% dvdI
thf(fact_4447_dvdI,axiom,
! [A: rat,B: rat,K: rat] :
( ( A
= ( times_times_rat @ B @ K ) )
=> ( dvd_dvd_rat @ B @ A ) ) ).
% dvdI
thf(fact_4448_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_4449_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_4450_dvdE,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ~ ! [K2: code_integer] :
( A
!= ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).
% dvdE
thf(fact_4451_dvdE,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ~ ! [K2: real] :
( A
!= ( times_times_real @ B @ K2 ) ) ) ).
% dvdE
thf(fact_4452_dvdE,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ~ ! [K2: rat] :
( A
!= ( times_times_rat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_4453_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K2: nat] :
( A
!= ( times_times_nat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_4454_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K2: int] :
( A
!= ( times_times_int @ B @ K2 ) ) ) ).
% dvdE
thf(fact_4455_dvd__unit__imp__unit,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% dvd_unit_imp_unit
thf(fact_4456_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_4457_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_4458_unit__imp__dvd,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( dvd_dvd_Code_integer @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_4459_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_4460_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_4461_one__dvd,axiom,
! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% one_dvd
thf(fact_4462_one__dvd,axiom,
! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% one_dvd
thf(fact_4463_one__dvd,axiom,
! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% one_dvd
thf(fact_4464_one__dvd,axiom,
! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% one_dvd
thf(fact_4465_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_4466_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_4467_dvd__add__right__iff,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_4468_dvd__add__right__iff,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
= ( dvd_dvd_real @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_4469_dvd__add__right__iff,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( dvd_dvd_rat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_4470_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_4471_dvd__add__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_4472_dvd__add__left__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ C )
=> ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_4473_dvd__add__left__iff,axiom,
! [A: real,C: real,B: real] :
( ( dvd_dvd_real @ A @ C )
=> ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
= ( dvd_dvd_real @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_4474_dvd__add__left__iff,axiom,
! [A: rat,C: rat,B: rat] :
( ( dvd_dvd_rat @ A @ C )
=> ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
= ( dvd_dvd_rat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_4475_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_4476_dvd__add__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_4477_dvd__add,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ A @ C )
=> ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_4478_dvd__add,axiom,
! [A: real,B: real,C: real] :
( ( dvd_dvd_real @ A @ B )
=> ( ( dvd_dvd_real @ A @ C )
=> ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_4479_dvd__add,axiom,
! [A: rat,B: rat,C: rat] :
( ( dvd_dvd_rat @ A @ B )
=> ( ( dvd_dvd_rat @ A @ C )
=> ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_4480_dvd__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_4481_dvd__add,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_4482_dvd__diff__commute,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
= ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% dvd_diff_commute
thf(fact_4483_dvd__diff__commute,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
= ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% dvd_diff_commute
thf(fact_4484_dvd__div__eq__iff,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( ( divide6298287555418463151nteger @ A @ C )
= ( divide6298287555418463151nteger @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4485_dvd__div__eq__iff,axiom,
! [C: complex,A: complex,B: complex] :
( ( dvd_dvd_complex @ C @ A )
=> ( ( dvd_dvd_complex @ C @ B )
=> ( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4486_dvd__div__eq__iff,axiom,
! [C: real,A: real,B: real] :
( ( dvd_dvd_real @ C @ A )
=> ( ( dvd_dvd_real @ C @ B )
=> ( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4487_dvd__div__eq__iff,axiom,
! [C: rat,A: rat,B: rat] :
( ( dvd_dvd_rat @ C @ A )
=> ( ( dvd_dvd_rat @ C @ B )
=> ( ( ( divide_divide_rat @ A @ C )
= ( divide_divide_rat @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4488_dvd__div__eq__iff,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4489_dvd__div__eq__iff,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B @ C ) )
= ( A = B ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_4490_dvd__div__eq__cancel,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ( divide6298287555418463151nteger @ A @ C )
= ( divide6298287555418463151nteger @ B @ C ) )
=> ( ( dvd_dvd_Code_integer @ C @ A )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4491_dvd__div__eq__cancel,axiom,
! [A: complex,C: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ C )
= ( divide1717551699836669952omplex @ B @ C ) )
=> ( ( dvd_dvd_complex @ C @ A )
=> ( ( dvd_dvd_complex @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4492_dvd__div__eq__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
=> ( ( dvd_dvd_real @ C @ A )
=> ( ( dvd_dvd_real @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4493_dvd__div__eq__cancel,axiom,
! [A: rat,C: rat,B: rat] :
( ( ( divide_divide_rat @ A @ C )
= ( divide_divide_rat @ B @ C ) )
=> ( ( dvd_dvd_rat @ C @ A )
=> ( ( dvd_dvd_rat @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4494_dvd__div__eq__cancel,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( divide_divide_nat @ A @ C )
= ( divide_divide_nat @ B @ C ) )
=> ( ( dvd_dvd_nat @ C @ A )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4495_dvd__div__eq__cancel,axiom,
! [A: int,C: int,B: int] :
( ( ( divide_divide_int @ A @ C )
= ( divide_divide_int @ B @ C ) )
=> ( ( dvd_dvd_int @ C @ A )
=> ( ( dvd_dvd_int @ C @ B )
=> ( A = B ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_4496_div__div__div__same,axiom,
! [D: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ D @ B )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_4497_div__div__div__same,axiom,
! [D: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ D @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_4498_div__div__div__same,axiom,
! [D: int,B: int,A: int] :
( ( dvd_dvd_int @ D @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
= ( divide_divide_int @ A @ B ) ) ) ) ).
% div_div_div_same
thf(fact_4499_dvd__power__same,axiom,
! [X: code_integer,Y4: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ X @ Y4 )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y4 @ N ) ) ) ).
% dvd_power_same
thf(fact_4500_dvd__power__same,axiom,
! [X: nat,Y4: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y4 )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y4 @ N ) ) ) ).
% dvd_power_same
thf(fact_4501_dvd__power__same,axiom,
! [X: real,Y4: real,N: nat] :
( ( dvd_dvd_real @ X @ Y4 )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ N ) ) ) ).
% dvd_power_same
thf(fact_4502_dvd__power__same,axiom,
! [X: int,Y4: int,N: nat] :
( ( dvd_dvd_int @ X @ Y4 )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y4 @ N ) ) ) ).
% dvd_power_same
thf(fact_4503_dvd__power__same,axiom,
! [X: complex,Y4: complex,N: nat] :
( ( dvd_dvd_complex @ X @ Y4 )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y4 @ N ) ) ) ).
% dvd_power_same
thf(fact_4504_dvd__mod,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_4505_dvd__mod,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ M )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_4506_dvd__mod,axiom,
! [K: code_integer,M: code_integer,N: code_integer] :
( ( dvd_dvd_Code_integer @ K @ M )
=> ( ( dvd_dvd_Code_integer @ K @ N )
=> ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% dvd_mod
thf(fact_4507_mod__mod__cancel,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_4508_mod__mod__cancel,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
= ( modulo_modulo_int @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_4509_mod__mod__cancel,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
= ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% mod_mod_cancel
thf(fact_4510_signed__take__bit__mult,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% signed_take_bit_mult
thf(fact_4511_signed__take__bit__add,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
= ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L2 ) ) ) ).
% signed_take_bit_add
thf(fact_4512_even__signed__take__bit__iff,axiom,
! [M: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% even_signed_take_bit_iff
thf(fact_4513_even__signed__take__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_signed_take_bit_iff
thf(fact_4514_not__is__unit__0,axiom,
~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% not_is_unit_0
thf(fact_4515_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_4516_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_4517_minf_I10_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ X5 @ Z )
=> ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ) ).
% minf(10)
thf(fact_4518_minf_I10_J,axiom,
! [D: real,S2: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ) ).
% minf(10)
thf(fact_4519_minf_I10_J,axiom,
! [D: rat,S2: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ) ).
% minf(10)
thf(fact_4520_minf_I10_J,axiom,
! [D: nat,S2: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ) ).
% minf(10)
thf(fact_4521_minf_I10_J,axiom,
! [D: int,S2: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ) ).
% minf(10)
thf(fact_4522_minf_I9_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ X5 @ Z )
=> ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ).
% minf(9)
thf(fact_4523_minf_I9_J,axiom,
! [D: real,S2: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ X5 @ Z )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ).
% minf(9)
thf(fact_4524_minf_I9_J,axiom,
! [D: rat,S2: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ X5 @ Z )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ).
% minf(9)
thf(fact_4525_minf_I9_J,axiom,
! [D: nat,S2: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ).
% minf(9)
thf(fact_4526_minf_I9_J,axiom,
! [D: int,S2: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ).
% minf(9)
thf(fact_4527_pinf_I10_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ Z @ X5 )
=> ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ) ).
% pinf(10)
thf(fact_4528_pinf_I10_J,axiom,
! [D: real,S2: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ Z @ X5 )
=> ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ) ).
% pinf(10)
thf(fact_4529_pinf_I10_J,axiom,
! [D: rat,S2: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z @ X5 )
=> ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ) ).
% pinf(10)
thf(fact_4530_pinf_I10_J,axiom,
! [D: nat,S2: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ) ).
% pinf(10)
thf(fact_4531_pinf_I10_J,axiom,
! [D: int,S2: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ) ).
% pinf(10)
thf(fact_4532_pinf_I9_J,axiom,
! [D: code_integer,S2: code_integer] :
? [Z: code_integer] :
! [X5: code_integer] :
( ( ord_le6747313008572928689nteger @ Z @ X5 )
=> ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S2 ) ) ) ) ).
% pinf(9)
thf(fact_4533_pinf_I9_J,axiom,
! [D: real,S2: real] :
? [Z: real] :
! [X5: real] :
( ( ord_less_real @ Z @ X5 )
=> ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ).
% pinf(9)
thf(fact_4534_pinf_I9_J,axiom,
! [D: rat,S2: rat] :
? [Z: rat] :
! [X5: rat] :
( ( ord_less_rat @ Z @ X5 )
=> ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ).
% pinf(9)
thf(fact_4535_pinf_I9_J,axiom,
! [D: nat,S2: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ).
% pinf(9)
thf(fact_4536_pinf_I9_J,axiom,
! [D: int,S2: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ).
% pinf(9)
thf(fact_4537_dvd__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( ( divide6298287555418463151nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4538_dvd__div__eq__0__iff,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ( ( ( divide1717551699836669952omplex @ A @ B )
= zero_zero_complex )
= ( A = zero_zero_complex ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4539_dvd__div__eq__0__iff,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( A = zero_zero_real ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4540_dvd__div__eq__0__iff,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( ( divide_divide_rat @ A @ B )
= zero_zero_rat )
= ( A = zero_zero_rat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4541_dvd__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4542_dvd__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_4543_unit__mult__right__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( ( times_3573771949741848930nteger @ B @ A )
= ( times_3573771949741848930nteger @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_4544_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_4545_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_4546_unit__mult__left__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( ( times_3573771949741848930nteger @ A @ B )
= ( times_3573771949741848930nteger @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_4547_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_4548_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_4549_mult__unit__dvd__iff_H,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_4550_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_4551_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_4552_dvd__mult__unit__iff_H,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_4553_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_4554_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_4555_mult__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_4556_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_4557_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_4558_dvd__mult__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_4559_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_4560_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_4561_is__unit__mult__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
& ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% is_unit_mult_iff
thf(fact_4562_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_4563_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_4564_dvd__div__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_4565_dvd__div__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
= ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_4566_dvd__div__mult,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
= ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% dvd_div_mult
thf(fact_4567_div__mult__swap,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_4568_div__mult__swap,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_4569_div__mult__swap,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% div_mult_swap
thf(fact_4570_div__div__eq__right,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_4571_div__div__eq__right,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_4572_div__div__eq__right,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
= ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% div_div_eq_right
thf(fact_4573_dvd__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_4574_dvd__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_4575_dvd__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% dvd_div_mult2_eq
thf(fact_4576_dvd__mult__imp__div,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
=> ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_4577_dvd__mult__imp__div,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
=> ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_4578_dvd__mult__imp__div,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
=> ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% dvd_mult_imp_div
thf(fact_4579_div__mult__div__if__dvd,axiom,
! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( dvd_dvd_Code_integer @ D @ C )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_4580_div__mult__div__if__dvd,axiom,
! [B: nat,A: nat,D: nat,C: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ D @ C )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_4581_div__mult__div__if__dvd,axiom,
! [B: int,A: int,D: int,C: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ D @ C )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_4582_unit__div__cancel,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= ( divide6298287555418463151nteger @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_4583_unit__div__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_4584_unit__div__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_div_cancel
thf(fact_4585_div__unit__dvd__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_4586_div__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_4587_div__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% div_unit_dvd_iff
thf(fact_4588_dvd__div__unit__iff,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
= ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_4589_dvd__div__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_4590_dvd__div__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_div_unit_iff
thf(fact_4591_div__plus__div__distrib__dvd__right,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_4592_div__plus__div__distrib__dvd__right,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ B )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_4593_div__plus__div__distrib__dvd__right,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ B )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_4594_div__plus__div__distrib__dvd__left,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ A )
=> ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
= ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_4595_div__plus__div__distrib__dvd__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ A )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_4596_div__plus__div__distrib__dvd__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_4597_div__power,axiom,
! [B: code_integer,A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
= ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% div_power
thf(fact_4598_div__power,axiom,
! [B: nat,A: nat,N: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
= ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% div_power
thf(fact_4599_div__power,axiom,
! [B: int,A: int,N: nat] :
( ( dvd_dvd_int @ B @ A )
=> ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
= ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% div_power
thf(fact_4600_dvd__power__le,axiom,
! [X: code_integer,Y4: code_integer,N: nat,M: nat] :
( ( dvd_dvd_Code_integer @ X @ Y4 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y4 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_4601_dvd__power__le,axiom,
! [X: nat,Y4: nat,N: nat,M: nat] :
( ( dvd_dvd_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y4 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_4602_dvd__power__le,axiom,
! [X: real,Y4: real,N: nat,M: nat] :
( ( dvd_dvd_real @ X @ Y4 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_4603_dvd__power__le,axiom,
! [X: int,Y4: int,N: nat,M: nat] :
( ( dvd_dvd_int @ X @ Y4 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y4 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_4604_dvd__power__le,axiom,
! [X: complex,Y4: complex,N: nat,M: nat] :
( ( dvd_dvd_complex @ X @ Y4 )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y4 @ M ) ) ) ) ).
% dvd_power_le
thf(fact_4605_power__le__dvd,axiom,
! [A: code_integer,N: nat,B: code_integer,M: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_4606_power__le__dvd,axiom,
! [A: nat,N: nat,B: nat,M: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_4607_power__le__dvd,axiom,
! [A: real,N: nat,B: real,M: nat] :
( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_4608_power__le__dvd,axiom,
! [A: int,N: nat,B: int,M: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_4609_power__le__dvd,axiom,
! [A: complex,N: nat,B: complex,M: nat] :
( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_4610_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: code_integer] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_4611_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_4612_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_4613_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_4614_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: complex] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_4615_mod__eq__dvd__iff,axiom,
! [A: int,C: int,B: int] :
( ( ( modulo_modulo_int @ A @ C )
= ( modulo_modulo_int @ B @ C ) )
= ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% mod_eq_dvd_iff
thf(fact_4616_mod__eq__dvd__iff,axiom,
! [A: code_integer,C: code_integer,B: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ C )
= ( modulo364778990260209775nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% mod_eq_dvd_iff
thf(fact_4617_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_4618_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_4619_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_4620_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_4621_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_4622_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_4623_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_4624_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_4625_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) )
| ( ( times_times_nat @ B @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_4626_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X: nat,Y4: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) )
=> ? [X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_4627_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
= D3 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_4628_zdvd__period,axiom,
! [A: int,D: int,X: int,T: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_4629_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_4630_unit__dvdE,axiom,
! [A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [C3: code_integer] :
( B
!= ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_4631_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C3: nat] :
( B
!= ( times_times_nat @ A @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_4632_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C3: int] :
( B
!= ( times_times_int @ A @ C3 ) ) ) ) ).
% unit_dvdE
thf(fact_4633_unity__coeff__ex,axiom,
! [P: code_integer > $o,L2: code_integer] :
( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X2 ) ) )
= ( ? [X2: code_integer] :
( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4634_unity__coeff__ex,axiom,
! [P: complex > $o,L2: complex] :
( ( ? [X2: complex] : ( P @ ( times_times_complex @ L2 @ X2 ) ) )
= ( ? [X2: complex] :
( ( dvd_dvd_complex @ L2 @ ( plus_plus_complex @ X2 @ zero_zero_complex ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4635_unity__coeff__ex,axiom,
! [P: real > $o,L2: real] :
( ( ? [X2: real] : ( P @ ( times_times_real @ L2 @ X2 ) ) )
= ( ? [X2: real] :
( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X2 @ zero_zero_real ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4636_unity__coeff__ex,axiom,
! [P: rat > $o,L2: rat] :
( ( ? [X2: rat] : ( P @ ( times_times_rat @ L2 @ X2 ) ) )
= ( ? [X2: rat] :
( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4637_unity__coeff__ex,axiom,
! [P: nat > $o,L2: nat] :
( ( ? [X2: nat] : ( P @ ( times_times_nat @ L2 @ X2 ) ) )
= ( ? [X2: nat] :
( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4638_unity__coeff__ex,axiom,
! [P: int > $o,L2: int] :
( ( ? [X2: int] : ( P @ ( times_times_int @ L2 @ X2 ) ) )
= ( ? [X2: int] :
( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X2 @ zero_zero_int ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_4639_dvd__div__eq__mult,axiom,
! [A: code_integer,B: code_integer,C: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= C )
= ( B
= ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_4640_dvd__div__eq__mult,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( ( divide_divide_nat @ B @ A )
= C )
= ( B
= ( times_times_nat @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_4641_dvd__div__eq__mult,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( ( divide_divide_int @ B @ A )
= C )
= ( B
= ( times_times_int @ C @ A ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_4642_div__dvd__iff__mult,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( B != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
= ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_4643_div__dvd__iff__mult,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_4644_div__dvd__iff__mult,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_4645_dvd__div__iff__mult,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( C != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ C @ B )
=> ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_4646_dvd__div__iff__mult,axiom,
! [C: nat,B: nat,A: nat] :
( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ C @ B )
=> ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_4647_dvd__div__iff__mult,axiom,
! [C: int,B: int,A: int] :
( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ C @ B )
=> ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
= ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_4648_dvd__div__div__eq__mult,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( C != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ A @ B )
=> ( ( dvd_dvd_Code_integer @ C @ D )
=> ( ( ( divide6298287555418463151nteger @ B @ A )
= ( divide6298287555418463151nteger @ D @ C ) )
= ( ( times_3573771949741848930nteger @ B @ C )
= ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_4649_dvd__div__div__eq__mult,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( A != zero_zero_nat )
=> ( ( C != zero_zero_nat )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( ( ( divide_divide_nat @ B @ A )
= ( divide_divide_nat @ D @ C ) )
= ( ( times_times_nat @ B @ C )
= ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_4650_dvd__div__div__eq__mult,axiom,
! [A: int,C: int,B: int,D: int] :
( ( A != zero_zero_int )
=> ( ( C != zero_zero_int )
=> ( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( ( ( divide_divide_int @ B @ A )
= ( divide_divide_int @ D @ C ) )
= ( ( times_times_int @ B @ C )
= ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_4651_unit__div__eq__0__iff,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ) ).
% unit_div_eq_0_iff
thf(fact_4652_unit__div__eq__0__iff,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= zero_zero_nat )
= ( A = zero_zero_nat ) ) ) ).
% unit_div_eq_0_iff
thf(fact_4653_unit__div__eq__0__iff,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= zero_zero_int )
= ( A = zero_zero_int ) ) ) ).
% unit_div_eq_0_iff
thf(fact_4654_even__numeral,axiom,
! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_4655_even__numeral,axiom,
! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_4656_even__numeral,axiom,
! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% even_numeral
thf(fact_4657_inf__period_I4_J,axiom,
! [D: code_integer,D4: code_integer,T: code_integer] :
( ( dvd_dvd_Code_integer @ D @ D4 )
=> ! [X5: code_integer,K4: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4658_inf__period_I4_J,axiom,
! [D: real,D4: real,T: real] :
( ( dvd_dvd_real @ D @ D4 )
=> ! [X5: real,K4: real] :
( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4659_inf__period_I4_J,axiom,
! [D: rat,D4: rat,T: rat] :
( ( dvd_dvd_rat @ D @ D4 )
=> ! [X5: rat,K4: rat] :
( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4660_inf__period_I4_J,axiom,
! [D: int,D4: int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int,K4: int] :
( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% inf_period(4)
thf(fact_4661_inf__period_I3_J,axiom,
! [D: code_integer,D4: code_integer,T: code_integer] :
( ( dvd_dvd_Code_integer @ D @ D4 )
=> ! [X5: code_integer,K4: code_integer] :
( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
= ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4662_inf__period_I3_J,axiom,
! [D: real,D4: real,T: real] :
( ( dvd_dvd_real @ D @ D4 )
=> ! [X5: real,K4: real] :
( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
= ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4663_inf__period_I3_J,axiom,
! [D: rat,D4: rat,T: rat] :
( ( dvd_dvd_rat @ D @ D4 )
=> ! [X5: rat,K4: rat] :
( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
= ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4664_inf__period_I3_J,axiom,
! [D: int,D4: int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int,K4: int] :
( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).
% inf_period(3)
thf(fact_4665_unit__eq__div1,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ A @ B )
= C )
= ( A
= ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_4666_unit__eq__div1,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( ( divide_divide_nat @ A @ B )
= C )
= ( A
= ( times_times_nat @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_4667_unit__eq__div1,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( ( divide_divide_int @ A @ B )
= C )
= ( A
= ( times_times_int @ C @ B ) ) ) ) ).
% unit_eq_div1
thf(fact_4668_unit__eq__div2,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( A
= ( divide6298287555418463151nteger @ C @ B ) )
= ( ( times_3573771949741848930nteger @ A @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_4669_unit__eq__div2,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( A
= ( divide_divide_nat @ C @ B ) )
= ( ( times_times_nat @ A @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_4670_unit__eq__div2,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( A
= ( divide_divide_int @ C @ B ) )
= ( ( times_times_int @ A @ B )
= C ) ) ) ).
% unit_eq_div2
thf(fact_4671_div__mult__unit2,axiom,
! [C: code_integer,B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_4672_div__mult__unit2,axiom,
! [C: nat,B: nat,A: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_4673_div__mult__unit2,axiom,
! [C: int,B: int,A: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% div_mult_unit2
thf(fact_4674_unit__div__commute,axiom,
! [B: code_integer,A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_4675_unit__div__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
= ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_4676_unit__div__commute,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
= ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% unit_div_commute
thf(fact_4677_unit__div__mult__swap,axiom,
! [C: code_integer,A: code_integer,B: code_integer] :
( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_4678_unit__div__mult__swap,axiom,
! [C: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
= ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_4679_unit__div__mult__swap,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
= ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% unit_div_mult_swap
thf(fact_4680_is__unit__div__mult2__eq,axiom,
! [B: code_integer,C: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
= ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_4681_is__unit__div__mult2__eq,axiom,
! [B: nat,C: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ C @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_4682_is__unit__div__mult2__eq,axiom,
! [B: int,C: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ C @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_4683_unit__imp__mod__eq__0,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( modulo_modulo_nat @ A @ B )
= zero_zero_nat ) ) ).
% unit_imp_mod_eq_0
thf(fact_4684_unit__imp__mod__eq__0,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( modulo_modulo_int @ A @ B )
= zero_zero_int ) ) ).
% unit_imp_mod_eq_0
thf(fact_4685_unit__imp__mod__eq__0,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( modulo364778990260209775nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% unit_imp_mod_eq_0
thf(fact_4686_is__unit__power__iff,axiom,
! [A: code_integer,N: nat] :
( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
= ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_4687_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_4688_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_4689_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_4690_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_4691_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_4692_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D3: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_4693_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_4694_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect_nat
@ ^ [N2: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_4695_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q2: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( ( modulo_modulo_nat @ M @ Q2 )
= ( modulo_modulo_nat @ N @ Q2 ) )
= ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_4696_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_4697_even__zero,axiom,
dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% even_zero
thf(fact_4698_even__zero,axiom,
dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% even_zero
thf(fact_4699_even__zero,axiom,
dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% even_zero
thf(fact_4700_is__unitE,axiom,
! [A: code_integer,C: code_integer] :
( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
=> ~ ( ( A != zero_z3403309356797280102nteger )
=> ! [B2: code_integer] :
( ( B2 != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
= B2 )
=> ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 )
= A )
=> ( ( ( times_3573771949741848930nteger @ A @ B2 )
= one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ C @ A )
!= ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_4701_is__unitE,axiom,
! [A: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [B2: nat] :
( ( B2 != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B2 @ one_one_nat )
=> ( ( ( divide_divide_nat @ one_one_nat @ A )
= B2 )
=> ( ( ( divide_divide_nat @ one_one_nat @ B2 )
= A )
=> ( ( ( times_times_nat @ A @ B2 )
= one_one_nat )
=> ( ( divide_divide_nat @ C @ A )
!= ( times_times_nat @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_4702_is__unitE,axiom,
! [A: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [B2: int] :
( ( B2 != zero_zero_int )
=> ( ( dvd_dvd_int @ B2 @ one_one_int )
=> ( ( ( divide_divide_int @ one_one_int @ A )
= B2 )
=> ( ( ( divide_divide_int @ one_one_int @ B2 )
= A )
=> ( ( ( times_times_int @ A @ B2 )
= one_one_int )
=> ( ( divide_divide_int @ C @ A )
!= ( times_times_int @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_4703_is__unit__div__mult__cancel__left,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_4704_is__unit__div__mult__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_4705_is__unit__div__mult__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_4706_is__unit__div__mult__cancel__right,axiom,
! [A: code_integer,B: code_integer] :
( ( A != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
=> ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
= ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_4707_is__unit__div__mult__cancel__right,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
= ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_4708_is__unit__div__mult__cancel__right,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
= ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_4709_evenE,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: code_integer] :
( A
!= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% evenE
thf(fact_4710_evenE,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: nat] :
( A
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% evenE
thf(fact_4711_evenE,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: int] :
( A
!= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% evenE
thf(fact_4712_odd__one,axiom,
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% odd_one
thf(fact_4713_odd__one,axiom,
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% odd_one
thf(fact_4714_odd__one,axiom,
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% odd_one
thf(fact_4715_odd__even__add,axiom,
! [A: code_integer,B: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_4716_odd__even__add,axiom,
! [A: nat,B: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_4717_odd__even__add,axiom,
! [A: int,B: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
=> ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% odd_even_add
thf(fact_4718_bit__eq__rec,axiom,
( ( ^ [Y6: code_integer,Z3: code_integer] : Y6 = Z3 )
= ( ^ [A4: code_integer,B3: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
& ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( divide6298287555418463151nteger @ B3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_4719_bit__eq__rec,axiom,
( ( ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
= ( ^ [A4: nat,B3: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
& ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_4720_bit__eq__rec,axiom,
( ( ^ [Y6: int,Z3: int] : Y6 = Z3 )
= ( ^ [A4: int,B3: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
& ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( divide_divide_int @ B3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_4721_dvd__power__iff,axiom,
! [X: code_integer,M: nat,N: nat] :
( ( X != zero_z3403309356797280102nteger )
=> ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
= ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_4722_dvd__power__iff,axiom,
! [X: nat,M: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_4723_dvd__power__iff,axiom,
! [X: int,M: nat,N: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_4724_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% subset_decode_imp_le
thf(fact_4725_dvd__power,axiom,
! [N: nat,X: code_integer] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_Code_integer ) )
=> ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% dvd_power
thf(fact_4726_dvd__power,axiom,
! [N: nat,X: rat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_rat ) )
=> ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% dvd_power
thf(fact_4727_dvd__power,axiom,
! [N: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% dvd_power
thf(fact_4728_dvd__power,axiom,
! [N: nat,X: real] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_real ) )
=> ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% dvd_power
thf(fact_4729_dvd__power,axiom,
! [N: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% dvd_power
thf(fact_4730_dvd__power,axiom,
! [N: nat,X: complex] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_complex ) )
=> ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% dvd_power
thf(fact_4731_div2__even__ext__nat,axiom,
! [X: nat,Y4: nat] :
( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y4 ) )
=> ( X = Y4 ) ) ) ).
% div2_even_ext_nat
thf(fact_4732_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_4733_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_4734_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_4735_dvd__minus__add,axiom,
! [Q2: nat,N: nat,R2: nat,M: nat] :
( ( ord_less_eq_nat @ Q2 @ N )
=> ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
=> ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
= ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_4736_power__dvd__imp__le,axiom,
! [I2: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_4737_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M: nat] :
( ( ord_less_nat @ R2 @ N )
=> ( ( ord_less_eq_nat @ R2 @ M )
=> ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
=> ( ( modulo_modulo_nat @ M @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_4738_bset_I9_J,axiom,
! [D: int,D4: int,B5: set_int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% bset(9)
thf(fact_4739_bset_I10_J,axiom,
! [D: int,D4: int,B5: set_int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ B5 )
=> ( X5
!= ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% bset(10)
thf(fact_4740_aset_I9_J,axiom,
! [D: int,D4: int,A2: set_int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% aset(9)
thf(fact_4741_aset_I10_J,axiom,
! [D: int,D4: int,A2: set_int,T: int] :
( ( dvd_dvd_int @ D @ D4 )
=> ! [X5: int] :
( ! [Xa3: int] :
( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
=> ! [Xb3: int] :
( ( member_int @ Xb3 @ A2 )
=> ( X5
!= ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
=> ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% aset(10)
thf(fact_4742_even__two__times__div__two,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_4743_even__two__times__div__two,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_4744_even__two__times__div__two,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= A ) ) ).
% even_two_times_div_two
thf(fact_4745_even__iff__mod__2__eq__zero,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_4746_even__iff__mod__2__eq__zero,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_4747_even__iff__mod__2__eq__zero,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_4748_odd__iff__mod__2__eq__one,axiom,
! [A: nat] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_4749_odd__iff__mod__2__eq__one,axiom,
! [A: int] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_4750_odd__iff__mod__2__eq__one,axiom,
! [A: code_integer] :
( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
= ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_4751_power__mono__odd,axiom,
! [N: nat,A: real,B: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_4752_power__mono__odd,axiom,
! [N: nat,A: rat,B: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_4753_power__mono__odd,axiom,
! [N: nat,A: int,B: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono_odd
thf(fact_4754_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_4755_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_4756_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_4757_even__unset__bit__iff,axiom,
! [M: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_4758_even__unset__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_4759_even__unset__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
| ( M = zero_zero_nat ) ) ) ).
% even_unset_bit_iff
thf(fact_4760_even__set__bit__iff,axiom,
! [M: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_4761_even__set__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_4762_even__set__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
& ( M != zero_zero_nat ) ) ) ).
% even_set_bit_iff
thf(fact_4763_even__flip__bit__iff,axiom,
! [M: nat,A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
= ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_4764_even__flip__bit__iff,axiom,
! [M: nat,A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_4765_even__flip__bit__iff,axiom,
! [M: nat,A: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
!= ( M = zero_zero_nat ) ) ) ).
% even_flip_bit_iff
thf(fact_4766_even__diff__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% even_diff_iff
thf(fact_4767_oddE,axiom,
! [A: code_integer] :
( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: code_integer] :
( A
!= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ one_one_Code_integer ) ) ) ).
% oddE
thf(fact_4768_oddE,axiom,
! [A: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: nat] :
( A
!= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ one_one_nat ) ) ) ).
% oddE
thf(fact_4769_oddE,axiom,
! [A: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ~ ! [B2: int] :
( A
!= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) ) ) ).
% oddE
thf(fact_4770_mod2__eq__if,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_nat ) ) ) ).
% mod2_eq_if
thf(fact_4771_mod2__eq__if,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ) ) ).
% mod2_eq_if
thf(fact_4772_mod2__eq__if,axiom,
! [A: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) )
& ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ) ) ).
% mod2_eq_if
thf(fact_4773_parity__cases,axiom,
! [A: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= zero_zero_nat ) )
=> ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
!= one_one_nat ) ) ) ).
% parity_cases
thf(fact_4774_parity__cases,axiom,
! [A: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= zero_zero_int ) )
=> ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
=> ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
!= one_one_int ) ) ) ).
% parity_cases
thf(fact_4775_parity__cases,axiom,
! [A: code_integer] :
( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= zero_z3403309356797280102nteger ) )
=> ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
=> ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
!= one_one_Code_integer ) ) ) ).
% parity_cases
thf(fact_4776_zero__le__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_4777_zero__le__power__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_4778_zero__le__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% zero_le_power_eq
thf(fact_4779_zero__le__odd__power,axiom,
! [N: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% zero_le_odd_power
thf(fact_4780_zero__le__odd__power,axiom,
! [N: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% zero_le_odd_power
thf(fact_4781_zero__le__odd__power,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% zero_le_odd_power
thf(fact_4782_zero__le__even__power,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_4783_zero__le__even__power,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_4784_zero__le__even__power,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_even_power
thf(fact_4785_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
= ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_4786_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_4787_zero__less__power__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_real ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_4788_zero__less__power__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_rat ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_4789_zero__less__power__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
= ( ( N = zero_zero_nat )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A != zero_zero_int ) )
| ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% zero_less_power_eq
thf(fact_4790_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
=> ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_4791_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_4792_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_4793_even__mask__div__iff_H,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% even_mask_div_iff'
thf(fact_4794_power__le__zero__eq,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_real @ A @ zero_zero_real ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_real ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_4795_power__le__zero__eq,axiom,
! [A: rat,N: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_rat @ A @ zero_zero_rat ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_rat ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_4796_power__le__zero__eq,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
= ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( ord_less_eq_int @ A @ zero_zero_int ) )
| ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
& ( A = zero_zero_int ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_4797_option_Osize__gen_I1_J,axiom,
! [X: nat > nat] :
( ( size_option_nat @ X @ none_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_4798_option_Osize__gen_I1_J,axiom,
! [X: product_prod_nat_nat > nat] :
( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_4799_option_Osize__gen_I1_J,axiom,
! [X: num > nat] :
( ( size_option_num @ X @ none_num )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_4800_even__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suc @ zero_zero_nat ) )
=> ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_4801_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
= zero_z3403309356797280102nteger )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4802_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4803_even__mask__div__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ).
% even_mask_div_iff
thf(fact_4804_even__mult__exp__div__exp__iff,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
= zero_z3403309356797280102nteger )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_4805_even__mult__exp__div__exp__iff,axiom,
! [A: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= zero_zero_nat )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_4806_even__mult__exp__div__exp__iff,axiom,
! [A: int,M: nat,N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
= ( ( ord_less_nat @ N @ M )
| ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
= zero_zero_int )
| ( ( ord_less_eq_nat @ M @ N )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_4807_ex__min__if__finite,axiom,
! [S3: set_real] :
( ( finite_finite_real @ S3 )
=> ( ( S3 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ S3 )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S3 )
& ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4808_ex__min__if__finite,axiom,
! [S3: set_rat] :
( ( finite_finite_rat @ S3 )
=> ( ( S3 != bot_bot_set_rat )
=> ? [X3: rat] :
( ( member_rat @ X3 @ S3 )
& ~ ? [Xa: rat] :
( ( member_rat @ Xa @ S3 )
& ( ord_less_rat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4809_ex__min__if__finite,axiom,
! [S3: set_num] :
( ( finite_finite_num @ S3 )
=> ( ( S3 != bot_bot_set_num )
=> ? [X3: num] :
( ( member_num @ X3 @ S3 )
& ~ ? [Xa: num] :
( ( member_num @ Xa @ S3 )
& ( ord_less_num @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4810_ex__min__if__finite,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S3 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S3 )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4811_ex__min__if__finite,axiom,
! [S3: set_int] :
( ( finite_finite_int @ S3 )
=> ( ( S3 != bot_bot_set_int )
=> ? [X3: int] :
( ( member_int @ X3 @ S3 )
& ~ ? [Xa: int] :
( ( member_int @ Xa @ S3 )
& ( ord_less_int @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4812_vebt__buildup_Oelims,axiom,
! [X: nat,Y4: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y4 )
=> ( ( ( X = zero_zero_nat )
=> ( Y4
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ zero_zero_nat ) )
=> ( Y4
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va3: nat] :
( ( X
= ( suc @ ( suc @ Va3 ) ) )
=> ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y4
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y4
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_4813_signed__take__bit__rec,axiom,
( bit_ri6519982836138164636nteger
= ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_4814_signed__take__bit__rec,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_4815_flip__bit__0,axiom,
! [A: code_integer] :
( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
= ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_4816_flip__bit__0,axiom,
! [A: int] :
( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
= ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_4817_flip__bit__0,axiom,
! [A: nat] :
( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
= ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_4818_diff__shunt__var,axiom,
! [X: set_int,Y4: set_int] :
( ( ( minus_minus_set_int @ X @ Y4 )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X @ Y4 ) ) ).
% diff_shunt_var
thf(fact_4819_diff__shunt__var,axiom,
! [X: set_real,Y4: set_real] :
( ( ( minus_minus_set_real @ X @ Y4 )
= bot_bot_set_real )
= ( ord_less_eq_set_real @ X @ Y4 ) ) ).
% diff_shunt_var
thf(fact_4820_diff__shunt__var,axiom,
! [X: set_nat,Y4: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y4 ) ) ).
% diff_shunt_var
thf(fact_4821_add__scale__eq__noteq,axiom,
! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
( ( R2 != zero_zero_complex )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
!= ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4822_add__scale__eq__noteq,axiom,
! [R2: real,A: real,B: real,C: real,D: real] :
( ( R2 != zero_zero_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
!= ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4823_add__scale__eq__noteq,axiom,
! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
( ( R2 != zero_zero_rat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
!= ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4824_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4825_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C: int,D: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_4826_artanh__def,axiom,
( artanh_real
= ( ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% artanh_def
thf(fact_4827_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ N )
=> ( ( groups4538972089207619220nt_int
@ ^ [X2: int] : X2
@ ( set_or1266510415728281911st_int @ M @ N ) )
= ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_4828_intind,axiom,
! [I2: nat,N: nat,P: nat > $o,X: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( P @ X )
=> ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_4829_intind,axiom,
! [I2: nat,N: nat,P: int > $o,X: int] :
( ( ord_less_nat @ I2 @ N )
=> ( ( P @ X )
=> ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_4830_intind,axiom,
! [I2: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ N )
=> ( ( P @ X )
=> ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_4831_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_4832_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_4833_verit__minus__simplify_I4_J,axiom,
! [B: complex] :
( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_4834_verit__minus__simplify_I4_J,axiom,
! [B: rat] :
( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_4835_verit__minus__simplify_I4_J,axiom,
! [B: code_integer] :
( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_4836_compl__le__compl__iff,axiom,
! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y4 ) )
= ( ord_less_eq_set_nat @ Y4 @ X ) ) ).
% compl_le_compl_iff
thf(fact_4837_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4838_neg__le__iff__le,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4839_neg__le__iff__le,axiom,
! [B: rat,A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4840_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_4841_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4842_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4843_neg__less__iff__less,axiom,
! [B: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4844_neg__less__iff__less,axiom,
! [B: code_integer,A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_4845_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_4846_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_4847_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_4848_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_4849_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_4850_mult__minus__left,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4851_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4852_mult__minus__left,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4853_mult__minus__left,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4854_mult__minus__left,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_4855_minus__mult__minus,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( times_times_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4856_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4857_minus__mult__minus,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( times_times_complex @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4858_minus__mult__minus,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( times_times_rat @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4859_minus__mult__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( times_3573771949741848930nteger @ A @ B ) ) ).
% minus_mult_minus
thf(fact_4860_mult__minus__right,axiom,
! [A: real,B: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4861_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4862_mult__minus__right,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4863_mult__minus__right,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4864_mult__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_4865_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4866_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4867_add__minus__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4868_add__minus__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4869_add__minus__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_4870_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4871_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4872_minus__add__cancel,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4873_minus__add__cancel,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4874_minus__add__cancel,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_4875_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_4876_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_4877_minus__add__distrib,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_add_distrib
thf(fact_4878_minus__add__distrib,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_add_distrib
thf(fact_4879_minus__add__distrib,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% minus_add_distrib
thf(fact_4880_div__minus__minus,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ A @ B ) ) ).
% div_minus_minus
thf(fact_4881_div__minus__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( divide6298287555418463151nteger @ A @ B ) ) ).
% div_minus_minus
thf(fact_4882_mod__minus__minus,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% mod_minus_minus
thf(fact_4883_mod__minus__minus,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% mod_minus_minus
thf(fact_4884_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_4885_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_4886_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_4887_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_4888_real__add__minus__iff,axiom,
! [X: real,A: real] :
( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X = A ) ) ).
% real_add_minus_iff
thf(fact_4889_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_4890_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_4891_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_4892_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_4893_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_4894_of__bool__eq_I2_J,axiom,
( ( zero_n1201886186963655149omplex @ $true )
= one_one_complex ) ).
% of_bool_eq(2)
thf(fact_4895_of__bool__eq_I2_J,axiom,
( ( zero_n3304061248610475627l_real @ $true )
= one_one_real ) ).
% of_bool_eq(2)
thf(fact_4896_of__bool__eq_I2_J,axiom,
( ( zero_n2052037380579107095ol_rat @ $true )
= one_one_rat ) ).
% of_bool_eq(2)
thf(fact_4897_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_4898_of__bool__eq_I2_J,axiom,
( ( zero_n2684676970156552555ol_int @ $true )
= one_one_int ) ).
% of_bool_eq(2)
thf(fact_4899_of__bool__eq_I2_J,axiom,
( ( zero_n356916108424825756nteger @ $true )
= one_one_Code_integer ) ).
% of_bool_eq(2)
thf(fact_4900_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n1201886186963655149omplex @ P )
= one_one_complex )
= P ) ).
% of_bool_eq_1_iff
thf(fact_4901_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n3304061248610475627l_real @ P )
= one_one_real )
= P ) ).
% of_bool_eq_1_iff
thf(fact_4902_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2052037380579107095ol_rat @ P )
= one_one_rat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_4903_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= one_one_nat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_4904_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2684676970156552555ol_int @ P )
= one_one_int )
= P ) ).
% of_bool_eq_1_iff
thf(fact_4905_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n356916108424825756nteger @ P )
= one_one_Code_integer )
= P ) ).
% of_bool_eq_1_iff
thf(fact_4906_length__replicate,axiom,
! [N: nat,X: vEBT_VEBT] :
( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_4907_length__replicate,axiom,
! [N: nat,X: $o] :
( ( size_size_list_o @ ( replicate_o @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_4908_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_4909_length__replicate,axiom,
! [N: nat,X: int] :
( ( size_size_list_int @ ( replicate_int @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_4910_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_4911_neg__0__le__iff__le,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% neg_0_le_iff_le
thf(fact_4912_neg__0__le__iff__le,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% neg_0_le_iff_le
thf(fact_4913_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_4914_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4915_neg__le__0__iff__le,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4916_neg__le__0__iff__le,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4917_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_4918_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_4919_less__eq__neg__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_eq_neg_nonpos
thf(fact_4920_less__eq__neg__nonpos,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% less_eq_neg_nonpos
thf(fact_4921_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_4922_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4923_neg__less__eq__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4924_neg__less__eq__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4925_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_4926_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4927_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4928_neg__less__0__iff__less,axiom,
! [A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4929_neg__less__0__iff__less,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_0_iff_less
thf(fact_4930_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_4931_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_4932_neg__0__less__iff__less,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% neg_0_less_iff_less
thf(fact_4933_neg__0__less__iff__less,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% neg_0_less_iff_less
thf(fact_4934_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_4935_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_4936_neg__less__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% neg_less_pos
thf(fact_4937_neg__less__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% neg_less_pos
thf(fact_4938_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_4939_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_4940_less__neg__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% less_neg_neg
thf(fact_4941_less__neg__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% less_neg_neg
thf(fact_4942_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_4943_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_4944_add_Oright__inverse,axiom,
! [A: complex] :
( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
= zero_zero_complex ) ).
% add.right_inverse
thf(fact_4945_add_Oright__inverse,axiom,
! [A: rat] :
( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
= zero_zero_rat ) ).
% add.right_inverse
thf(fact_4946_add_Oright__inverse,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
= zero_z3403309356797280102nteger ) ).
% add.right_inverse
thf(fact_4947_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_4948_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_4949_ab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_left_minus
thf(fact_4950_ab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_left_minus
thf(fact_4951_ab__left__minus,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= zero_z3403309356797280102nteger ) ).
% ab_left_minus
thf(fact_4952_verit__minus__simplify_I3_J,axiom,
! [B: real] :
( ( minus_minus_real @ zero_zero_real @ B )
= ( uminus_uminus_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4953_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4954_verit__minus__simplify_I3_J,axiom,
! [B: complex] :
( ( minus_minus_complex @ zero_zero_complex @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4955_verit__minus__simplify_I3_J,axiom,
! [B: rat] :
( ( minus_minus_rat @ zero_zero_rat @ B )
= ( uminus_uminus_rat @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4956_verit__minus__simplify_I3_J,axiom,
! [B: code_integer] :
( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
= ( uminus1351360451143612070nteger @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_4957_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4958_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4959_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4960_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4961_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_4962_mult__minus1__right,axiom,
! [Z2: real] :
( ( times_times_real @ Z2 @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ Z2 ) ) ).
% mult_minus1_right
thf(fact_4963_mult__minus1__right,axiom,
! [Z2: int] :
( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1_right
thf(fact_4964_mult__minus1__right,axiom,
! [Z2: complex] :
( ( times_times_complex @ Z2 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ Z2 ) ) ).
% mult_minus1_right
thf(fact_4965_mult__minus1__right,axiom,
! [Z2: rat] :
( ( times_times_rat @ Z2 @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ Z2 ) ) ).
% mult_minus1_right
thf(fact_4966_mult__minus1__right,axiom,
! [Z2: code_integer] :
( ( times_3573771949741848930nteger @ Z2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ Z2 ) ) ).
% mult_minus1_right
thf(fact_4967_mult__minus1,axiom,
! [Z2: real] :
( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z2 )
= ( uminus_uminus_real @ Z2 ) ) ).
% mult_minus1
thf(fact_4968_mult__minus1,axiom,
! [Z2: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
= ( uminus_uminus_int @ Z2 ) ) ).
% mult_minus1
thf(fact_4969_mult__minus1,axiom,
! [Z2: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z2 )
= ( uminus1482373934393186551omplex @ Z2 ) ) ).
% mult_minus1
thf(fact_4970_mult__minus1,axiom,
! [Z2: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z2 )
= ( uminus_uminus_rat @ Z2 ) ) ).
% mult_minus1
thf(fact_4971_mult__minus1,axiom,
! [Z2: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z2 )
= ( uminus1351360451143612070nteger @ Z2 ) ) ).
% mult_minus1
thf(fact_4972_diff__minus__eq__add,axiom,
! [A: real,B: real] :
( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
= ( plus_plus_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4973_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4974_diff__minus__eq__add,axiom,
! [A: complex,B: complex] :
( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( plus_plus_complex @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4975_diff__minus__eq__add,axiom,
! [A: rat,B: rat] :
( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( plus_plus_rat @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4976_diff__minus__eq__add,axiom,
! [A: code_integer,B: code_integer] :
( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_4977_uminus__add__conv__diff,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
= ( minus_minus_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4978_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4979_uminus__add__conv__diff,axiom,
! [A: complex,B: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( minus_minus_complex @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4980_uminus__add__conv__diff,axiom,
! [A: rat,B: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( minus_minus_rat @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4981_uminus__add__conv__diff,axiom,
! [A: code_integer,B: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( minus_8373710615458151222nteger @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_4982_div__minus1__right,axiom,
! [A: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ A ) ) ).
% div_minus1_right
thf(fact_4983_div__minus1__right,axiom,
! [A: code_integer] :
( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ A ) ) ).
% div_minus1_right
thf(fact_4984_divide__minus1,axiom,
! [X: real] :
( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ X ) ) ).
% divide_minus1
thf(fact_4985_divide__minus1,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% divide_minus1
thf(fact_4986_divide__minus1,axiom,
! [X: rat] :
( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ X ) ) ).
% divide_minus1
thf(fact_4987_sum_Oempty,axiom,
! [G: nat > complex] :
( ( groups2073611262835488442omplex @ G @ bot_bot_set_nat )
= zero_zero_complex ) ).
% sum.empty
thf(fact_4988_sum_Oempty,axiom,
! [G: nat > rat] :
( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
= zero_zero_rat ) ).
% sum.empty
thf(fact_4989_sum_Oempty,axiom,
! [G: nat > int] :
( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
= zero_zero_int ) ).
% sum.empty
thf(fact_4990_sum_Oempty,axiom,
! [G: int > complex] :
( ( groups3049146728041665814omplex @ G @ bot_bot_set_int )
= zero_zero_complex ) ).
% sum.empty
thf(fact_4991_sum_Oempty,axiom,
! [G: int > real] :
( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
= zero_zero_real ) ).
% sum.empty
thf(fact_4992_sum_Oempty,axiom,
! [G: int > rat] :
( ( groups3906332499630173760nt_rat @ G @ bot_bot_set_int )
= zero_zero_rat ) ).
% sum.empty
thf(fact_4993_sum_Oempty,axiom,
! [G: int > nat] :
( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
= zero_zero_nat ) ).
% sum.empty
thf(fact_4994_sum_Oempty,axiom,
! [G: real > complex] :
( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
= zero_zero_complex ) ).
% sum.empty
thf(fact_4995_sum_Oempty,axiom,
! [G: real > real] :
( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
= zero_zero_real ) ).
% sum.empty
thf(fact_4996_sum_Oempty,axiom,
! [G: real > rat] :
( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
= zero_zero_rat ) ).
% sum.empty
thf(fact_4997_minus__mod__self1,axiom,
! [B: int,A: int] :
( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_mod_self1
thf(fact_4998_minus__mod__self1,axiom,
! [B: code_integer,A: code_integer] :
( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% minus_mod_self1
thf(fact_4999_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_5000_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_5001_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_5002_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_5003_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_5004_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_5005_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_5006_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_5007_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_5008_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_5009_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_5010_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_5011_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_5012_ln__one,axiom,
( ( ln_ln_real @ one_one_real )
= zero_zero_real ) ).
% ln_one
thf(fact_5013_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n1201886186963655149omplex @ ~ P )
= ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% of_bool_not_iff
thf(fact_5014_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n3304061248610475627l_real @ ~ P )
= ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% of_bool_not_iff
thf(fact_5015_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n2052037380579107095ol_rat @ ~ P )
= ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% of_bool_not_iff
thf(fact_5016_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n2684676970156552555ol_int @ ~ P )
= ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% of_bool_not_iff
thf(fact_5017_of__bool__not__iff,axiom,
! [P: $o] :
( ( zero_n356916108424825756nteger @ ~ P )
= ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% of_bool_not_iff
thf(fact_5018_Suc__0__mod__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
= ( zero_n2687167440665602831ol_nat
@ ( N
!= ( suc @ zero_zero_nat ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_5019_signed__take__bit__of__minus__1,axiom,
! [N: nat] :
( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% signed_take_bit_of_minus_1
thf(fact_5020_signed__take__bit__of__minus__1,axiom,
! [N: nat] :
( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% signed_take_bit_of_minus_1
thf(fact_5021_in__set__replicate,axiom,
! [X: real,N: nat,Y4: real] :
( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y4 ) ) )
= ( ( X = Y4 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5022_in__set__replicate,axiom,
! [X: complex,N: nat,Y4: complex] :
( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N @ Y4 ) ) )
= ( ( X = Y4 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5023_in__set__replicate,axiom,
! [X: product_prod_nat_nat,N: nat,Y4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y4 ) ) )
= ( ( X = Y4 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5024_in__set__replicate,axiom,
! [X: int,N: nat,Y4: int] :
( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y4 ) ) )
= ( ( X = Y4 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5025_in__set__replicate,axiom,
! [X: nat,N: nat,Y4: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y4 ) ) )
= ( ( X = Y4 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5026_in__set__replicate,axiom,
! [X: vEBT_VEBT,N: nat,Y4: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y4 ) ) )
= ( ( X = Y4 )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_5027_Bex__set__replicate,axiom,
! [N: nat,A: int,P: int > $o] :
( ( ? [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_5028_Bex__set__replicate,axiom,
! [N: nat,A: nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_5029_Bex__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ? [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_5030_Ball__set__replicate,axiom,
! [N: nat,A: int,P: int > $o] :
( ( ! [X2: int] :
( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_5031_Ball__set__replicate,axiom,
! [N: nat,A: nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_5032_Ball__set__replicate,axiom,
! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
( ( ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_5033_nth__replicate,axiom,
! [I2: nat,N: nat,X: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_5034_nth__replicate,axiom,
! [I2: nat,N: nat,X: int] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_int @ ( replicate_int @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_5035_nth__replicate,axiom,
! [I2: nat,N: nat,X: vEBT_VEBT] :
( ( ord_less_nat @ I2 @ N )
=> ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_5036_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
= ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_5037_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_5038_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_5039_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_5040_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_5041_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_5042_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_5043_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= zero_zero_complex ) ).
% add_neg_numeral_special(8)
thf(fact_5044_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
= zero_zero_rat ) ).
% add_neg_numeral_special(8)
thf(fact_5045_add__neg__numeral__special_I8_J,axiom,
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
= zero_z3403309356797280102nteger ) ).
% add_neg_numeral_special(8)
thf(fact_5046_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_5047_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_5048_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% add_neg_numeral_special(7)
thf(fact_5049_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% add_neg_numeral_special(7)
thf(fact_5050_add__neg__numeral__special_I7_J,axiom,
( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% add_neg_numeral_special(7)
thf(fact_5051_diff__numeral__special_I12_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% diff_numeral_special(12)
thf(fact_5052_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_5053_diff__numeral__special_I12_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= zero_zero_complex ) ).
% diff_numeral_special(12)
thf(fact_5054_diff__numeral__special_I12_J,axiom,
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= zero_zero_rat ) ).
% diff_numeral_special(12)
thf(fact_5055_diff__numeral__special_I12_J,axiom,
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% diff_numeral_special(12)
thf(fact_5056_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ one_one_real ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5057_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5058_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5059_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5060_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_5061_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5062_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5063_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus1482373934393186551omplex @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5064_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_rat @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5065_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_5066_left__minus__one__mult__self,axiom,
! [N: nat,A: real] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5067_left__minus__one__mult__self,axiom,
! [N: nat,A: int] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5068_left__minus__one__mult__self,axiom,
! [N: nat,A: complex] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5069_left__minus__one__mult__self,axiom,
! [N: nat,A: rat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5070_left__minus__one__mult__self,axiom,
! [N: nat,A: code_integer] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
= A ) ).
% left_minus_one_mult_self
thf(fact_5071_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
= one_one_real ) ).
% minus_one_mult_self
thf(fact_5072_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
= one_one_int ) ).
% minus_one_mult_self
thf(fact_5073_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
= one_one_complex ) ).
% minus_one_mult_self
thf(fact_5074_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
= one_one_rat ) ).
% minus_one_mult_self
thf(fact_5075_minus__one__mult__self,axiom,
! [N: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
= one_one_Code_integer ) ).
% minus_one_mult_self
thf(fact_5076_mod__minus1__right,axiom,
! [A: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% mod_minus1_right
thf(fact_5077_mod__minus1__right,axiom,
! [A: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= zero_z3403309356797280102nteger ) ).
% mod_minus1_right
thf(fact_5078_max__number__of_I2_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
& ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( numeral_numeral_real @ U2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( numeral_numeral_real @ U2 ) ) ) ) ).
% max_number_of(2)
thf(fact_5079_max__number__of_I2_J,axiom,
! [U2: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( numera6620942414471956472nteger @ U2 ) ) ) ) ).
% max_number_of(2)
thf(fact_5080_max__number__of_I2_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
& ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( numeral_numeral_rat @ U2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( numeral_numeral_rat @ U2 ) ) ) ) ).
% max_number_of(2)
thf(fact_5081_max__number__of_I2_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( numeral_numeral_int @ U2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( numeral_numeral_int @ U2 ) ) ) ) ).
% max_number_of(2)
thf(fact_5082_max__number__of_I3_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U2 ) ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U2 ) ) @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_real @ V ) ) )
& ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U2 ) ) @ ( numeral_numeral_real @ V ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U2 ) ) @ ( numeral_numeral_real @ V ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ U2 ) ) ) ) ) ).
% max_number_of(3)
thf(fact_5083_max__number__of_I3_J,axiom,
! [U2: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U2 ) ) @ ( numera6620942414471956472nteger @ V ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U2 ) ) @ ( numera6620942414471956472nteger @ V ) )
= ( numera6620942414471956472nteger @ V ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U2 ) ) @ ( numera6620942414471956472nteger @ V ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U2 ) ) @ ( numera6620942414471956472nteger @ V ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U2 ) ) ) ) ) ).
% max_number_of(3)
thf(fact_5084_max__number__of_I3_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U2 ) ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U2 ) ) @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_rat @ V ) ) )
& ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U2 ) ) @ ( numeral_numeral_rat @ V ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U2 ) ) @ ( numeral_numeral_rat @ V ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ U2 ) ) ) ) ) ).
% max_number_of(3)
thf(fact_5085_max__number__of_I3_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U2 ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U2 ) ) @ ( numeral_numeral_int @ V ) )
= ( numeral_numeral_int @ V ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U2 ) ) @ ( numeral_numeral_int @ V ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U2 ) ) @ ( numeral_numeral_int @ V ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U2 ) ) ) ) ) ).
% max_number_of(3)
thf(fact_5086_max__number__of_I4_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
& ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
=> ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ U2 ) ) ) ) ) ).
% max_number_of(4)
thf(fact_5087_max__number__of_I4_J,axiom,
! [U2: num,V: num] :
( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
& ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
=> ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U2 ) ) ) ) ) ).
% max_number_of(4)
thf(fact_5088_max__number__of_I4_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
& ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
=> ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ U2 ) ) ) ) ) ).
% max_number_of(4)
thf(fact_5089_max__number__of_I4_J,axiom,
! [U2: num,V: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
=> ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ U2 ) ) ) ) ) ).
% max_number_of(4)
thf(fact_5090_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_5091_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_5092_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y4: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y4 ) )
= ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(168)
thf(fact_5093_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y4: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y4 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(168)
thf(fact_5094_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y4: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ Y4 ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(168)
thf(fact_5095_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y4: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y4 ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(168)
thf(fact_5096_semiring__norm_I168_J,axiom,
! [V: num,W2: num,Y4: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y4 ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(168)
thf(fact_5097_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5098_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5099_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5100_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5101_diff__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_5102_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5103_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5104_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5105_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5106_diff__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_5107_semiring__norm_I172_J,axiom,
! [V: num,W2: num,Y4: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y4 ) )
= ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) @ Y4 ) ) ).
% semiring_norm(172)
thf(fact_5108_semiring__norm_I172_J,axiom,
! [V: num,W2: num,Y4: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y4 ) )
= ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) @ Y4 ) ) ).
% semiring_norm(172)
thf(fact_5109_semiring__norm_I172_J,axiom,
! [V: num,W2: num,Y4: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ Y4 ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W2 ) ) @ Y4 ) ) ).
% semiring_norm(172)
thf(fact_5110_semiring__norm_I172_J,axiom,
! [V: num,W2: num,Y4: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y4 ) )
= ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) @ Y4 ) ) ).
% semiring_norm(172)
thf(fact_5111_semiring__norm_I172_J,axiom,
! [V: num,W2: num,Y4: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y4 ) )
= ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) @ Y4 ) ) ).
% semiring_norm(172)
thf(fact_5112_semiring__norm_I171_J,axiom,
! [V: num,W2: num,Y4: real] :
( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y4 ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(171)
thf(fact_5113_semiring__norm_I171_J,axiom,
! [V: num,W2: num,Y4: int] :
( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y4 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(171)
thf(fact_5114_semiring__norm_I171_J,axiom,
! [V: num,W2: num,Y4: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ Y4 ) )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(171)
thf(fact_5115_semiring__norm_I171_J,axiom,
! [V: num,W2: num,Y4: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y4 ) )
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(171)
thf(fact_5116_semiring__norm_I171_J,axiom,
! [V: num,W2: num,Y4: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y4 ) )
= ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(171)
thf(fact_5117_semiring__norm_I170_J,axiom,
! [V: num,W2: num,Y4: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Y4 ) )
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(170)
thf(fact_5118_semiring__norm_I170_J,axiom,
! [V: num,W2: num,Y4: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Y4 ) )
= ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(170)
thf(fact_5119_semiring__norm_I170_J,axiom,
! [V: num,W2: num,Y4: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ Y4 ) )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(170)
thf(fact_5120_semiring__norm_I170_J,axiom,
! [V: num,W2: num,Y4: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Y4 ) )
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(170)
thf(fact_5121_semiring__norm_I170_J,axiom,
! [V: num,W2: num,Y4: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ Y4 ) )
= ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W2 ) ) ) @ Y4 ) ) ).
% semiring_norm(170)
thf(fact_5122_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5123_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5124_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5125_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5126_mult__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_5127_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5128_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5129_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5130_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5131_mult__neg__numeral__simps_I2_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_5132_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5133_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5134_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5135_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5136_mult__neg__numeral__simps_I1_J,axiom,
! [M: num,N: num] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_5137_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5138_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5139_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5140_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_5141_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5142_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5143_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5144_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_5145_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_5146_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_5147_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_5148_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_5149_le__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_5150_le__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_5151_divide__le__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_5152_divide__le__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_5153_eq__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( A
= ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
!= zero_zero_real )
=> ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= B ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_5154_eq__divide__eq__numeral1_I2_J,axiom,
! [A: complex,B: complex,W2: num] :
( ( A
= ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
!= zero_zero_complex )
=> ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= B ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_5155_eq__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( A
= ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
!= zero_zero_rat )
=> ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= B ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_5156_divide__eq__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= A )
= ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
!= zero_zero_real )
=> ( B
= ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
& ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_5157_divide__eq__eq__numeral1_I2_J,axiom,
! [B: complex,W2: num,A: complex] :
( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= A )
= ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
!= zero_zero_complex )
=> ( B
= ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
& ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_5158_divide__eq__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= A )
= ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
!= zero_zero_rat )
=> ( B
= ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
& ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_5159_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_5160_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_5161_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_5162_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_5163_less__divide__eq__numeral1_I2_J,axiom,
! [A: real,B: real,W2: num] :
( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
= ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_5164_less__divide__eq__numeral1_I2_J,axiom,
! [A: rat,B: rat,W2: num] :
( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
= ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_5165_divide__less__eq__numeral1_I2_J,axiom,
! [B: real,W2: num,A: real] :
( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_5166_divide__less__eq__numeral1_I2_J,axiom,
! [B: rat,W2: num,A: rat] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_5167_power2__minus,axiom,
! [A: real] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5168_power2__minus,axiom,
! [A: int] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5169_power2__minus,axiom,
! [A: complex] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5170_power2__minus,axiom,
! [A: rat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5171_power2__minus,axiom,
! [A: code_integer] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_minus
thf(fact_5172_odd__of__bool__self,axiom,
! [P6: $o] :
( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P6 ) ) )
= P6 ) ).
% odd_of_bool_self
thf(fact_5173_odd__of__bool__self,axiom,
! [P6: $o] :
( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P6 ) ) )
= P6 ) ).
% odd_of_bool_self
thf(fact_5174_odd__of__bool__self,axiom,
! [P6: $o] :
( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P6 ) ) )
= P6 ) ).
% odd_of_bool_self
thf(fact_5175_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_5176_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_5177_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_5178_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_5179_add__neg__numeral__special_I9_J,axiom,
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_5180_diff__numeral__special_I10_J,axiom,
( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_5181_diff__numeral__special_I10_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_5182_diff__numeral__special_I10_J,axiom,
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_5183_diff__numeral__special_I10_J,axiom,
( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_5184_diff__numeral__special_I10_J,axiom,
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_5185_diff__numeral__special_I11_J,axiom,
( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_5186_diff__numeral__special_I11_J,axiom,
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_5187_diff__numeral__special_I11_J,axiom,
( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_5188_diff__numeral__special_I11_J,axiom,
( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_5189_diff__numeral__special_I11_J,axiom,
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% diff_numeral_special(11)
thf(fact_5190_minus__1__div__2__eq,axiom,
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_2_eq
thf(fact_5191_minus__1__div__2__eq,axiom,
( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% minus_1_div_2_eq
thf(fact_5192_minus__1__mod__2__eq,axiom,
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% minus_1_mod_2_eq
thf(fact_5193_minus__1__mod__2__eq,axiom,
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% minus_1_mod_2_eq
thf(fact_5194_bits__minus__1__mod__2__eq,axiom,
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= one_one_int ) ).
% bits_minus_1_mod_2_eq
thf(fact_5195_bits__minus__1__mod__2__eq,axiom,
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= one_one_Code_integer ) ).
% bits_minus_1_mod_2_eq
thf(fact_5196_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5197_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5198_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5199_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5200_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_5201_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= zero_zero_nat ) ).
% of_bool_half_eq_0
thf(fact_5202_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int ) ).
% of_bool_half_eq_0
thf(fact_5203_of__bool__half__eq__0,axiom,
! [B: $o] :
( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= zero_z3403309356797280102nteger ) ).
% of_bool_half_eq_0
thf(fact_5204_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( power_power_real @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5205_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5206_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: complex] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( power_power_complex @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5207_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( power_power_rat @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5208_Parity_Oring__1__class_Opower__minus__even,axiom,
! [N: nat,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_5209_power__minus__odd,axiom,
! [N: nat,A: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5210_power__minus__odd,axiom,
! [N: nat,A: int] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5211_power__minus__odd,axiom,
! [N: nat,A: complex] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5212_power__minus__odd,axiom,
! [N: nat,A: rat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5213_power__minus__odd,axiom,
! [N: nat,A: code_integer] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% power_minus_odd
thf(fact_5214_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5215_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5216_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
= ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5217_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5218_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_5219_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_5220_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_5221_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_5222_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_5223_diff__numeral__special_I4_J,axiom,
! [M: num] :
( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_5224_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_5225_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_5226_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_5227_dbl__simps_I4_J,axiom,
( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_5228_dbl__simps_I4_J,axiom,
( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_5229_dbl__simps_I4_J,axiom,
( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% dbl_simps(4)
thf(fact_5230_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_real ) ).
% power_minus1_even
thf(fact_5231_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_int ) ).
% power_minus1_even
thf(fact_5232_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_complex ) ).
% power_minus1_even
thf(fact_5233_power__minus1__even,axiom,
! [N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_rat ) ).
% power_minus1_even
thf(fact_5234_power__minus1__even,axiom,
! [N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= one_one_Code_integer ) ).
% power_minus1_even
thf(fact_5235_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= one_one_real ) ) ).
% neg_one_even_power
thf(fact_5236_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) ) ).
% neg_one_even_power
thf(fact_5237_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= one_one_complex ) ) ).
% neg_one_even_power
thf(fact_5238_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= one_one_rat ) ) ).
% neg_one_even_power
thf(fact_5239_neg__one__even__power,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= one_one_Code_integer ) ) ).
% neg_one_even_power
thf(fact_5240_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= ( uminus_uminus_real @ one_one_real ) ) ) ).
% neg_one_odd_power
thf(fact_5241_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% neg_one_odd_power
thf(fact_5242_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% neg_one_odd_power
thf(fact_5243_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% neg_one_odd_power
thf(fact_5244_neg__one__odd__power,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% neg_one_odd_power
thf(fact_5245_signed__take__bit__0,axiom,
! [A: code_integer] :
( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_5246_signed__take__bit__0,axiom,
! [A: int] :
( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
= ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_5247_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_5248_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_5249_one__div__2__pow__eq,axiom,
! [N: nat] :
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% one_div_2_pow_eq
thf(fact_5250_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_5251_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_5252_bits__1__div__exp,axiom,
! [N: nat] :
( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% bits_1_div_exp
thf(fact_5253_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_5254_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_5255_one__mod__2__pow__eq,axiom,
! [N: nat] :
( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% one_mod_2_pow_eq
thf(fact_5256_compl__le__swap2,axiom,
! [Y4: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y4 ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y4 ) ) ).
% compl_le_swap2
thf(fact_5257_compl__le__swap1,axiom,
! [Y4: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y4 ) ) ) ).
% compl_le_swap1
thf(fact_5258_compl__mono,axiom,
! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y4 ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_5259_verit__negate__coefficient_I3_J,axiom,
! [A: real,B: real] :
( ( A = B )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_5260_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_5261_verit__negate__coefficient_I3_J,axiom,
! [A: rat,B: rat] :
( ( A = B )
=> ( ( uminus_uminus_rat @ A )
= ( uminus_uminus_rat @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_5262_verit__negate__coefficient_I3_J,axiom,
! [A: code_integer,B: code_integer] :
( ( A = B )
=> ( ( uminus1351360451143612070nteger @ A )
= ( uminus1351360451143612070nteger @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_5263_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n3304061248610475627l_real
@ ( P
& Q ) )
= ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% of_bool_conj
thf(fact_5264_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n2052037380579107095ol_rat
@ ( P
& Q ) )
= ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% of_bool_conj
thf(fact_5265_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n2687167440665602831ol_nat
@ ( P
& Q ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% of_bool_conj
thf(fact_5266_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n2684676970156552555ol_int
@ ( P
& Q ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% of_bool_conj
thf(fact_5267_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n356916108424825756nteger
@ ( P
& Q ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% of_bool_conj
thf(fact_5268_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_5269_le__imp__neg__le,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% le_imp_neg_le
thf(fact_5270_le__imp__neg__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% le_imp_neg_le
thf(fact_5271_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_5272_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_5273_minus__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_5274_minus__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_5275_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_5276_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_5277_le__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% le_minus_iff
thf(fact_5278_le__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% le_minus_iff
thf(fact_5279_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_5280_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_5281_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_5282_less__minus__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% less_minus_iff
thf(fact_5283_less__minus__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% less_minus_iff
thf(fact_5284_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_5285_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_5286_minus__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_5287_minus__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_5288_verit__negate__coefficient_I2_J,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_5289_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_5290_verit__negate__coefficient_I2_J,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_5291_verit__negate__coefficient_I2_J,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ B )
=> ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_5292_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_real @ M )
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5293_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5294_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numera6690914467698888265omplex @ M )
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5295_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_rat @ M )
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5296_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numera6620942414471956472nteger @ M )
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_5297_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
!= ( numeral_numeral_real @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5298_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5299_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
!= ( numera6690914467698888265omplex @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5300_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
!= ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5301_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
!= ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_5302_square__eq__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ A )
= ( times_times_real @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_5303_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_5304_square__eq__iff,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ A )
= ( times_times_complex @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% square_eq_iff
thf(fact_5305_square__eq__iff,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ A )
= ( times_times_rat @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_rat @ B ) ) ) ) ).
% square_eq_iff
thf(fact_5306_square__eq__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( times_3573771949741848930nteger @ A @ A )
= ( times_3573771949741848930nteger @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% square_eq_iff
thf(fact_5307_minus__mult__commute,axiom,
! [A: real,B: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
= ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_5308_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_5309_minus__mult__commute,axiom,
! [A: complex,B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_mult_commute
thf(fact_5310_minus__mult__commute,axiom,
! [A: rat,B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_mult_commute
thf(fact_5311_minus__mult__commute,axiom,
! [A: code_integer,B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% minus_mult_commute
thf(fact_5312_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_5313_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_5314_one__neq__neg__one,axiom,
( one_one_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% one_neq_neg_one
thf(fact_5315_one__neq__neg__one,axiom,
( one_one_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% one_neq_neg_one
thf(fact_5316_one__neq__neg__one,axiom,
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% one_neq_neg_one
thf(fact_5317_group__cancel_Oneg1,axiom,
! [A2: real,K: real,A: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_5318_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_5319_group__cancel_Oneg1,axiom,
! [A2: complex,K: complex,A: complex] :
( ( A2
= ( plus_plus_complex @ K @ A ) )
=> ( ( uminus1482373934393186551omplex @ A2 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_5320_group__cancel_Oneg1,axiom,
! [A2: rat,K: rat,A: rat] :
( ( A2
= ( plus_plus_rat @ K @ A ) )
=> ( ( uminus_uminus_rat @ A2 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_5321_group__cancel_Oneg1,axiom,
! [A2: code_integer,K: code_integer,A: code_integer] :
( ( A2
= ( plus_p5714425477246183910nteger @ K @ A ) )
=> ( ( uminus1351360451143612070nteger @ A2 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_5322_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_5323_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_5324_add_Oinverse__distrib__swap,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_5325_add_Oinverse__distrib__swap,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_5326_add_Oinverse__distrib__swap,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_5327_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_5328_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_5329_is__num__normalize_I8_J,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_5330_is__num__normalize_I8_J,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
= ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_5331_is__num__normalize_I8_J,axiom,
! [A: code_integer,B: code_integer] :
( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_5332_div__minus__right,axiom,
! [A: int,B: int] :
( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% div_minus_right
thf(fact_5333_div__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% div_minus_right
thf(fact_5334_minus__divide__left,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_5335_minus__divide__left,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_5336_minus__divide__left,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_5337_minus__divide__divide,axiom,
! [A: real,B: real] :
( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ).
% minus_divide_divide
thf(fact_5338_minus__divide__divide,axiom,
! [A: complex,B: complex] :
( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ).
% minus_divide_divide
thf(fact_5339_minus__divide__divide,axiom,
! [A: rat,B: rat] :
( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( divide_divide_rat @ A @ B ) ) ).
% minus_divide_divide
thf(fact_5340_minus__divide__right,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% minus_divide_right
thf(fact_5341_minus__divide__right,axiom,
! [A: complex,B: complex] :
( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% minus_divide_right
thf(fact_5342_minus__divide__right,axiom,
! [A: rat,B: rat] :
( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% minus_divide_right
thf(fact_5343_mod__minus__right,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% mod_minus_right
thf(fact_5344_mod__minus__right,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% mod_minus_right
thf(fact_5345_mod__minus__cong,axiom,
! [A: int,B: int,A5: int] :
( ( ( modulo_modulo_int @ A @ B )
= ( modulo_modulo_int @ A5 @ B ) )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_5346_mod__minus__cong,axiom,
! [A: code_integer,B: code_integer,A5: code_integer] :
( ( ( modulo364778990260209775nteger @ A @ B )
= ( modulo364778990260209775nteger @ A5 @ B ) )
=> ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A5 ) @ B ) ) ) ).
% mod_minus_cong
thf(fact_5347_mod__minus__eq,axiom,
! [A: int,B: int] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
= ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% mod_minus_eq
thf(fact_5348_mod__minus__eq,axiom,
! [A: code_integer,B: code_integer] :
( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
= ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% mod_minus_eq
thf(fact_5349_sum__mono,axiom,
! [K5: set_nat,F: nat > rat,G: nat > rat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5350_sum__mono,axiom,
! [K5: set_int,F: int > rat,G: int > rat] :
( ! [I3: int] :
( ( member_int @ I3 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5351_sum__mono,axiom,
! [K5: set_real,F: real > rat,G: real > rat] :
( ! [I3: real] :
( ( member_real @ I3 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5352_sum__mono,axiom,
! [K5: set_complex,F: complex > rat,G: complex > rat] :
( ! [I3: complex] :
( ( member_complex @ I3 @ K5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5353_sum__mono,axiom,
! [K5: set_int,F: int > nat,G: int > nat] :
( ! [I3: int] :
( ( member_int @ I3 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5354_sum__mono,axiom,
! [K5: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5355_sum__mono,axiom,
! [K5: set_complex,F: complex > nat,G: complex > nat] :
( ! [I3: complex] :
( ( member_complex @ I3 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5356_sum__mono,axiom,
! [K5: set_nat,F: nat > int,G: nat > int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K5 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5357_sum__mono,axiom,
! [K5: set_real,F: real > int,G: real > int] :
( ! [I3: real] :
( ( member_real @ I3 @ K5 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5358_sum__mono,axiom,
! [K5: set_complex,F: complex > int,G: complex > int] :
( ! [I3: complex] :
( ( member_complex @ I3 @ K5 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_5359_sum__product,axiom,
! [F: int > int,A2: set_int,G: int > int,B5: set_int] :
( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B5 ) )
= ( groups4538972089207619220nt_int
@ ^ [I4: int] :
( groups4538972089207619220nt_int
@ ^ [J3: int] : ( times_times_int @ ( F @ I4 ) @ ( G @ J3 ) )
@ B5 )
@ A2 ) ) ).
% sum_product
thf(fact_5360_sum__product,axiom,
! [F: complex > complex,A2: set_complex,G: complex > complex,B5: set_complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B5 ) )
= ( groups7754918857620584856omplex
@ ^ [I4: complex] :
( groups7754918857620584856omplex
@ ^ [J3: complex] : ( times_times_complex @ ( F @ I4 ) @ ( G @ J3 ) )
@ B5 )
@ A2 ) ) ).
% sum_product
thf(fact_5361_sum__product,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B5: set_nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] :
( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
@ B5 )
@ A2 ) ) ).
% sum_product
thf(fact_5362_sum__product,axiom,
! [F: nat > real,A2: set_nat,G: nat > real,B5: set_nat] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B5 ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] :
( groups6591440286371151544t_real
@ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
@ B5 )
@ A2 ) ) ).
% sum_product
thf(fact_5363_sum__distrib__right,axiom,
! [F: int > int,A2: set_int,R2: int] :
( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R2 )
= ( groups4538972089207619220nt_int
@ ^ [N2: int] : ( times_times_int @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_5364_sum__distrib__right,axiom,
! [F: complex > complex,A2: set_complex,R2: complex] :
( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_5365_sum__distrib__right,axiom,
! [F: nat > nat,A2: set_nat,R2: nat] :
( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R2 )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_5366_sum__distrib__right,axiom,
! [F: nat > real,A2: set_nat,R2: real] :
( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_5367_sum__distrib__left,axiom,
! [R2: int,F: int > int,A2: set_int] :
( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A2 ) )
= ( groups4538972089207619220nt_int
@ ^ [N2: int] : ( times_times_int @ R2 @ ( F @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_5368_sum__distrib__left,axiom,
! [R2: complex,F: complex > complex,A2: set_complex] :
( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A2 ) )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( times_times_complex @ R2 @ ( F @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_5369_sum__distrib__left,axiom,
! [R2: nat,F: nat > nat,A2: set_nat] :
( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups3542108847815614940at_nat
@ ^ [N2: nat] : ( times_times_nat @ R2 @ ( F @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_5370_sum__distrib__left,axiom,
! [R2: real,F: nat > real,A2: set_nat] :
( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A2 ) )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( times_times_real @ R2 @ ( F @ N2 ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_5371_sum_Odistrib,axiom,
! [G: int > int,H2: int > int,A2: set_int] :
( ( groups4538972089207619220nt_int
@ ^ [X2: int] : ( plus_plus_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_5372_sum_Odistrib,axiom,
! [G: complex > complex,H2: complex > complex,A2: set_complex] :
( ( groups7754918857620584856omplex
@ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_5373_sum_Odistrib,axiom,
! [G: nat > nat,H2: nat > nat,A2: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_5374_sum_Odistrib,axiom,
! [G: nat > real,H2: nat > real,A2: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% sum.distrib
thf(fact_5375_sum__divide__distrib,axiom,
! [F: complex > complex,A2: set_complex,R2: complex] :
( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
= ( groups7754918857620584856omplex
@ ^ [N2: complex] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_5376_sum__divide__distrib,axiom,
! [F: nat > real,A2: set_nat,R2: real] :
( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
= ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ R2 )
@ A2 ) ) ).
% sum_divide_distrib
thf(fact_5377_mod__sum__eq,axiom,
! [F: int > int,A: int,A2: set_int] :
( ( modulo_modulo_int
@ ( groups4538972089207619220nt_int
@ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).
% mod_sum_eq
thf(fact_5378_mod__sum__eq,axiom,
! [F: nat > nat,A: nat,A2: set_nat] :
( ( modulo_modulo_nat
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).
% mod_sum_eq
thf(fact_5379_sum__nonpos,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_5380_sum__nonpos,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_5381_sum__nonpos,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_5382_sum__nonpos,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5383_sum__nonpos,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5384_sum__nonpos,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5385_sum__nonpos,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% sum_nonpos
thf(fact_5386_sum__nonpos,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_5387_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_5388_sum__nonpos,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_5389_sum__nonneg,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_5390_sum__nonneg,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_5391_sum__nonneg,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_5392_sum__nonneg,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_5393_sum__nonneg,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_5394_sum__nonneg,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_5395_sum__nonneg,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_5396_sum__nonneg,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_5397_sum__nonneg,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_5398_sum__nonneg,axiom,
! [A2: set_complex,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_5399_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_5400_sum__mono__inv,axiom,
! [F: int > rat,I5: set_int,G: int > rat,I2: int] :
( ( ( groups3906332499630173760nt_rat @ F @ I5 )
= ( groups3906332499630173760nt_rat @ G @ I5 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_int @ I2 @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5401_sum__mono__inv,axiom,
! [F: real > rat,I5: set_real,G: real > rat,I2: real] :
( ( ( groups1300246762558778688al_rat @ F @ I5 )
= ( groups1300246762558778688al_rat @ G @ I5 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_real @ I2 @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5402_sum__mono__inv,axiom,
! [F: nat > rat,I5: set_nat,G: nat > rat,I2: nat] :
( ( ( groups2906978787729119204at_rat @ F @ I5 )
= ( groups2906978787729119204at_rat @ G @ I5 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_nat @ I2 @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5403_sum__mono__inv,axiom,
! [F: complex > rat,I5: set_complex,G: complex > rat,I2: complex] :
( ( ( groups5058264527183730370ex_rat @ F @ I5 )
= ( groups5058264527183730370ex_rat @ G @ I5 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5404_sum__mono__inv,axiom,
! [F: int > nat,I5: set_int,G: int > nat,I2: int] :
( ( ( groups4541462559716669496nt_nat @ F @ I5 )
= ( groups4541462559716669496nt_nat @ G @ I5 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_int @ I2 @ I5 )
=> ( ( finite_finite_int @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5405_sum__mono__inv,axiom,
! [F: real > nat,I5: set_real,G: real > nat,I2: real] :
( ( ( groups1935376822645274424al_nat @ F @ I5 )
= ( groups1935376822645274424al_nat @ G @ I5 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_real @ I2 @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5406_sum__mono__inv,axiom,
! [F: complex > nat,I5: set_complex,G: complex > nat,I2: complex] :
( ( ( groups5693394587270226106ex_nat @ F @ I5 )
= ( groups5693394587270226106ex_nat @ G @ I5 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5407_sum__mono__inv,axiom,
! [F: real > int,I5: set_real,G: real > int,I2: real] :
( ( ( groups1932886352136224148al_int @ F @ I5 )
= ( groups1932886352136224148al_int @ G @ I5 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_real @ I2 @ I5 )
=> ( ( finite_finite_real @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5408_sum__mono__inv,axiom,
! [F: nat > int,I5: set_nat,G: nat > int,I2: nat] :
( ( ( groups3539618377306564664at_int @ F @ I5 )
= ( groups3539618377306564664at_int @ G @ I5 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_nat @ I2 @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5409_sum__mono__inv,axiom,
! [F: complex > int,I5: set_complex,G: complex > int,I2: complex] :
( ( ( groups5690904116761175830ex_int @ F @ I5 )
= ( groups5690904116761175830ex_int @ G @ I5 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( finite3207457112153483333omplex @ I5 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_5410_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_5411_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_5412_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_5413_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_5414_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_5415_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% of_bool_less_eq_one
thf(fact_5416_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% of_bool_less_eq_one
thf(fact_5417_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% of_bool_less_eq_one
thf(fact_5418_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% of_bool_less_eq_one
thf(fact_5419_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% of_bool_less_eq_one
thf(fact_5420_split__of__bool__asm,axiom,
! [P: complex > $o,P6: $o] :
( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_complex ) )
| ( ~ P6
& ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_5421_split__of__bool__asm,axiom,
! [P: real > $o,P6: $o] :
( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_real ) )
| ( ~ P6
& ~ ( P @ zero_zero_real ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_5422_split__of__bool__asm,axiom,
! [P: rat > $o,P6: $o] :
( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_rat ) )
| ( ~ P6
& ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_5423_split__of__bool__asm,axiom,
! [P: nat > $o,P6: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_nat ) )
| ( ~ P6
& ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_5424_split__of__bool__asm,axiom,
! [P: int > $o,P6: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_int ) )
| ( ~ P6
& ~ ( P @ zero_zero_int ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_5425_split__of__bool__asm,axiom,
! [P: code_integer > $o,P6: $o] :
( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
= ( ~ ( ( P6
& ~ ( P @ one_one_Code_integer ) )
| ( ~ P6
& ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_5426_split__of__bool,axiom,
! [P: complex > $o,P6: $o] :
( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
= ( ( P6
=> ( P @ one_one_complex ) )
& ( ~ P6
=> ( P @ zero_zero_complex ) ) ) ) ).
% split_of_bool
thf(fact_5427_split__of__bool,axiom,
! [P: real > $o,P6: $o] :
( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
= ( ( P6
=> ( P @ one_one_real ) )
& ( ~ P6
=> ( P @ zero_zero_real ) ) ) ) ).
% split_of_bool
thf(fact_5428_split__of__bool,axiom,
! [P: rat > $o,P6: $o] :
( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
= ( ( P6
=> ( P @ one_one_rat ) )
& ( ~ P6
=> ( P @ zero_zero_rat ) ) ) ) ).
% split_of_bool
thf(fact_5429_split__of__bool,axiom,
! [P: nat > $o,P6: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
= ( ( P6
=> ( P @ one_one_nat ) )
& ( ~ P6
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_5430_split__of__bool,axiom,
! [P: int > $o,P6: $o] :
( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
= ( ( P6
=> ( P @ one_one_int ) )
& ( ~ P6
=> ( P @ zero_zero_int ) ) ) ) ).
% split_of_bool
thf(fact_5431_split__of__bool,axiom,
! [P: code_integer > $o,P6: $o] :
( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
= ( ( P6
=> ( P @ one_one_Code_integer ) )
& ( ~ P6
=> ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% split_of_bool
thf(fact_5432_of__bool__def,axiom,
( zero_n1201886186963655149omplex
= ( ^ [P4: $o] : ( if_complex @ P4 @ one_one_complex @ zero_zero_complex ) ) ) ).
% of_bool_def
thf(fact_5433_of__bool__def,axiom,
( zero_n3304061248610475627l_real
= ( ^ [P4: $o] : ( if_real @ P4 @ one_one_real @ zero_zero_real ) ) ) ).
% of_bool_def
thf(fact_5434_of__bool__def,axiom,
( zero_n2052037380579107095ol_rat
= ( ^ [P4: $o] : ( if_rat @ P4 @ one_one_rat @ zero_zero_rat ) ) ) ).
% of_bool_def
thf(fact_5435_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P4: $o] : ( if_nat @ P4 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_5436_of__bool__def,axiom,
( zero_n2684676970156552555ol_int
= ( ^ [P4: $o] : ( if_int @ P4 @ one_one_int @ zero_zero_int ) ) ) ).
% of_bool_def
thf(fact_5437_of__bool__def,axiom,
( zero_n356916108424825756nteger
= ( ^ [P4: $o] : ( if_Code_integer @ P4 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% of_bool_def
thf(fact_5438_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5439_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5440_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5441_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_5442_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5443_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5444_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5445_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_5446_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5447_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5448_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5449_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5450_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_5451_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5452_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5453_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5454_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_5455_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5456_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5457_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5458_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_5459_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_5460_le__minus__one__simps_I4_J,axiom,
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(4)
thf(fact_5461_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(4)
thf(fact_5462_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_5463_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_5464_le__minus__one__simps_I2_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% le_minus_one_simps(2)
thf(fact_5465_le__minus__one__simps_I2_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% le_minus_one_simps(2)
thf(fact_5466_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_5467_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_5468_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_5469_zero__neq__neg__one,axiom,
( zero_zero_complex
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% zero_neq_neg_one
thf(fact_5470_zero__neq__neg__one,axiom,
( zero_zero_rat
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% zero_neq_neg_one
thf(fact_5471_zero__neq__neg__one,axiom,
( zero_z3403309356797280102nteger
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% zero_neq_neg_one
thf(fact_5472_add__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5473_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5474_add__eq__0__iff,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
= ( B
= ( uminus1482373934393186551omplex @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5475_add__eq__0__iff,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= zero_zero_rat )
= ( B
= ( uminus_uminus_rat @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5476_add__eq__0__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger )
= ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% add_eq_0_iff
thf(fact_5477_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5478_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5479_ab__group__add__class_Oab__left__minus,axiom,
! [A: complex] :
( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
= zero_zero_complex ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5480_ab__group__add__class_Oab__left__minus,axiom,
! [A: rat] :
( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
= zero_zero_rat ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5481_ab__group__add__class_Oab__left__minus,axiom,
! [A: code_integer] :
( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
= zero_z3403309356797280102nteger ) ).
% ab_group_add_class.ab_left_minus
thf(fact_5482_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5483_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5484_add_Oinverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( plus_plus_complex @ A @ B )
= zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5485_add_Oinverse__unique,axiom,
! [A: rat,B: rat] :
( ( ( plus_plus_rat @ A @ B )
= zero_zero_rat )
=> ( ( uminus_uminus_rat @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5486_add_Oinverse__unique,axiom,
! [A: code_integer,B: code_integer] :
( ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger )
=> ( ( uminus1351360451143612070nteger @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_5487_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5488_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5489_eq__neg__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( A
= ( uminus1482373934393186551omplex @ B ) )
= ( ( plus_plus_complex @ A @ B )
= zero_zero_complex ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5490_eq__neg__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( A
= ( uminus_uminus_rat @ B ) )
= ( ( plus_plus_rat @ A @ B )
= zero_zero_rat ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5491_eq__neg__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( uminus1351360451143612070nteger @ B ) )
= ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_5492_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5493_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5494_neg__eq__iff__add__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( uminus1482373934393186551omplex @ A )
= B )
= ( ( plus_plus_complex @ A @ B )
= zero_zero_complex ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5495_neg__eq__iff__add__eq__0,axiom,
! [A: rat,B: rat] :
( ( ( uminus_uminus_rat @ A )
= B )
= ( ( plus_plus_rat @ A @ B )
= zero_zero_rat ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5496_neg__eq__iff__add__eq__0,axiom,
! [A: code_integer,B: code_integer] :
( ( ( uminus1351360451143612070nteger @ A )
= B )
= ( ( plus_p5714425477246183910nteger @ A @ B )
= zero_z3403309356797280102nteger ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_5497_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_5498_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_5499_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(4)
thf(fact_5500_less__minus__one__simps_I4_J,axiom,
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(4)
thf(fact_5501_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_5502_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_5503_less__minus__one__simps_I2_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% less_minus_one_simps(2)
thf(fact_5504_less__minus__one__simps_I2_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% less_minus_one_simps(2)
thf(fact_5505_numeral__times__minus__swap,axiom,
! [W2: num,X: real] :
( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ ( uminus_uminus_real @ X ) )
= ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5506_numeral__times__minus__swap,axiom,
! [W2: num,X: int] :
( ( times_times_int @ ( numeral_numeral_int @ W2 ) @ ( uminus_uminus_int @ X ) )
= ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5507_numeral__times__minus__swap,axiom,
! [W2: num,X: complex] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ ( uminus1482373934393186551omplex @ X ) )
= ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5508_numeral__times__minus__swap,axiom,
! [W2: num,X: rat] :
( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ ( uminus_uminus_rat @ X ) )
= ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5509_numeral__times__minus__swap,axiom,
! [W2: num,X: code_integer] :
( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ ( uminus1351360451143612070nteger @ X ) )
= ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_5510_nonzero__minus__divide__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
= ( divide_divide_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_5511_nonzero__minus__divide__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
= ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_5512_nonzero__minus__divide__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
= ( divide_divide_rat @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_5513_nonzero__minus__divide__right,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_5514_nonzero__minus__divide__right,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_5515_nonzero__minus__divide__right,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_5516_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_real @ N )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_5517_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_5518_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ N )
!= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% numeral_neq_neg_one
thf(fact_5519_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_rat @ N )
!= ( uminus_uminus_rat @ one_one_rat ) ) ).
% numeral_neq_neg_one
thf(fact_5520_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numera6620942414471956472nteger @ N )
!= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% numeral_neq_neg_one
thf(fact_5521_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5522_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5523_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_complex
!= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5524_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_rat
!= ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5525_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_Code_integer
!= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_5526_square__eq__1__iff,axiom,
! [X: real] :
( ( ( times_times_real @ X @ X )
= one_one_real )
= ( ( X = one_one_real )
| ( X
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_5527_square__eq__1__iff,axiom,
! [X: int] :
( ( ( times_times_int @ X @ X )
= one_one_int )
= ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% square_eq_1_iff
thf(fact_5528_square__eq__1__iff,axiom,
! [X: complex] :
( ( ( times_times_complex @ X @ X )
= one_one_complex )
= ( ( X = one_one_complex )
| ( X
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% square_eq_1_iff
thf(fact_5529_square__eq__1__iff,axiom,
! [X: rat] :
( ( ( times_times_rat @ X @ X )
= one_one_rat )
= ( ( X = one_one_rat )
| ( X
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% square_eq_1_iff
thf(fact_5530_square__eq__1__iff,axiom,
! [X: code_integer] :
( ( ( times_3573771949741848930nteger @ X @ X )
= one_one_Code_integer )
= ( ( X = one_one_Code_integer )
| ( X
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% square_eq_1_iff
thf(fact_5531_group__cancel_Osub2,axiom,
! [B5: real,K: real,B: real,A: real] :
( ( B5
= ( plus_plus_real @ K @ B ) )
=> ( ( minus_minus_real @ A @ B5 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5532_group__cancel_Osub2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5533_group__cancel_Osub2,axiom,
! [B5: complex,K: complex,B: complex,A: complex] :
( ( B5
= ( plus_plus_complex @ K @ B ) )
=> ( ( minus_minus_complex @ A @ B5 )
= ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5534_group__cancel_Osub2,axiom,
! [B5: rat,K: rat,B: rat,A: rat] :
( ( B5
= ( plus_plus_rat @ K @ B ) )
=> ( ( minus_minus_rat @ A @ B5 )
= ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5535_group__cancel_Osub2,axiom,
! [B5: code_integer,K: code_integer,B: code_integer,A: code_integer] :
( ( B5
= ( plus_p5714425477246183910nteger @ K @ B ) )
=> ( ( minus_8373710615458151222nteger @ A @ B5 )
= ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_5536_diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A4: real,B3: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5537_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5538_diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A4: complex,B3: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5539_diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5540_diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A4: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_5541_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_real
= ( ^ [A4: real,B3: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5542_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5543_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_complex
= ( ^ [A4: complex,B3: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5544_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_rat
= ( ^ [A4: rat,B3: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5545_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_8373710615458151222nteger
= ( ^ [A4: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5546_replicate__length__same,axiom,
! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
( ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( X3 = X ) )
=> ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_5547_replicate__length__same,axiom,
! [Xs2: list_o,X: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
=> ( X3 = X ) )
=> ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_5548_replicate__length__same,axiom,
! [Xs2: list_nat,X: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( X3 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_5549_replicate__length__same,axiom,
! [Xs2: list_int,X: int] :
( ! [X3: int] :
( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
=> ( X3 = X ) )
=> ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_5550_replicate__eqI,axiom,
! [Xs2: list_real,N: nat,X: real] :
( ( ( size_size_list_real @ Xs2 )
= N )
=> ( ! [Y3: real] :
( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_real @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5551_replicate__eqI,axiom,
! [Xs2: list_complex,N: nat,X: complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
= N )
=> ( ! [Y3: complex] :
( ( member_complex @ Y3 @ ( set_complex2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_complex @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5552_replicate__eqI,axiom,
! [Xs2: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= N )
=> ( ! [Y3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replic4235873036481779905at_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5553_replicate__eqI,axiom,
! [Xs2: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
= N )
=> ( ! [Y3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5554_replicate__eqI,axiom,
! [Xs2: list_o,N: nat,X: $o] :
( ( ( size_size_list_o @ Xs2 )
= N )
=> ( ! [Y3: $o] :
( ( member_o @ Y3 @ ( set_o2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_o @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5555_replicate__eqI,axiom,
! [Xs2: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs2 )
= N )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5556_replicate__eqI,axiom,
! [Xs2: list_int,N: nat,X: int] :
( ( ( size_size_list_int @ Xs2 )
= N )
=> ( ! [Y3: int] :
( ( member_int @ Y3 @ ( set_int2 @ Xs2 ) )
=> ( Y3 = X ) )
=> ( Xs2
= ( replicate_int @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_5557_dvd__div__neg,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5558_dvd__div__neg,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5559_dvd__div__neg,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5560_dvd__div__neg,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5561_dvd__div__neg,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
= ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% dvd_div_neg
thf(fact_5562_dvd__neg__div,axiom,
! [B: real,A: real] :
( ( dvd_dvd_real @ B @ A )
=> ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5563_dvd__neg__div,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5564_dvd__neg__div,axiom,
! [B: complex,A: complex] :
( ( dvd_dvd_complex @ B @ A )
=> ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5565_dvd__neg__div,axiom,
! [B: rat,A: rat] :
( ( dvd_dvd_rat @ B @ A )
=> ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5566_dvd__neg__div,axiom,
! [B: code_integer,A: code_integer] :
( ( dvd_dvd_Code_integer @ B @ A )
=> ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
= ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% dvd_neg_div
thf(fact_5567_subset__Compl__self__eq,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% subset_Compl_self_eq
thf(fact_5568_subset__Compl__self__eq,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
= ( A2 = bot_bot_set_real ) ) ).
% subset_Compl_self_eq
thf(fact_5569_subset__Compl__self__eq,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_5570_real__minus__mult__self__le,axiom,
! [U2: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U2 @ U2 ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_5571_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_5572_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_5573_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X2: real,Y: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y ) ) ) ) ).
% minus_real_def
thf(fact_5574_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_5575_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F: int > real] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ A2 )
= zero_zero_real )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5576_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: real > real] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ A2 )
= zero_zero_real )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5577_sum__nonneg__eq__0__iff,axiom,
! [A2: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ A2 )
= zero_zero_real )
= ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5578_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
= zero_zero_rat )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5579_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: real > rat] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ A2 )
= zero_zero_rat )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5580_sum__nonneg__eq__0__iff,axiom,
! [A2: set_nat,F: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ A2 )
= zero_zero_rat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5581_sum__nonneg__eq__0__iff,axiom,
! [A2: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
= zero_zero_rat )
= ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_rat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5582_sum__nonneg__eq__0__iff,axiom,
! [A2: set_int,F: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5583_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: real > nat] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X2: real] :
( ( member_real @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5584_sum__nonneg__eq__0__iff,axiom,
! [A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( F @ X2 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_5585_sum__le__included,axiom,
! [S2: set_complex,T: set_complex,G: complex > real,I2: complex > complex,F: complex > real] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5586_sum__le__included,axiom,
! [S2: set_nat,T: set_nat,G: nat > rat,I2: nat > nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5587_sum__le__included,axiom,
! [S2: set_nat,T: set_complex,G: complex > rat,I2: complex > nat,F: nat > rat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5588_sum__le__included,axiom,
! [S2: set_complex,T: set_nat,G: nat > rat,I2: nat > complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5589_sum__le__included,axiom,
! [S2: set_complex,T: set_complex,G: complex > rat,I2: complex > complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5590_sum__le__included,axiom,
! [S2: set_complex,T: set_complex,G: complex > nat,I2: complex > complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5591_sum__le__included,axiom,
! [S2: set_nat,T: set_nat,G: nat > int,I2: nat > nat,F: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S2 ) @ ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5592_sum__le__included,axiom,
! [S2: set_nat,T: set_complex,G: complex > int,I2: complex > nat,F: nat > int] :
( ( finite_finite_nat @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S2 ) @ ( groups5690904116761175830ex_int @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5593_sum__le__included,axiom,
! [S2: set_complex,T: set_nat,G: nat > int,I2: nat > complex,F: complex > int] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ S2 ) @ ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5594_sum__le__included,axiom,
! [S2: set_complex,T: set_complex,G: complex > int,I2: complex > complex,F: complex > int] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ( finite3207457112153483333omplex @ T )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ T )
=> ( ord_less_eq_int @ zero_zero_int @ ( G @ X3 ) ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S2 )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ T )
& ( ( I2 @ Xa )
= X3 )
& ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ S2 ) @ ( groups5690904116761175830ex_int @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_5595_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A2 )
& ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5596_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5597_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A2 )
& ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5598_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A2 )
& ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5599_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F: nat > int,G: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5600_sum__strict__mono__ex1,axiom,
! [A2: set_complex,F: complex > int,G: complex > int] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: complex] :
( ( member_complex @ X5 @ A2 )
& ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5601_sum__strict__mono__ex1,axiom,
! [A2: set_int,F: int > int,G: int > int] :
( ( finite_finite_int @ A2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: int] :
( ( member_int @ X5 @ A2 )
& ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5602_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5603_sum__strict__mono__ex1,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_5604_sum_Orelated,axiom,
! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups2073611262835488442omplex @ H2 @ S3 ) @ ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_5605_sum_Orelated,axiom,
! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
( ( R @ zero_zero_real @ zero_zero_real )
=> ( ! [X15: real,Y15: real,X23: real,Y23: real] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_5606_sum_Orelated,axiom,
! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_5607_sum_Orelated,axiom,
! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
( ( R @ zero_zero_rat @ zero_zero_rat )
=> ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_5608_sum_Orelated,axiom,
! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_5609_sum_Orelated,axiom,
! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X15: int,Y15: int,X23: int,Y23: int] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_5610_sum_Orelated,axiom,
! [R: int > int > $o,S3: set_complex,H2: complex > int,G: complex > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X15: int,Y15: int,X23: int,Y23: int] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups5690904116761175830ex_int @ H2 @ S3 ) @ ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_5611_sum_Orelated,axiom,
! [R: int > int > $o,S3: set_int,H2: int > int,G: int > int] :
( ( R @ zero_zero_int @ zero_zero_int )
=> ( ! [X15: int,Y15: int,X23: int,Y23: int] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
=> ( ( finite_finite_int @ S3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups4538972089207619220nt_int @ H2 @ S3 ) @ ( groups4538972089207619220nt_int @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_5612_sum_Orelated,axiom,
! [R: complex > complex > $o,S3: set_complex,H2: complex > complex,G: complex > complex] :
( ( R @ zero_zero_complex @ zero_zero_complex )
=> ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups7754918857620584856omplex @ H2 @ S3 ) @ ( groups7754918857620584856omplex @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_5613_sum_Orelated,axiom,
! [R: nat > nat > $o,S3: set_nat,H2: nat > nat,G: nat > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups3542108847815614940at_nat @ H2 @ S3 ) @ ( groups3542108847815614940at_nat @ G @ S3 ) ) ) ) ) ) ).
% sum.related
thf(fact_5614_sum__strict__mono,axiom,
! [A2: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( A2 != bot_bot_set_complex )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5615_sum__strict__mono,axiom,
! [A2: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5616_sum__strict__mono,axiom,
! [A2: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5617_sum__strict__mono,axiom,
! [A2: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( A2 != bot_bot_set_complex )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5618_sum__strict__mono,axiom,
! [A2: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5619_sum__strict__mono,axiom,
! [A2: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5620_sum__strict__mono,axiom,
! [A2: set_real,F: real > rat,G: real > rat] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5621_sum__strict__mono,axiom,
! [A2: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( A2 != bot_bot_set_complex )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5622_sum__strict__mono,axiom,
! [A2: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ( A2 != bot_bot_set_int )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5623_sum__strict__mono,axiom,
! [A2: set_real,F: real > nat,G: real > nat] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5624_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_5625_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_5626_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_5627_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_ge_zero
thf(fact_5628_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5629_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5630_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5631_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_5632_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_le_zero
thf(fact_5633_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_le_zero
thf(fact_5634_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_le_zero
thf(fact_5635_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_5636_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_less_zero
thf(fact_5637_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_5638_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% neg_numeral_less_zero
thf(fact_5639_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% neg_numeral_less_zero
thf(fact_5640_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5641_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5642_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5643_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_5644_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_5645_le__minus__one__simps_I3_J,axiom,
~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% le_minus_one_simps(3)
thf(fact_5646_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% le_minus_one_simps(3)
thf(fact_5647_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_5648_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_5649_le__minus__one__simps_I1_J,axiom,
ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% le_minus_one_simps(1)
thf(fact_5650_le__minus__one__simps_I1_J,axiom,
ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% le_minus_one_simps(1)
thf(fact_5651_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_5652_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_5653_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_5654_less__minus__one__simps_I1_J,axiom,
ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% less_minus_one_simps(1)
thf(fact_5655_less__minus__one__simps_I1_J,axiom,
ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% less_minus_one_simps(1)
thf(fact_5656_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_5657_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_5658_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% less_minus_one_simps(3)
thf(fact_5659_less__minus__one__simps_I3_J,axiom,
~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% less_minus_one_simps(3)
thf(fact_5660_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5661_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5662_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5663_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_5664_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_le_neg_one
thf(fact_5665_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_le_neg_one
thf(fact_5666_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_numeral_le_neg_one
thf(fact_5667_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_5668_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% neg_numeral_le_neg_one
thf(fact_5669_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% neg_numeral_le_neg_one
thf(fact_5670_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% neg_numeral_le_neg_one
thf(fact_5671_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_5672_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% neg_one_le_numeral
thf(fact_5673_neg__one__le__numeral,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_le_numeral
thf(fact_5674_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% neg_one_le_numeral
thf(fact_5675_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_5676_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% neg_numeral_le_one
thf(fact_5677_neg__numeral__le__one,axiom,
! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_le_one
thf(fact_5678_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% neg_numeral_le_one
thf(fact_5679_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_5680_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% neg_numeral_less_one
thf(fact_5681_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_5682_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% neg_numeral_less_one
thf(fact_5683_neg__numeral__less__one,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% neg_numeral_less_one
thf(fact_5684_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% neg_one_less_numeral
thf(fact_5685_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_5686_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% neg_one_less_numeral
thf(fact_5687_neg__one__less__numeral,axiom,
! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% neg_one_less_numeral
thf(fact_5688_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_less_neg_one
thf(fact_5689_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_5690_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% not_numeral_less_neg_one
thf(fact_5691_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% not_numeral_less_neg_one
thf(fact_5692_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5693_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5694_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5695_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_5696_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5697_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5698_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5699_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_5700_nonzero__neg__divide__eq__eq2,axiom,
! [B: real,C: real,A: real] :
( ( B != zero_zero_real )
=> ( ( C
= ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
= ( ( times_times_real @ C @ B )
= ( uminus_uminus_real @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5701_nonzero__neg__divide__eq__eq2,axiom,
! [B: complex,C: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( C
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ( times_times_complex @ C @ B )
= ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5702_nonzero__neg__divide__eq__eq2,axiom,
! [B: rat,C: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( C
= ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
= ( ( times_times_rat @ C @ B )
= ( uminus_uminus_rat @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_5703_nonzero__neg__divide__eq__eq,axiom,
! [B: real,A: real,C: real] :
( ( B != zero_zero_real )
=> ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
= C )
= ( ( uminus_uminus_real @ A )
= ( times_times_real @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5704_nonzero__neg__divide__eq__eq,axiom,
! [B: complex,A: complex,C: complex] :
( ( B != zero_zero_complex )
=> ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= C )
= ( ( uminus1482373934393186551omplex @ A )
= ( times_times_complex @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5705_nonzero__neg__divide__eq__eq,axiom,
! [B: rat,A: rat,C: rat] :
( ( B != zero_zero_rat )
=> ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
= C )
= ( ( uminus_uminus_rat @ A )
= ( times_times_rat @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_5706_minus__divide__eq__eq,axiom,
! [B: real,C: real,A: real] :
( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
= A )
= ( ( ( C != zero_zero_real )
=> ( ( uminus_uminus_real @ B )
= ( times_times_real @ A @ C ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5707_minus__divide__eq__eq,axiom,
! [B: complex,C: complex,A: complex] :
( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
= A )
= ( ( ( C != zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ B )
= ( times_times_complex @ A @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5708_minus__divide__eq__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
= A )
= ( ( ( C != zero_zero_rat )
=> ( ( uminus_uminus_rat @ B )
= ( times_times_rat @ A @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_5709_eq__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( A
= ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ A @ C )
= ( uminus_uminus_real @ B ) ) )
& ( ( C = zero_zero_real )
=> ( A = zero_zero_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5710_eq__minus__divide__eq,axiom,
! [A: complex,B: complex,C: complex] :
( ( A
= ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ A @ C )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( C = zero_zero_complex )
=> ( A = zero_zero_complex ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5711_eq__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( A
= ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ A @ C )
= ( uminus_uminus_rat @ B ) ) )
& ( ( C = zero_zero_rat )
=> ( A = zero_zero_rat ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_5712_mult__1s__ring__1_I1_J,axiom,
! [B: real] :
( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
= ( uminus_uminus_real @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_5713_mult__1s__ring__1_I1_J,axiom,
! [B: int] :
( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_5714_mult__1s__ring__1_I1_J,axiom,
! [B: complex] :
( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_5715_mult__1s__ring__1_I1_J,axiom,
! [B: rat] :
( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
= ( uminus_uminus_rat @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_5716_mult__1s__ring__1_I1_J,axiom,
! [B: code_integer] :
( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
= ( uminus1351360451143612070nteger @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_5717_mult__1s__ring__1_I2_J,axiom,
! [B: real] :
( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
= ( uminus_uminus_real @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_5718_mult__1s__ring__1_I2_J,axiom,
! [B: int] :
( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
= ( uminus_uminus_int @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_5719_mult__1s__ring__1_I2_J,axiom,
! [B: complex] :
( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
= ( uminus1482373934393186551omplex @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_5720_mult__1s__ring__1_I2_J,axiom,
! [B: rat] :
( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
= ( uminus_uminus_rat @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_5721_mult__1s__ring__1_I2_J,axiom,
! [B: code_integer] :
( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
= ( uminus1351360451143612070nteger @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_5722_divide__eq__minus__1__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= ( uminus_uminus_real @ one_one_real ) )
= ( ( B != zero_zero_real )
& ( A
= ( uminus_uminus_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5723_divide__eq__minus__1__iff,axiom,
! [A: complex,B: complex] :
( ( ( divide1717551699836669952omplex @ A @ B )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( ( B != zero_zero_complex )
& ( A
= ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5724_divide__eq__minus__1__iff,axiom,
! [A: rat,B: rat] :
( ( ( divide_divide_rat @ A @ B )
= ( uminus_uminus_rat @ one_one_rat ) )
= ( ( B != zero_zero_rat )
& ( A
= ( uminus_uminus_rat @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_5725_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_5726_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_5727_uminus__numeral__One,axiom,
( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% uminus_numeral_One
thf(fact_5728_uminus__numeral__One,axiom,
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% uminus_numeral_One
thf(fact_5729_uminus__numeral__One,axiom,
( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% uminus_numeral_One
thf(fact_5730_power__minus,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% power_minus
thf(fact_5731_power__minus,axiom,
! [A: int,N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% power_minus
thf(fact_5732_power__minus,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% power_minus
thf(fact_5733_power__minus,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% power_minus
thf(fact_5734_power__minus,axiom,
! [A: code_integer,N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% power_minus
thf(fact_5735_power__minus__Bit0,axiom,
! [X: real,K: num] :
( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5736_power__minus__Bit0,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5737_power__minus__Bit0,axiom,
! [X: complex,K: num] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5738_power__minus__Bit0,axiom,
! [X: rat,K: num] :
( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5739_power__minus__Bit0,axiom,
! [X: code_integer,K: num] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
= ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% power_minus_Bit0
thf(fact_5740_sum__nonneg__0,axiom,
! [S2: set_int,F: int > real,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_int @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5741_sum__nonneg__0,axiom,
! [S2: set_real,F: real > real,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_real @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5742_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > real,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S2 )
= zero_zero_real )
=> ( ( member_complex @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5743_sum__nonneg__0,axiom,
! [S2: set_int,F: int > rat,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
= zero_zero_rat )
=> ( ( member_int @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5744_sum__nonneg__0,axiom,
! [S2: set_real,F: real > rat,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ S2 )
= zero_zero_rat )
=> ( ( member_real @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5745_sum__nonneg__0,axiom,
! [S2: set_nat,F: nat > rat,I2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ S2 )
= zero_zero_rat )
=> ( ( member_nat @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5746_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > rat,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
= zero_zero_rat )
=> ( ( member_complex @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_rat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5747_sum__nonneg__0,axiom,
! [S2: set_int,F: int > nat,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_int @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5748_sum__nonneg__0,axiom,
! [S2: set_real,F: real > nat,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_real @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5749_sum__nonneg__0,axiom,
! [S2: set_complex,F: complex > nat,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_complex @ I2 @ S2 )
=> ( ( F @ I2 )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_5750_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > real,B5: real,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8778361861064173332t_real @ F @ S2 )
= B5 )
=> ( ( member_int @ I2 @ S2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5751_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > real,B5: real,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S2 )
= B5 )
=> ( ( member_real @ I2 @ S2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5752_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > real,B5: real,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ( ( groups5808333547571424918x_real @ F @ S2 )
= B5 )
=> ( ( member_complex @ I2 @ S2 )
=> ( ord_less_eq_real @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5753_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > rat,B5: rat,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
= B5 )
=> ( ( member_int @ I2 @ S2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5754_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > rat,B5: rat,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ F @ S2 )
= B5 )
=> ( ( member_real @ I2 @ S2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5755_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: nat > rat,B5: rat,I2: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ F @ S2 )
= B5 )
=> ( ( member_nat @ I2 @ S2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5756_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > rat,B5: rat,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
= B5 )
=> ( ( member_complex @ I2 @ S2 )
=> ( ord_less_eq_rat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5757_sum__nonneg__leq__bound,axiom,
! [S2: set_int,F: int > nat,B5: nat,I2: int] :
( ( finite_finite_int @ S2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups4541462559716669496nt_nat @ F @ S2 )
= B5 )
=> ( ( member_int @ I2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5758_sum__nonneg__leq__bound,axiom,
! [S2: set_real,F: real > nat,B5: nat,I2: real] :
( ( finite_finite_real @ S2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S2 )
= B5 )
=> ( ( member_real @ I2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5759_sum__nonneg__leq__bound,axiom,
! [S2: set_complex,F: complex > nat,B5: nat,I2: complex] :
( ( finite3207457112153483333omplex @ S2 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups5693394587270226106ex_nat @ F @ S2 )
= B5 )
=> ( ( member_complex @ I2 @ S2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ B5 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_5760_real__0__less__add__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y4 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y4 ) ) ).
% real_0_less_add_iff
thf(fact_5761_real__add__less__0__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ ( plus_plus_real @ X @ Y4 ) @ zero_zero_real )
= ( ord_less_real @ Y4 @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_5762_real__add__le__0__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y4 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y4 @ ( uminus_uminus_real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_5763_real__0__le__add__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y4 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y4 ) ) ).
% real_0_le_add_iff
thf(fact_5764_sum__pos2,axiom,
! [I5: set_int,I2: int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I2 @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5765_sum__pos2,axiom,
! [I5: set_real,I2: real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5766_sum__pos2,axiom,
! [I5: set_complex,I2: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5767_sum__pos2,axiom,
! [I5: set_int,I2: int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I2 @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5768_sum__pos2,axiom,
! [I5: set_real,I2: real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5769_sum__pos2,axiom,
! [I5: set_nat,I2: nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I2 @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5770_sum__pos2,axiom,
! [I5: set_complex,I2: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5771_sum__pos2,axiom,
! [I5: set_int,I2: int,F: int > nat] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I2 @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5772_sum__pos2,axiom,
! [I5: set_real,I2: real,F: real > nat] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5773_sum__pos2,axiom,
! [I5: set_complex,I2: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_5774_sum__pos,axiom,
! [I5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_5775_sum__pos,axiom,
! [I5: set_int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_5776_sum__pos,axiom,
! [I5: set_real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_5777_sum__pos,axiom,
! [I5: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_5778_sum__pos,axiom,
! [I5: set_nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_5779_sum__pos,axiom,
! [I5: set_int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_5780_sum__pos,axiom,
! [I5: set_real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_5781_sum__pos,axiom,
! [I5: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_5782_sum__pos,axiom,
! [I5: set_int,F: int > nat] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_5783_sum__pos,axiom,
! [I5: set_real,F: real > nat] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).
% sum_pos
thf(fact_5784_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_5785_sum_Osubset__diff,axiom,
! [B5: set_complex,A2: set_complex,G: complex > real] :
( ( ord_le211207098394363844omplex @ B5 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups5808333547571424918x_real @ G @ A2 )
= ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5786_sum_Osubset__diff,axiom,
! [B5: set_complex,A2: set_complex,G: complex > rat] :
( ( ord_le211207098394363844omplex @ B5 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups5058264527183730370ex_rat @ G @ A2 )
= ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5787_sum_Osubset__diff,axiom,
! [B5: set_complex,A2: set_complex,G: complex > nat] :
( ( ord_le211207098394363844omplex @ B5 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups5693394587270226106ex_nat @ G @ A2 )
= ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5788_sum_Osubset__diff,axiom,
! [B5: set_complex,A2: set_complex,G: complex > int] :
( ( ord_le211207098394363844omplex @ B5 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups5690904116761175830ex_int @ G @ A2 )
= ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5789_sum_Osubset__diff,axiom,
! [B5: set_nat,A2: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups2906978787729119204at_rat @ G @ A2 )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5790_sum_Osubset__diff,axiom,
! [B5: set_nat,A2: set_nat,G: nat > int] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups3539618377306564664at_int @ G @ A2 )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5791_sum_Osubset__diff,axiom,
! [B5: set_int,A2: set_int,G: int > int] :
( ( ord_less_eq_set_int @ B5 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups4538972089207619220nt_int @ G @ A2 )
= ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5792_sum_Osubset__diff,axiom,
! [B5: set_complex,A2: set_complex,G: complex > complex] :
( ( ord_le211207098394363844omplex @ B5 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups7754918857620584856omplex @ G @ A2 )
= ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups7754918857620584856omplex @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5793_sum_Osubset__diff,axiom,
! [B5: set_nat,A2: set_nat,G: nat > nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups3542108847815614940at_nat @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5794_sum_Osubset__diff,axiom,
! [B5: set_nat,A2: set_nat,G: nat > real] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups6591440286371151544t_real @ G @ B5 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_5795_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_5796_ln__mult,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ln_ln_real @ ( times_times_real @ X @ Y4 ) )
= ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y4 ) ) ) ) ) ).
% ln_mult
thf(fact_5797_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= ( minus_minus_real @ X @ one_one_real ) )
=> ( X = one_one_real ) ) ) ).
% ln_eq_minus_one
thf(fact_5798_pos__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_5799_pos__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_5800_pos__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_5801_pos__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_5802_neg__minus__divide__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_5803_neg__minus__divide__less__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_5804_neg__less__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_5805_neg__less__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_5806_minus__divide__less__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_5807_minus__divide__less__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_5808_less__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_5809_less__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_5810_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C: real] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= ( divide_divide_real @ B @ C ) )
= ( ( ( C != zero_zero_real )
=> ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
= B ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5811_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: complex,C: complex] :
( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= ( divide1717551699836669952omplex @ B @ C ) )
= ( ( ( C != zero_zero_complex )
=> ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
= B ) )
& ( ( C = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5812_eq__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= ( divide_divide_rat @ B @ C ) )
= ( ( ( C != zero_zero_rat )
=> ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
= B ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_5813_divide__eq__eq__numeral_I2_J,axiom,
! [B: real,C: real,W2: num] :
( ( ( divide_divide_real @ B @ C )
= ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( C != zero_zero_real )
=> ( B
= ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ( C = zero_zero_real )
=> ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
= zero_zero_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5814_divide__eq__eq__numeral_I2_J,axiom,
! [B: complex,C: complex,W2: num] :
( ( ( divide1717551699836669952omplex @ B @ C )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( ( ( C != zero_zero_complex )
=> ( B
= ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
& ( ( C = zero_zero_complex )
=> ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
= zero_zero_complex ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5815_divide__eq__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ( divide_divide_rat @ B @ C )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( C != zero_zero_rat )
=> ( B
= ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ( C = zero_zero_rat )
=> ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
= zero_zero_rat ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_5816_add__divide__eq__if__simps_I3_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= B ) )
& ( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5817_add__divide__eq__if__simps_I3_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= B ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5818_add__divide__eq__if__simps_I3_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= B ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_5819_minus__divide__add__eq__iff,axiom,
! [Z2: real,X: real,Y4: real] :
( ( Z2 != zero_zero_real )
=> ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y4 )
= ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5820_minus__divide__add__eq__iff,axiom,
! [Z2: complex,X: complex,Y4: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y4 )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5821_minus__divide__add__eq__iff,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y4 )
= ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_5822_add__divide__eq__if__simps_I6_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5823_add__divide__eq__if__simps_I6_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5824_add__divide__eq__if__simps_I6_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_5825_add__divide__eq__if__simps_I5_J,axiom,
! [Z2: real,A: real,B: real] :
( ( ( Z2 = zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= ( uminus_uminus_real @ B ) ) )
& ( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
= ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5826_add__divide__eq__if__simps_I5_J,axiom,
! [Z2: complex,A: complex,B: complex] :
( ( ( Z2 = zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= ( uminus1482373934393186551omplex @ B ) ) )
& ( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5827_add__divide__eq__if__simps_I5_J,axiom,
! [Z2: rat,A: rat,B: rat] :
( ( ( Z2 = zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= ( uminus_uminus_rat @ B ) ) )
& ( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
= ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_5828_minus__divide__diff__eq__iff,axiom,
! [Z2: real,X: real,Y4: real] :
( ( Z2 != zero_zero_real )
=> ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y4 )
= ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5829_minus__divide__diff__eq__iff,axiom,
! [Z2: complex,X: complex,Y4: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y4 )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5830_minus__divide__diff__eq__iff,axiom,
! [Z2: rat,X: rat,Y4: rat] :
( ( Z2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y4 )
= ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y4 @ Z2 ) ) @ Z2 ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_5831_even__minus,axiom,
! [A: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% even_minus
thf(fact_5832_even__minus,axiom,
! [A: code_integer] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
= ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% even_minus
thf(fact_5833_power2__eq__iff,axiom,
! [X: real,Y4: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y4 )
| ( X
= ( uminus_uminus_real @ Y4 ) ) ) ) ).
% power2_eq_iff
thf(fact_5834_power2__eq__iff,axiom,
! [X: int,Y4: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y4 )
| ( X
= ( uminus_uminus_int @ Y4 ) ) ) ) ).
% power2_eq_iff
thf(fact_5835_power2__eq__iff,axiom,
! [X: complex,Y4: complex] :
( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_complex @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y4 )
| ( X
= ( uminus1482373934393186551omplex @ Y4 ) ) ) ) ).
% power2_eq_iff
thf(fact_5836_power2__eq__iff,axiom,
! [X: rat,Y4: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y4 )
| ( X
= ( uminus_uminus_rat @ Y4 ) ) ) ) ).
% power2_eq_iff
thf(fact_5837_power2__eq__iff,axiom,
! [X: code_integer,Y4: code_integer] :
( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_8256067586552552935nteger @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ( X = Y4 )
| ( X
= ( uminus1351360451143612070nteger @ Y4 ) ) ) ) ).
% power2_eq_iff
thf(fact_5838_uminus__power__if,axiom,
! [N: nat,A: real] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( power_power_real @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
= ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5839_uminus__power__if,axiom,
! [N: nat,A: int] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
= ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5840_uminus__power__if,axiom,
! [N: nat,A: complex] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( power_power_complex @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
= ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5841_uminus__power__if,axiom,
! [N: nat,A: rat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( power_power_rat @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
= ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5842_uminus__power__if,axiom,
! [N: nat,A: code_integer] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( power_8256067586552552935nteger @ A @ N ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% uminus_power_if
thf(fact_5843_verit__less__mono__div__int2,axiom,
! [A2: int,B5: int,N: int] :
( ( ord_less_eq_int @ A2 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_5844_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_5845_sum__mono2,axiom,
! [B5: set_int,A2: set_int,F: int > real] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
=> ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5846_sum__mono2,axiom,
! [B5: set_real,A2: set_real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5847_sum__mono2,axiom,
! [B5: set_complex,A2: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
=> ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5848_sum__mono2,axiom,
! [B5: set_int,A2: set_int,F: int > rat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A2 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
=> ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5849_sum__mono2,axiom,
! [B5: set_real,A2: set_real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
=> ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5850_sum__mono2,axiom,
! [B5: set_complex,A2: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
=> ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5851_sum__mono2,axiom,
! [B5: set_int,A2: set_int,F: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5852_sum__mono2,axiom,
! [B5: set_real,A2: set_real,F: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5853_sum__mono2,axiom,
! [B5: set_complex,A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5854_sum__mono2,axiom,
! [B5: set_real,A2: set_real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ B2 ) ) )
=> ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ).
% sum_mono2
thf(fact_5855_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% ln_le_minus_one
thf(fact_5856_of__bool__odd__eq__mod__2,axiom,
! [A: nat] :
( ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
= ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_5857_of__bool__odd__eq__mod__2,axiom,
! [A: int] :
( ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_5858_of__bool__odd__eq__mod__2,axiom,
! [A: code_integer] :
( ( zero_n356916108424825756nteger
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
= ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_5859_le__minus__divide__eq,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_5860_le__minus__divide__eq,axiom,
! [A: rat,B: rat,C: rat] :
( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_5861_minus__divide__le__eq,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_5862_minus__divide__le__eq,axiom,
! [B: rat,C: rat,A: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_5863_neg__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_5864_neg__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_5865_neg__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_5866_neg__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_5867_pos__le__minus__divide__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
= ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_5868_pos__le__minus__divide__eq,axiom,
! [C: rat,A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
= ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_5869_pos__minus__divide__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_5870_pos__minus__divide__le__eq,axiom,
! [C: rat,B: rat,A: rat] :
( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
= ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_5871_divide__less__eq__numeral_I2_J,axiom,
! [B: real,C: real,W2: num] :
( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5872_divide__less__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_5873_less__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5874_less__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_5875_power2__eq__1__iff,axiom,
! [A: real] :
( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real )
= ( ( A = one_one_real )
| ( A
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% power2_eq_1_iff
thf(fact_5876_power2__eq__1__iff,axiom,
! [A: int] :
( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( A = one_one_int )
| ( A
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% power2_eq_1_iff
thf(fact_5877_power2__eq__1__iff,axiom,
! [A: complex] :
( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex )
= ( ( A = one_one_complex )
| ( A
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% power2_eq_1_iff
thf(fact_5878_power2__eq__1__iff,axiom,
! [A: rat] :
( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat )
= ( ( A = one_one_rat )
| ( A
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% power2_eq_1_iff
thf(fact_5879_power2__eq__1__iff,axiom,
! [A: code_integer] :
( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_Code_integer )
= ( ( A = one_one_Code_integer )
| ( A
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% power2_eq_1_iff
thf(fact_5880_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= one_one_real ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% minus_one_power_iff
thf(fact_5881_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= one_one_int ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% minus_one_power_iff
thf(fact_5882_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= one_one_complex ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% minus_one_power_iff
thf(fact_5883_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= one_one_rat ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
= ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% minus_one_power_iff
thf(fact_5884_minus__one__power__iff,axiom,
! [N: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= one_one_Code_integer ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% minus_one_power_iff
thf(fact_5885_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5886_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5887_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5888_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5889_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5890_realpow__square__minus__le,axiom,
! [U2: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% realpow_square_minus_le
thf(fact_5891_ln__one__minus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_5892_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_5893_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_5894_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
= ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_5895_minus__mod__int__eq,axiom,
! [L2: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ L2 )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
= ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).
% minus_mod_int_eq
thf(fact_5896_zmod__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( minus_minus_int @ B @ one_one_int ) ) ) ).
% zmod_minus1
thf(fact_5897_sum__strict__mono2,axiom,
! [B5: set_int,A2: set_int,B: int,F: int > real] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( member_int @ B @ ( minus_minus_set_int @ B5 @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5898_sum__strict__mono2,axiom,
! [B5: set_real,A2: set_real,B: real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5899_sum__strict__mono2,axiom,
! [B5: set_complex,A2: set_complex,B: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ B5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5900_sum__strict__mono2,axiom,
! [B5: set_int,A2: set_int,B: int,F: int > rat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( member_int @ B @ ( minus_minus_set_int @ B5 @ A2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5901_sum__strict__mono2,axiom,
! [B5: set_real,A2: set_real,B: real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5902_sum__strict__mono2,axiom,
! [B5: set_complex,A2: set_complex,B: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
=> ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ B5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
=> ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5903_sum__strict__mono2,axiom,
! [B5: set_int,A2: set_int,B: int,F: int > nat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( member_int @ B @ ( minus_minus_set_int @ B5 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5904_sum__strict__mono2,axiom,
! [B5: set_real,A2: set_real,B: real,F: real > nat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5905_sum__strict__mono2,axiom,
! [B5: set_complex,A2: set_complex,B: complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ B5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5906_sum__strict__mono2,axiom,
! [B5: set_real,A2: set_real,B: real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ B5 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_5907_zdiv__zminus1__eq__if,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_5908_zdiv__zminus2__eq__if,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( ( ( modulo_modulo_int @ A @ B )
= zero_zero_int )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
& ( ( ( modulo_modulo_int @ A @ B )
!= zero_zero_int )
=> ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
= ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_5909_zminus1__lemma,axiom,
! [A: int,B: int,Q2: int,R2: int] :
( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
=> ( ( B != zero_zero_int )
=> ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_5910_bits__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [A3: nat] :
( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ A3 ) )
=> ( ! [A3: nat,B2: $o] :
( ( P @ A3 )
=> ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_5911_bits__induct,axiom,
! [P: int > $o,A: int] :
( ! [A3: int] :
( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ A3 ) )
=> ( ! [A3: int,B2: $o] :
( ( P @ A3 )
=> ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_5912_bits__induct,axiom,
! [P: code_integer > $o,A: code_integer] :
( ! [A3: code_integer] :
( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ A3 ) )
=> ( ! [A3: code_integer,B2: $o] :
( ( P @ A3 )
=> ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
= A3 )
=> ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
=> ( P @ A ) ) ) ).
% bits_induct
thf(fact_5913_le__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: real,C: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_5914_le__divide__eq__numeral_I2_J,axiom,
! [W2: num,B: rat,C: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_5915_divide__le__eq__numeral_I2_J,axiom,
! [B: real,C: real,W2: num] :
( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_real @ zero_zero_real @ C )
=> ( ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_5916_divide__le__eq__numeral_I2_J,axiom,
! [B: rat,C: rat,W2: num] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ C )
=> ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
& ( ~ ( ord_less_rat @ zero_zero_rat @ C )
=> ( ( ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
& ( ~ ( ord_less_rat @ C @ zero_zero_rat )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_5917_square__le__1,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% square_le_1
thf(fact_5918_square__le__1,axiom,
! [X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
=> ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% square_le_1
thf(fact_5919_square__le__1,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
=> ( ( ord_less_eq_rat @ X @ one_one_rat )
=> ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% square_le_1
thf(fact_5920_square__le__1,axiom,
! [X: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
=> ( ( ord_less_eq_int @ X @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% square_le_1
thf(fact_5921_minus__power__mult__self,axiom,
! [A: real,N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
= ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_5922_minus__power__mult__self,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_5923_minus__power__mult__self,axiom,
! [A: complex,N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
= ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_5924_minus__power__mult__self,axiom,
! [A: rat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
= ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_5925_minus__power__mult__self,axiom,
! [A: code_integer,N: nat] :
( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
= ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% minus_power_mult_self
thf(fact_5926_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri631733984087533419it_int @ N @ K )
= K )
= ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
& ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_5927_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
=> ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_ri631733984087533419it_int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_5928_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% minus_1_div_exp_eq_int
thf(fact_5929_div__pos__neg__trivial,axiom,
! [K: int,L2: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L2 )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_5930_add__0__iff,axiom,
! [B: complex,A: complex] :
( ( B
= ( plus_plus_complex @ B @ A ) )
= ( A = zero_zero_complex ) ) ).
% add_0_iff
thf(fact_5931_add__0__iff,axiom,
! [B: real,A: real] :
( ( B
= ( plus_plus_real @ B @ A ) )
= ( A = zero_zero_real ) ) ).
% add_0_iff
thf(fact_5932_add__0__iff,axiom,
! [B: rat,A: rat] :
( ( B
= ( plus_plus_rat @ B @ A ) )
= ( A = zero_zero_rat ) ) ).
% add_0_iff
thf(fact_5933_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_5934_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_5935_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_5936_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_5937_exp__mod__exp,axiom,
! [M: nat,N: nat] :
( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% exp_mod_exp
thf(fact_5938_crossproduct__eq,axiom,
! [W2: real,Y4: real,X: real,Z2: real] :
( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y4 ) @ ( times_times_real @ X @ Z2 ) )
= ( plus_plus_real @ ( times_times_real @ W2 @ Z2 ) @ ( times_times_real @ X @ Y4 ) ) )
= ( ( W2 = X )
| ( Y4 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_5939_crossproduct__eq,axiom,
! [W2: rat,Y4: rat,X: rat,Z2: rat] :
( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y4 ) @ ( times_times_rat @ X @ Z2 ) )
= ( plus_plus_rat @ ( times_times_rat @ W2 @ Z2 ) @ ( times_times_rat @ X @ Y4 ) ) )
= ( ( W2 = X )
| ( Y4 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_5940_crossproduct__eq,axiom,
! [W2: nat,Y4: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y4 ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W2 @ Z2 ) @ ( times_times_nat @ X @ Y4 ) ) )
= ( ( W2 = X )
| ( Y4 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_5941_crossproduct__eq,axiom,
! [W2: int,Y4: int,X: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y4 ) @ ( times_times_int @ X @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z2 ) @ ( times_times_int @ X @ Y4 ) ) )
= ( ( W2 = X )
| ( Y4 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_5942_crossproduct__noteq,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
!= ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_5943_crossproduct__noteq,axiom,
! [A: rat,B: rat,C: rat,D: rat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
!= ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_5944_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_5945_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_5946_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% power_minus1_odd
thf(fact_5947_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% power_minus1_odd
thf(fact_5948_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% power_minus1_odd
thf(fact_5949_power__minus1__odd,axiom,
! [N: nat] :
( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% power_minus1_odd
thf(fact_5950_power__minus1__odd,axiom,
! [N: nat] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% power_minus1_odd
thf(fact_5951_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_5952_ln__one__plus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_5953_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_5954_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= zero_zero_nat )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_5955_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
= ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
!= zero_zero_int )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_5956_exp__div__exp__eq,axiom,
! [M: nat,N: nat] :
( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
= ( times_3573771949741848930nteger
@ ( zero_n356916108424825756nteger
@ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
!= zero_z3403309356797280102nteger )
& ( ord_less_eq_nat @ N @ M ) ) )
@ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% exp_div_exp_eq
thf(fact_5957_vebt__buildup_Osimps_I3_J,axiom,
! [Va: nat] :
( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_5958_ln__2__less__1,axiom,
ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% ln_2_less_1
thf(fact_5959_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_5960_tanh__ln__real,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( tanh_real @ ( ln_ln_real @ X ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% tanh_ln_real
thf(fact_5961_Divides_Oadjust__div__eq,axiom,
! [Q2: int,R2: int] :
( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R2 ) )
= ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_5962_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_5963_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_5964_vebt__buildup_Opelims,axiom,
! [X: nat,Y4: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y4 )
=> ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X = zero_zero_nat )
=> ( ( Y4
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
=> ( ( ( X
= ( suc @ zero_zero_nat ) )
=> ( ( Y4
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
=> ~ ! [Va3: nat] :
( ( X
= ( suc @ ( suc @ Va3 ) ) )
=> ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y4
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
& ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( Y4
= ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_5965_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_5966_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_5967_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_5968_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_5969_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_5970_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_5971_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% abs_numeral
thf(fact_5972_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% abs_numeral
thf(fact_5973_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_rat @ N ) ) ).
% abs_numeral
thf(fact_5974_abs__numeral,axiom,
! [N: num] :
( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% abs_numeral
thf(fact_5975_abs__mult__self__eq,axiom,
! [A: code_integer] :
( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
= ( times_3573771949741848930nteger @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_5976_abs__mult__self__eq,axiom,
! [A: real] :
( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
= ( times_times_real @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_5977_abs__mult__self__eq,axiom,
! [A: rat] :
( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
= ( times_times_rat @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_5978_abs__mult__self__eq,axiom,
! [A: int] :
( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
= ( times_times_int @ A @ A ) ) ).
% abs_mult_self_eq
thf(fact_5979_abs__1,axiom,
( ( abs_abs_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% abs_1
thf(fact_5980_abs__1,axiom,
( ( abs_abs_complex @ one_one_complex )
= one_one_complex ) ).
% abs_1
thf(fact_5981_abs__1,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_1
thf(fact_5982_abs__1,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
% abs_1
thf(fact_5983_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_5984_abs__add__abs,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
= ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_add_abs
thf(fact_5985_abs__add__abs,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
= ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_add_abs
thf(fact_5986_abs__add__abs,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
= ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_add_abs
thf(fact_5987_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_5988_abs__divide,axiom,
! [A: complex,B: complex] :
( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% abs_divide
thf(fact_5989_abs__divide,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_divide
thf(fact_5990_abs__divide,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_divide
thf(fact_5991_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_5992_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_5993_abs__le__zero__iff,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
= ( A = zero_z3403309356797280102nteger ) ) ).
% abs_le_zero_iff
thf(fact_5994_abs__le__zero__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
= ( A = zero_zero_real ) ) ).
% abs_le_zero_iff
thf(fact_5995_abs__le__zero__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
= ( A = zero_zero_rat ) ) ).
% abs_le_zero_iff
thf(fact_5996_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_5997_abs__le__self__iff,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
= ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% abs_le_self_iff
thf(fact_5998_abs__le__self__iff,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% abs_le_self_iff
thf(fact_5999_abs__le__self__iff,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% abs_le_self_iff
thf(fact_6000_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_6001_abs__of__nonneg,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_6002_abs__of__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_6003_abs__of__nonneg,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_6004_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_6005_zero__less__abs__iff,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
= ( A != zero_z3403309356797280102nteger ) ) ).
% zero_less_abs_iff
thf(fact_6006_zero__less__abs__iff,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
= ( A != zero_zero_real ) ) ).
% zero_less_abs_iff
thf(fact_6007_zero__less__abs__iff,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
= ( A != zero_zero_rat ) ) ).
% zero_less_abs_iff
thf(fact_6008_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_6009_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( numeral_numeral_real @ N ) ) ).
% abs_neg_numeral
thf(fact_6010_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ N ) ) ).
% abs_neg_numeral
thf(fact_6011_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( numeral_numeral_rat @ N ) ) ).
% abs_neg_numeral
thf(fact_6012_abs__neg__numeral,axiom,
! [N: num] :
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% abs_neg_numeral
thf(fact_6013_abs__neg__one,axiom,
( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
= one_one_real ) ).
% abs_neg_one
thf(fact_6014_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_6015_abs__neg__one,axiom,
( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= one_one_rat ) ).
% abs_neg_one
thf(fact_6016_abs__neg__one,axiom,
( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= one_one_Code_integer ) ).
% abs_neg_one
thf(fact_6017_abs__power__minus,axiom,
! [A: real,N: nat] :
( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
= ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_6018_abs__power__minus,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_6019_abs__power__minus,axiom,
! [A: rat,N: nat] :
( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
= ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_6020_abs__power__minus,axiom,
! [A: code_integer,N: nat] :
( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
= ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_6021_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_6022_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_6023_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_6024_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_6025_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_6026_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_6027_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_6028_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_6029_sum__abs,axiom,
! [F: int > int,A2: set_int] :
( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
@ ( groups4538972089207619220nt_int
@ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
@ A2 ) ) ).
% sum_abs
thf(fact_6030_sum__abs,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
@ A2 ) ) ).
% sum_abs
thf(fact_6031_divide__le__0__abs__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
= ( ( ord_less_eq_real @ A @ zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_le_0_abs_iff
thf(fact_6032_divide__le__0__abs__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
= ( ( ord_less_eq_rat @ A @ zero_zero_rat )
| ( B = zero_zero_rat ) ) ) ).
% divide_le_0_abs_iff
thf(fact_6033_zero__le__divide__abs__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A )
| ( B = zero_zero_real ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_6034_zero__le__divide__abs__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
= ( ( ord_less_eq_rat @ zero_zero_rat @ A )
| ( B = zero_zero_rat ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_6035_abs__of__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_nonpos
thf(fact_6036_abs__of__nonpos,axiom,
! [A: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_nonpos
thf(fact_6037_abs__of__nonpos,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ A @ zero_zero_rat )
=> ( ( abs_abs_rat @ A )
= ( uminus_uminus_rat @ A ) ) ) ).
% abs_of_nonpos
thf(fact_6038_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_6039_zdiv__numeral__Bit1,axiom,
! [V: num,W2: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
= ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_6040_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_6041_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_6042_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_6043_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_6044_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_6045_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( artanh_real @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_6046_sum__abs__ge__zero,axiom,
! [F: int > int,A2: set_int] :
( ord_less_eq_int @ zero_zero_int
@ ( groups4538972089207619220nt_int
@ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_6047_sum__abs__ge__zero,axiom,
! [F: nat > real,A2: set_nat] :
( ord_less_eq_real @ zero_zero_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
@ A2 ) ) ).
% sum_abs_ge_zero
thf(fact_6048_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_6049_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_6050_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_6051_zero__less__power__abs__iff,axiom,
! [A: code_integer,N: nat] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
= ( ( A != zero_z3403309356797280102nteger )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_6052_zero__less__power__abs__iff,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
= ( ( A != zero_zero_real )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_6053_zero__less__power__abs__iff,axiom,
! [A: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
= ( ( A != zero_zero_rat )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_6054_zero__less__power__abs__iff,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
= ( ( A != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_6055_abs__power2,axiom,
! [A: rat] :
( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% abs_power2
thf(fact_6056_abs__power2,axiom,
! [A: code_integer] :
( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% abs_power2
thf(fact_6057_abs__power2,axiom,
! [A: real] :
( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% abs_power2
thf(fact_6058_abs__power2,axiom,
! [A: int] :
( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% abs_power2
thf(fact_6059_power2__abs,axiom,
! [A: rat] :
( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_abs
thf(fact_6060_power2__abs,axiom,
! [A: code_integer] :
( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_abs
thf(fact_6061_power2__abs,axiom,
! [A: real] :
( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_abs
thf(fact_6062_power2__abs,axiom,
! [A: int] :
( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% power2_abs
thf(fact_6063_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6064_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6065_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6066_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6067_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_6068_sum__zero__power,axiom,
! [A2: set_nat,C: nat > complex] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
@ A2 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
@ A2 )
= zero_zero_complex ) ) ) ).
% sum_zero_power
thf(fact_6069_sum__zero__power,axiom,
! [A2: set_nat,C: nat > rat] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
@ A2 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
@ A2 )
= zero_zero_rat ) ) ) ).
% sum_zero_power
thf(fact_6070_sum__zero__power,axiom,
! [A2: set_nat,C: nat > real] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
@ A2 )
= ( C @ zero_zero_nat ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
@ A2 )
= zero_zero_real ) ) ) ).
% sum_zero_power
thf(fact_6071_power__even__abs__numeral,axiom,
! [W2: num,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
=> ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W2 ) )
= ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_even_abs_numeral
thf(fact_6072_power__even__abs__numeral,axiom,
! [W2: num,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
=> ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W2 ) )
= ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_even_abs_numeral
thf(fact_6073_power__even__abs__numeral,axiom,
! [W2: num,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
=> ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W2 ) )
= ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_even_abs_numeral
thf(fact_6074_power__even__abs__numeral,axiom,
! [W2: num,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
=> ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W2 ) )
= ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_even_abs_numeral
thf(fact_6075_div__Suc__eq__div__add3,axiom,
! [M: nat,N: nat] :
( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_6076_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V: num] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_6077_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V: num] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_6078_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_6079_sum__zero__power_H,axiom,
! [A2: set_nat,C: nat > complex,D: nat > complex] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
@ A2 )
= ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
@ A2 )
= zero_zero_complex ) ) ) ).
% sum_zero_power'
thf(fact_6080_sum__zero__power_H,axiom,
! [A2: set_nat,C: nat > rat,D: nat > rat] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
@ A2 )
= ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
@ A2 )
= zero_zero_rat ) ) ) ).
% sum_zero_power'
thf(fact_6081_sum__zero__power_H,axiom,
! [A2: set_nat,C: nat > real,D: nat > real] :
( ( ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
@ A2 )
= ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
& ( ~ ( ( finite_finite_nat @ A2 )
& ( member_nat @ zero_zero_nat @ A2 ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
@ A2 )
= zero_zero_real ) ) ) ).
% sum_zero_power'
thf(fact_6082_zmod__numeral__Bit1,axiom,
! [V: num,W2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
= ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W2 ) ) ) @ one_one_int ) ) ).
% zmod_numeral_Bit1
thf(fact_6083_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_6084_abs__le__D1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ A @ B ) ) ).
% abs_le_D1
thf(fact_6085_abs__le__D1,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% abs_le_D1
thf(fact_6086_abs__le__D1,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
=> ( ord_less_eq_rat @ A @ B ) ) ).
% abs_le_D1
thf(fact_6087_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_6088_abs__ge__self,axiom,
! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% abs_ge_self
thf(fact_6089_abs__ge__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_self
thf(fact_6090_abs__ge__self,axiom,
! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% abs_ge_self
thf(fact_6091_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_6092_abs__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_mult
thf(fact_6093_abs__mult,axiom,
! [A: real,B: real] :
( ( abs_abs_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_mult
thf(fact_6094_abs__mult,axiom,
! [A: rat,B: rat] :
( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_mult
thf(fact_6095_abs__mult,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_mult
thf(fact_6096_abs__one,axiom,
( ( abs_abs_Code_integer @ one_one_Code_integer )
= one_one_Code_integer ) ).
% abs_one
thf(fact_6097_abs__one,axiom,
( ( abs_abs_real @ one_one_real )
= one_one_real ) ).
% abs_one
thf(fact_6098_abs__one,axiom,
( ( abs_abs_rat @ one_one_rat )
= one_one_rat ) ).
% abs_one
thf(fact_6099_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_6100_power__abs,axiom,
! [A: rat,N: nat] :
( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
= ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% power_abs
thf(fact_6101_power__abs,axiom,
! [A: code_integer,N: nat] :
( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
= ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% power_abs
thf(fact_6102_power__abs,axiom,
! [A: real,N: nat] :
( ( abs_abs_real @ ( power_power_real @ A @ N ) )
= ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% power_abs
thf(fact_6103_power__abs,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ A @ N ) )
= ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% power_abs
thf(fact_6104_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_6105_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_6106_abs__ge__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_zero
thf(fact_6107_abs__ge__zero,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% abs_ge_zero
thf(fact_6108_abs__ge__zero,axiom,
! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% abs_ge_zero
thf(fact_6109_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_6110_abs__not__less__zero,axiom,
! [A: code_integer] :
~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% abs_not_less_zero
thf(fact_6111_abs__not__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% abs_not_less_zero
thf(fact_6112_abs__not__less__zero,axiom,
! [A: rat] :
~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% abs_not_less_zero
thf(fact_6113_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_6114_abs__of__pos,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
=> ( ( abs_abs_Code_integer @ A )
= A ) ) ).
% abs_of_pos
thf(fact_6115_abs__of__pos,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( abs_abs_real @ A )
= A ) ) ).
% abs_of_pos
thf(fact_6116_abs__of__pos,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( abs_abs_rat @ A )
= A ) ) ).
% abs_of_pos
thf(fact_6117_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_6118_abs__triangle__ineq,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_6119_abs__triangle__ineq,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_6120_abs__triangle__ineq,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_6121_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_6122_abs__mult__less,axiom,
! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
=> ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
=> ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_6123_abs__mult__less,axiom,
! [A: real,C: real,B: real,D: real] :
( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
=> ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
=> ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_6124_abs__mult__less,axiom,
! [A: rat,C: rat,B: rat,D: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
=> ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
=> ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_6125_abs__mult__less,axiom,
! [A: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
=> ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_6126_abs__triangle__ineq2__sym,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6127_abs__triangle__ineq2__sym,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6128_abs__triangle__ineq2__sym,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6129_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_6130_abs__triangle__ineq3,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_6131_abs__triangle__ineq3,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_6132_abs__triangle__ineq3,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_6133_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_6134_abs__triangle__ineq2,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_6135_abs__triangle__ineq2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_6136_abs__triangle__ineq2,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_6137_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_6138_nonzero__abs__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% nonzero_abs_divide
thf(fact_6139_nonzero__abs__divide,axiom,
! [B: rat,A: rat] :
( ( B != zero_zero_rat )
=> ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% nonzero_abs_divide
thf(fact_6140_abs__ge__minus__self,axiom,
! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% abs_ge_minus_self
thf(fact_6141_abs__ge__minus__self,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% abs_ge_minus_self
thf(fact_6142_abs__ge__minus__self,axiom,
! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% abs_ge_minus_self
thf(fact_6143_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_6144_abs__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_6145_abs__le__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
= ( ( ord_le3102999989581377725nteger @ A @ B )
& ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_6146_abs__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
= ( ( ord_less_eq_rat @ A @ B )
& ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_6147_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_6148_abs__le__D2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_6149_abs__le__D2,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_6150_abs__le__D2,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
=> ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_6151_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_6152_abs__leI,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
=> ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_6153_abs__leI,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le3102999989581377725nteger @ A @ B )
=> ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_6154_abs__leI,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_6155_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_6156_abs__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
= ( ( ord_less_real @ A @ B )
& ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_6157_abs__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_int @ A @ B )
& ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_6158_abs__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
= ( ( ord_less_rat @ A @ B )
& ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_6159_abs__less__iff,axiom,
! [A: code_integer,B: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
= ( ( ord_le6747313008572928689nteger @ A @ B )
& ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_6160_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ ( suc @ X3 ) @ A2 )
=> ( ( F @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_6161_sum__cong__Suc,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ ( suc @ X3 ) @ A2 )
=> ( ( F @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ F @ A2 )
= ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% sum_cong_Suc
thf(fact_6162_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
=> ( ! [M4: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) )
=> ( ! [M4: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
=> ( ! [M4: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) )
=> ~ ! [M4: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_6163_num_Oexhaust,axiom,
! [Y4: num] :
( ( Y4 != one )
=> ( ! [X23: num] :
( Y4
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y4
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_6164_sin__bound__lemma,axiom,
! [X: real,Y4: real,U2: real,V: real] :
( ( X = Y4 )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ U2 ) @ V )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U2 ) @ Y4 ) ) @ V ) ) ) ).
% sin_bound_lemma
thf(fact_6165_sum__subtractf__nat,axiom,
! [A2: set_int,G: int > nat,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups4541462559716669496nt_nat
@ ^ [X2: int] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6166_sum__subtractf__nat,axiom,
! [A2: set_real,G: real > nat,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6167_sum__subtractf__nat,axiom,
! [A2: set_complex,G: complex > nat,F: complex > nat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups5693394587270226106ex_nat
@ ^ [X2: complex] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6168_sum__subtractf__nat,axiom,
! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups977919841031483927at_nat
@ ^ [X2: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6169_sum__subtractf__nat,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A2 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_6170_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_6171_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_6172_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_6173_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > real,M: nat,K: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_6174_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% tanh_real_lt_1
thf(fact_6175_tanh__real__gt__neg1,axiom,
! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% tanh_real_gt_neg1
thf(fact_6176_dense__eq0__I,axiom,
! [X: real] :
( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
=> ( X = zero_zero_real ) ) ).
% dense_eq0_I
thf(fact_6177_dense__eq0__I,axiom,
! [X: rat] :
( ! [E2: rat] :
( ( ord_less_rat @ zero_zero_rat @ E2 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
=> ( X = zero_zero_rat ) ) ).
% dense_eq0_I
thf(fact_6178_abs__eq__mult,axiom,
! [A: code_integer,B: code_integer] :
( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
| ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
& ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
| ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
=> ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
= ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_6179_abs__eq__mult,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
| ( ord_less_eq_real @ A @ zero_zero_real ) )
& ( ( ord_less_eq_real @ zero_zero_real @ B )
| ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_6180_abs__eq__mult,axiom,
! [A: rat,B: rat] :
( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
| ( ord_less_eq_rat @ A @ zero_zero_rat ) )
& ( ( ord_less_eq_rat @ zero_zero_rat @ B )
| ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
=> ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_6181_abs__eq__mult,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
| ( ord_less_eq_int @ A @ zero_zero_int ) )
& ( ( ord_less_eq_int @ zero_zero_int @ B )
| ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
= ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% abs_eq_mult
thf(fact_6182_abs__mult__pos,axiom,
! [X: code_integer,Y4: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
=> ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y4 ) @ X )
= ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y4 @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6183_abs__mult__pos,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ ( abs_abs_real @ Y4 ) @ X )
= ( abs_abs_real @ ( times_times_real @ Y4 @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6184_abs__mult__pos,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X )
=> ( ( times_times_rat @ ( abs_abs_rat @ Y4 ) @ X )
= ( abs_abs_rat @ ( times_times_rat @ Y4 @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6185_abs__mult__pos,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( times_times_int @ ( abs_abs_int @ Y4 ) @ X )
= ( abs_abs_int @ ( times_times_int @ Y4 @ X ) ) ) ) ).
% abs_mult_pos
thf(fact_6186_abs__eq__iff_H,axiom,
! [A: real,B: real] :
( ( ( abs_abs_real @ A )
= B )
= ( ( ord_less_eq_real @ zero_zero_real @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_real @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6187_abs__eq__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( ( abs_abs_Code_integer @ A )
= B )
= ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
& ( ( A = B )
| ( A
= ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6188_abs__eq__iff_H,axiom,
! [A: rat,B: rat] :
( ( ( abs_abs_rat @ A )
= B )
= ( ( ord_less_eq_rat @ zero_zero_rat @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_rat @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6189_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_6190_eq__abs__iff_H,axiom,
! [A: real,B: real] :
( ( A
= ( abs_abs_real @ B ) )
= ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_real @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6191_eq__abs__iff_H,axiom,
! [A: code_integer,B: code_integer] :
( ( A
= ( abs_abs_Code_integer @ B ) )
= ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
& ( ( B = A )
| ( B
= ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6192_eq__abs__iff_H,axiom,
! [A: rat,B: rat] :
( ( A
= ( abs_abs_rat @ B ) )
= ( ( ord_less_eq_rat @ zero_zero_rat @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_rat @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6193_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_6194_abs__minus__le__zero,axiom,
! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% abs_minus_le_zero
thf(fact_6195_abs__minus__le__zero,axiom,
! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% abs_minus_le_zero
thf(fact_6196_abs__minus__le__zero,axiom,
! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% abs_minus_le_zero
thf(fact_6197_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_6198_abs__div__pos,axiom,
! [Y4: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y4 )
= ( abs_abs_real @ ( divide_divide_real @ X @ Y4 ) ) ) ) ).
% abs_div_pos
thf(fact_6199_abs__div__pos,axiom,
! [Y4: rat,X: rat] :
( ( ord_less_rat @ zero_zero_rat @ Y4 )
=> ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y4 )
= ( abs_abs_rat @ ( divide_divide_rat @ X @ Y4 ) ) ) ) ).
% abs_div_pos
thf(fact_6200_zero__le__power__abs,axiom,
! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_6201_zero__le__power__abs,axiom,
! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_6202_zero__le__power__abs,axiom,
! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_6203_zero__le__power__abs,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_6204_abs__of__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( abs_abs_real @ A )
= ( uminus_uminus_real @ A ) ) ) ).
% abs_of_neg
thf(fact_6205_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_6206_abs__of__neg,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( abs_abs_rat @ A )
= ( uminus_uminus_rat @ A ) ) ) ).
% abs_of_neg
thf(fact_6207_abs__of__neg,axiom,
! [A: code_integer] :
( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
=> ( ( abs_abs_Code_integer @ A )
= ( uminus1351360451143612070nteger @ A ) ) ) ).
% abs_of_neg
thf(fact_6208_abs__if,axiom,
( abs_abs_real
= ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_6209_abs__if,axiom,
( abs_abs_int
= ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_6210_abs__if,axiom,
( abs_abs_rat
= ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_6211_abs__if,axiom,
( abs_abs_Code_integer
= ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_6212_abs__if__raw,axiom,
( abs_abs_real
= ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_6213_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_6214_abs__if__raw,axiom,
( abs_abs_rat
= ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_6215_abs__if__raw,axiom,
( abs_abs_Code_integer
= ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_6216_abs__diff__triangle__ineq,axiom,
! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6217_abs__diff__triangle__ineq,axiom,
! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6218_abs__diff__triangle__ineq,axiom,
! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6219_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_6220_abs__triangle__ineq4,axiom,
! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_6221_abs__triangle__ineq4,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_6222_abs__triangle__ineq4,axiom,
! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_6223_abs__triangle__ineq4,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_6224_abs__diff__le__iff,axiom,
! [X: code_integer,A: code_integer,R2: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
= ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
& ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_6225_abs__diff__le__iff,axiom,
! [X: real,A: real,R2: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
= ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
& ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_6226_abs__diff__le__iff,axiom,
! [X: rat,A: rat,R2: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
= ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
& ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_6227_abs__diff__le__iff,axiom,
! [X: int,A: int,R2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
& ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_6228_abs__diff__less__iff,axiom,
! [X: code_integer,A: code_integer,R2: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
= ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
& ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_6229_abs__diff__less__iff,axiom,
! [X: real,A: real,R2: real] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
= ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
& ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_6230_abs__diff__less__iff,axiom,
! [X: rat,A: rat,R2: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
= ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
& ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_6231_abs__diff__less__iff,axiom,
! [X: int,A: int,R2: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
= ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_6232_sum__eq__Suc0__iff,axiom,
! [A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( ( F @ X2 )
= ( suc @ zero_zero_nat ) )
& ! [Y: complex] :
( ( member_complex @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_6233_sum__eq__Suc0__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( F @ X2 )
= ( suc @ zero_zero_nat ) )
& ! [Y: nat] :
( ( member_nat @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_6234_sum__SucD,axiom,
! [F: nat > nat,A2: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A2 )
= ( suc @ N ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_6235_sum__eq__1__iff,axiom,
! [A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
= one_one_nat )
= ( ? [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( ( F @ X2 )
= one_one_nat )
& ! [Y: complex] :
( ( member_complex @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_6236_sum__eq__1__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ A2 )
= one_one_nat )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( F @ X2 )
= one_one_nat )
& ! [Y: nat] :
( ( member_nat @ Y @ A2 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_6237_numeral__Bit1,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% numeral_Bit1
thf(fact_6238_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_Bit1
thf(fact_6239_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_Bit1
thf(fact_6240_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_6241_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_Bit1
thf(fact_6242_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_6243_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_6244_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_6245_cong__exp__iff__simps_I13_J,axiom,
! [M: num,Q2: num,N: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_6246_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_6247_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_6248_cong__exp__iff__simps_I12_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_6249_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_6250_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_6251_cong__exp__iff__simps_I10_J,axiom,
! [M: num,Q2: num,N: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
!= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_6252_power__minus__Bit1,axiom,
! [X: real,K: num] :
( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_6253_power__minus__Bit1,axiom,
! [X: int,K: num] :
( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_6254_power__minus__Bit1,axiom,
! [X: complex,K: num] :
( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_6255_power__minus__Bit1,axiom,
! [X: rat,K: num] :
( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_6256_power__minus__Bit1,axiom,
! [X: code_integer,K: num] :
( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
= ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_6257_sum__power__add,axiom,
! [X: complex,M: nat,I5: set_nat] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I4 ) )
@ I5 )
= ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_6258_sum__power__add,axiom,
! [X: rat,M: nat,I5: set_nat] :
( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I4 ) )
@ I5 )
= ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_6259_sum__power__add,axiom,
! [X: int,M: nat,I5: set_nat] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I4 ) )
@ I5 )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_6260_sum__power__add,axiom,
! [X: real,M: nat,I5: set_nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I4 ) )
@ I5 )
= ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).
% sum_power_add
thf(fact_6261_sum_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_6262_sum_OatLeastAtMost__rev,axiom,
! [G: nat > real,N: nat,M: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_6263_numeral__code_I3_J,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
= ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% numeral_code(3)
thf(fact_6264_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit1 @ N ) )
= ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% numeral_code(3)
thf(fact_6265_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bit1 @ N ) )
= ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% numeral_code(3)
thf(fact_6266_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_code(3)
thf(fact_6267_numeral__code_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit1 @ N ) )
= ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% numeral_code(3)
thf(fact_6268_power__numeral__odd,axiom,
! [Z2: complex,W2: num] :
( ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
= ( times_times_complex @ ( times_times_complex @ Z2 @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_numeral_odd
thf(fact_6269_power__numeral__odd,axiom,
! [Z2: real,W2: num] :
( ( power_power_real @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
= ( times_times_real @ ( times_times_real @ Z2 @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_numeral_odd
thf(fact_6270_power__numeral__odd,axiom,
! [Z2: rat,W2: num] :
( ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
= ( times_times_rat @ ( times_times_rat @ Z2 @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_numeral_odd
thf(fact_6271_power__numeral__odd,axiom,
! [Z2: nat,W2: num] :
( ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
= ( times_times_nat @ ( times_times_nat @ Z2 @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_numeral_odd
thf(fact_6272_power__numeral__odd,axiom,
! [Z2: int,W2: num] :
( ( power_power_int @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
= ( times_times_int @ ( times_times_int @ Z2 @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).
% power_numeral_odd
thf(fact_6273_sum__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X2: complex] : X2
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= C ) ) )
= zero_zero_complex ) ) ).
% sum_nth_roots
thf(fact_6274_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( groups7754918857620584856omplex
@ ^ [X2: complex] : X2
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= one_one_complex ) ) )
= zero_zero_complex ) ) ).
% sum_roots_unity
thf(fact_6275_abs__add__one__gt__zero,axiom,
! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_6276_abs__add__one__gt__zero,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_6277_abs__add__one__gt__zero,axiom,
! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_6278_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_6279_sum__shift__lb__Suc0__0,axiom,
! [F: nat > complex,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6280_sum__shift__lb__Suc0__0,axiom,
! [F: nat > rat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_rat )
=> ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6281_sum__shift__lb__Suc0__0,axiom,
! [F: nat > int,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_int )
=> ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6282_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6283_sum__shift__lb__Suc0__0,axiom,
! [F: nat > real,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_6284_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6285_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6286_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6287_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_6288_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6289_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6290_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6291_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_6292_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6293_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6294_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6295_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_6296_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_6297_numeral__Bit1__div__2,axiom,
! [N: num] :
( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ N ) ) ).
% numeral_Bit1_div_2
thf(fact_6298_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_6299_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_6300_odd__numeral,axiom,
! [N: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% odd_numeral
thf(fact_6301_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] :
( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
!= zero_zero_nat ) ).
% cong_exp_iff_simps(3)
thf(fact_6302_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] :
( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
!= zero_zero_int ) ).
% cong_exp_iff_simps(3)
thf(fact_6303_cong__exp__iff__simps_I3_J,axiom,
! [N: num,Q2: num] :
( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
!= zero_z3403309356797280102nteger ) ).
% cong_exp_iff_simps(3)
thf(fact_6304_power3__eq__cube,axiom,
! [A: complex] :
( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_6305_power3__eq__cube,axiom,
! [A: real] :
( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_6306_power3__eq__cube,axiom,
! [A: rat] :
( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_6307_power3__eq__cube,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_6308_power3__eq__cube,axiom,
! [A: int] :
( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
= ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% power3_eq_cube
thf(fact_6309_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_6310_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_6311_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ M )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6312_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ M )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6313_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ M )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6314_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ M )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_6315_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_6316_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_6317_sum__Suc__diff,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% sum_Suc_diff
thf(fact_6318_mod__exhaust__less__4,axiom,
! [M: nat] :
( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= zero_zero_nat )
| ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= one_one_nat )
| ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
| ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% mod_exhaust_less_4
thf(fact_6319_abs__le__square__iff,axiom,
! [X: code_integer,Y4: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y4 ) )
= ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_6320_abs__le__square__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y4 ) )
= ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_6321_abs__le__square__iff,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y4 ) )
= ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_6322_abs__le__square__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y4 ) )
= ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_6323_abs__square__eq__1,axiom,
! [X: code_integer] :
( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_Code_integer )
= ( ( abs_abs_Code_integer @ X )
= one_one_Code_integer ) ) ).
% abs_square_eq_1
thf(fact_6324_abs__square__eq__1,axiom,
! [X: rat] :
( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_rat )
= ( ( abs_abs_rat @ X )
= one_one_rat ) ) ).
% abs_square_eq_1
thf(fact_6325_abs__square__eq__1,axiom,
! [X: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real )
= ( ( abs_abs_real @ X )
= one_one_real ) ) ).
% abs_square_eq_1
thf(fact_6326_abs__square__eq__1,axiom,
! [X: int] :
( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_int )
= ( ( abs_abs_int @ X )
= one_one_int ) ) ).
% abs_square_eq_1
thf(fact_6327_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(6)
thf(fact_6328_power__even__abs,axiom,
! [N: nat,A: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
= ( power_power_rat @ A @ N ) ) ) ).
% power_even_abs
thf(fact_6329_power__even__abs,axiom,
! [N: nat,A: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
= ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% power_even_abs
thf(fact_6330_power__even__abs,axiom,
! [N: nat,A: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
= ( power_power_real @ A @ N ) ) ) ).
% power_even_abs
thf(fact_6331_power__even__abs,axiom,
! [N: nat,A: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
= ( power_power_int @ A @ N ) ) ) ).
% power_even_abs
thf(fact_6332_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > rat,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6333_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > int,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6334_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > nat,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6335_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > real,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_6336_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q2: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(11)
thf(fact_6337_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q2: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(11)
thf(fact_6338_cong__exp__iff__simps_I11_J,axiom,
! [M: num,Q2: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(11)
thf(fact_6339_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
= zero_zero_nat ) ) ).
% cong_exp_iff_simps(7)
thf(fact_6340_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
= zero_zero_int ) ) ).
% cong_exp_iff_simps(7)
thf(fact_6341_cong__exp__iff__simps_I7_J,axiom,
! [Q2: num,N: num] :
( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
= ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
= zero_z3403309356797280102nteger ) ) ).
% cong_exp_iff_simps(7)
thf(fact_6342_Suc__div__eq__add3__div,axiom,
! [M: nat,N: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_6343_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_6344_power2__le__iff__abs__le,axiom,
! [Y4: code_integer,X: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y4 )
=> ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y4 ) ) ) ).
% power2_le_iff_abs_le
thf(fact_6345_power2__le__iff__abs__le,axiom,
! [Y4: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y4 ) ) ) ).
% power2_le_iff_abs_le
thf(fact_6346_power2__le__iff__abs__le,axiom,
! [Y4: rat,X: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ Y4 )
=> ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y4 ) ) ) ).
% power2_le_iff_abs_le
thf(fact_6347_power2__le__iff__abs__le,axiom,
! [Y4: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y4 ) ) ) ).
% power2_le_iff_abs_le
thf(fact_6348_abs__sqrt__wlog,axiom,
! [P: code_integer > code_integer > $o,X: code_integer] :
( ! [X3: code_integer] :
( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
=> ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6349_abs__sqrt__wlog,axiom,
! [P: real > real > $o,X: real] :
( ! [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
=> ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6350_abs__sqrt__wlog,axiom,
! [P: rat > rat > $o,X: rat] :
( ! [X3: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
=> ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6351_abs__sqrt__wlog,axiom,
! [P: int > int > $o,X: int] :
( ! [X3: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_6352_abs__square__le__1,axiom,
! [X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
= ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% abs_square_le_1
thf(fact_6353_abs__square__le__1,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
= ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% abs_square_le_1
thf(fact_6354_abs__square__le__1,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
= ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% abs_square_le_1
thf(fact_6355_abs__square__le__1,axiom,
! [X: int] :
( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
= ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% abs_square_le_1
thf(fact_6356_abs__square__less__1,axiom,
! [X: code_integer] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
= ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% abs_square_less_1
thf(fact_6357_abs__square__less__1,axiom,
! [X: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
= ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% abs_square_less_1
thf(fact_6358_abs__square__less__1,axiom,
! [X: rat] :
( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
= ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% abs_square_less_1
thf(fact_6359_abs__square__less__1,axiom,
! [X: int] :
( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
= ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% abs_square_less_1
thf(fact_6360_power__mono__even,axiom,
! [N: nat,A: code_integer,B: code_integer] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
=> ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_6361_power__mono__even,axiom,
! [N: nat,A: real,B: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_6362_power__mono__even,axiom,
! [N: nat,A: rat,B: rat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
=> ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_6363_power__mono__even,axiom,
! [N: nat,A: int,B: int] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono_even
thf(fact_6364_convex__sum__bound__le,axiom,
! [I5: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
=> ( ( ( groups7501900531339628137nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7501900531339628137nteger
@ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6365_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
=> ( ( ( groups7873554091576472773nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7873554091576472773nteger
@ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6366_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
=> ( ( ( groups7713935264441627589nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups7713935264441627589nteger
@ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6367_convex__sum__bound__le,axiom,
! [I5: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
=> ( ( ( groups6621422865394947399nteger @ X @ I5 )
= one_one_Code_integer )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
=> ( ord_le3102999989581377725nteger
@ ( abs_abs_Code_integer
@ ( minus_8373710615458151222nteger
@ ( groups6621422865394947399nteger
@ ^ [I4: complex] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6368_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > real,A: int > real,B: real,Delta: real] :
( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
=> ( ( ( groups8778361861064173332t_real @ X @ I5 )
= one_one_real )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups8778361861064173332t_real
@ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6369_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > real,A: real > real,B: real,Delta: real] :
( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ X @ I5 )
= one_one_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups8097168146408367636l_real
@ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6370_convex__sum__bound__le,axiom,
! [I5: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
=> ( ( ( groups5808333547571424918x_real @ X @ I5 )
= one_one_real )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real
@ ( groups5808333547571424918x_real
@ ^ [I4: complex] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6371_convex__sum__bound__le,axiom,
! [I5: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
=> ( ( ( groups2906978787729119204at_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6372_convex__sum__bound__le,axiom,
! [I5: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
=> ( ( ( groups3906332499630173760nt_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups3906332499630173760nt_rat
@ ^ [I4: int] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6373_convex__sum__bound__le,axiom,
! [I5: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
=> ( ( ( groups1300246762558778688al_rat @ X @ I5 )
= one_one_rat )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
=> ( ord_less_eq_rat
@ ( abs_abs_rat
@ ( minus_minus_rat
@ ( groups1300246762558778688al_rat
@ ^ [I4: real] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
@ I5 )
@ B ) )
@ Delta ) ) ) ) ).
% convex_sum_bound_le
thf(fact_6374_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > complex] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_complex ) ) ) ).
% sum_natinterval_diff
thf(fact_6375_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > rat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_rat ) ) ) ).
% sum_natinterval_diff
thf(fact_6376_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_int ) ) ) ).
% sum_natinterval_diff
thf(fact_6377_sum__natinterval__diff,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_real ) ) ) ).
% sum_natinterval_diff
thf(fact_6378_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_6379_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_6380_sum__telescope_H_H,axiom,
! [M: nat,N: nat,F: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% sum_telescope''
thf(fact_6381_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
= ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_6382_mask__eq__sum__exp,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_6383_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: complex] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6384_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6385_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6386_sum__gp__multiplied,axiom,
! [M: nat,N: nat,X: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
= ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_6387_sum_Oin__pairs,axiom,
! [G: nat > rat,M: nat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_6388_sum_Oin__pairs,axiom,
! [G: nat > int,M: nat,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_6389_sum_Oin__pairs,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_6390_sum_Oin__pairs,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.in_pairs
thf(fact_6391_eq__diff__eq_H,axiom,
! [X: real,Y4: real,Z2: real] :
( ( X
= ( minus_minus_real @ Y4 @ Z2 ) )
= ( Y4
= ( plus_plus_real @ X @ Z2 ) ) ) ).
% eq_diff_eq'
thf(fact_6392_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( collect_nat
@ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_6393_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_nat
thf(fact_6394_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_6395_arith__series__nat,axiom,
! [A: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series_nat
thf(fact_6396_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Icc_nat
thf(fact_6397_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_nat @ ( bit1 @ one ) ) )
=> ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_6398_signed__take__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_6399_dbl__dec__simps_I4_J,axiom,
( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6400_dbl__dec__simps_I4_J,axiom,
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6401_dbl__dec__simps_I4_J,axiom,
( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6402_dbl__dec__simps_I4_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6403_dbl__dec__simps_I4_J,axiom,
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_6404_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_6405_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_6406_divmod__algorithm__code_I8_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_6407_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_6408_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_6409_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_6410_signed__take__bit__numeral__bit1,axiom,
! [L2: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_6411_arctan__double,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
= ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_6412_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
= ( ( abs_abs_int @ X )
= one_one_int ) ) ).
% zdvd1_eq
thf(fact_6413_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6511756317524482435omplex @ one_one_complex )
= one_one_complex ) ).
% dbl_dec_simps(3)
thf(fact_6414_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_6415_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
= one_one_rat ) ).
% dbl_dec_simps(3)
thf(fact_6416_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_6417_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_6418_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one )
= zero_zero_nat ) ).
% pred_numeral_simps(1)
thf(fact_6419_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral_nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_6420_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral_nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_6421_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_6422_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% less_numeral_Suc
thf(fact_6423_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_6424_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% le_numeral_Suc
thf(fact_6425_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_6426_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_6427_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_6428_max__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_6429_max__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_6430_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_6431_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_6432_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_dec_simps(2)
thf(fact_6433_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% dbl_dec_simps(2)
thf(fact_6434_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_dec_simps(2)
thf(fact_6435_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_6436_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_6437_dvd__numeral__simp,axiom,
! [M: num,N: num] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
= ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% dvd_numeral_simp
thf(fact_6438_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique5052692396658037445od_int @ M @ one )
= ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% divmod_algorithm_code(2)
thf(fact_6439_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique5055182867167087721od_nat @ M @ one )
= ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% divmod_algorithm_code(2)
thf(fact_6440_divmod__algorithm__code_I2_J,axiom,
! [M: num] :
( ( unique3479559517661332726nteger @ M @ one )
= ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% divmod_algorithm_code(2)
thf(fact_6441_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_6442_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_6443_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_6444_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
= ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_6445_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_6446_divmod__algorithm__code_I4_J,axiom,
! [N: num] :
( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_6447_one__div__minus__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% one_div_minus_numeral
thf(fact_6448_minus__one__div__numeral,axiom,
! [N: num] :
( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% minus_one_div_numeral
thf(fact_6449_signed__take__bit__numeral__bit0,axiom,
! [L2: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_6450_signed__take__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_6451_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_6452_numeral__eq__Suc,axiom,
( numeral_numeral_nat
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_6453_abs__mod__less,axiom,
! [L2: int,K: int] :
( ( L2 != zero_zero_int )
=> ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).
% abs_mod_less
thf(fact_6454_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% pred_numeral_def
thf(fact_6455_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
= ( ( abs_abs_int @ N )
= one_one_int ) ) ) ).
% zdvd_mult_cancel1
thf(fact_6456_even__abs__add__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% even_abs_add_iff
thf(fact_6457_even__add__abs__iff,axiom,
! [K: int,L2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% even_add_abs_iff
thf(fact_6458_divmod__int__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M2: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% divmod_int_def
thf(fact_6459_divmod__def,axiom,
( unique5052692396658037445od_int
= ( ^ [M2: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_6460_divmod__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M2: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_6461_divmod__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M2: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% divmod_def
thf(fact_6462_divmod_H__nat__def,axiom,
( unique5055182867167087721od_nat
= ( ^ [M2: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_6463_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_nat @ I3 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_6464_dbl__dec__def,axiom,
( neg_nu6511756317524482435omplex
= ( ^ [X2: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% dbl_dec_def
thf(fact_6465_dbl__dec__def,axiom,
( neg_nu6075765906172075777c_real
= ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% dbl_dec_def
thf(fact_6466_dbl__dec__def,axiom,
( neg_nu3179335615603231917ec_rat
= ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% dbl_dec_def
thf(fact_6467_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_6468_incr__lemma,axiom,
! [D: int,Z2: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_6469_decr__lemma,axiom,
! [D: int,X: int,Z2: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_6470_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_6471_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_6472_arctan__add,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
=> ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y4 ) )
= ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y4 ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_6473_divmod__divmod__step,axiom,
( unique5055182867167087721od_nat
= ( ^ [M2: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6474_divmod__divmod__step,axiom,
( unique5052692396658037445od_int
= ( ^ [M2: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6475_divmod__divmod__step,axiom,
( unique3479559517661332726nteger
= ( ^ [M2: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_6476_sum__gp,axiom,
! [N: nat,M: nat,X: complex] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_complex ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6477_sum__gp,axiom,
! [N: nat,M: nat,X: rat] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_rat ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6478_sum__gp,axiom,
! [N: nat,M: nat,X: real] :
( ( ( ord_less_nat @ N @ M )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= zero_zero_real ) )
& ( ~ ( ord_less_nat @ N @ M )
=> ( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_6479_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8557863876264182079omplex @ one_one_complex )
= ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6480_dbl__inc__simps_I3_J,axiom,
( ( neg_nu8295874005876285629c_real @ one_one_real )
= ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6481_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
= ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6482_dbl__inc__simps_I3_J,axiom,
( ( neg_nu5851722552734809277nc_int @ one_one_int )
= ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_6483_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_6484_gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_6485_of__int__code__if,axiom,
( ring_1_of_int_real
= ( ^ [K3: int] :
( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
@ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_real
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6486_of__int__code__if,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] :
( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
@ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_int
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6487_of__int__code__if,axiom,
( ring_17405671764205052669omplex
= ( ^ [K3: int] :
( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
@ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_complex
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6488_of__int__code__if,axiom,
( ring_1_of_int_rat
= ( ^ [K3: int] :
( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
@ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_rat
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6489_of__int__code__if,axiom,
( ring_18347121197199848620nteger
= ( ^ [K3: int] :
( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_6490_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
@ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_6491_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
@ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_6492_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
@ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_6493_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
= ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_6494_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_6495_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_6496_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_6497_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_6498_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_6499_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6500_case__prod__conv,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6501_case__prod__conv,axiom,
! [F: int > int > product_prod_int_int,A: int,B: int] :
( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6502_case__prod__conv,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6503_case__prod__conv,axiom,
! [F: int > int > int,A: int,B: int] :
( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_6504_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_6505_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_6506_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_6507_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_6508_of__nat__numeral,axiom,
! [N: num] :
( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
= ( numera6690914467698888265omplex @ N ) ) ).
% of_nat_numeral
thf(fact_6509_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_6510_of__nat__numeral,axiom,
! [N: num] :
( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_real @ N ) ) ).
% of_nat_numeral
thf(fact_6511_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_6512_of__nat__numeral,axiom,
! [N: num] :
( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_rat @ N ) ) ).
% of_nat_numeral
thf(fact_6513_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_6514_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_6515_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_6516_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_6517_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_6518_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_add
thf(fact_6519_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_6520_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_add
thf(fact_6521_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_6522_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% of_nat_mult
thf(fact_6523_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_6524_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% of_nat_mult
thf(fact_6525_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri8010041392384452111omplex @ N )
= one_one_complex )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6526_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6527_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri5074537144036343181t_real @ N )
= one_one_real )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6528_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6529_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri681578069525770553at_rat @ N )
= one_one_rat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_6530_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6531_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6532_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6533_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6534_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_rat
= ( semiri681578069525770553at_rat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_6535_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_6536_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_6537_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_6538_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_6539_of__nat__1,axiom,
( ( semiri681578069525770553at_rat @ one_one_nat )
= one_one_rat ) ).
% of_nat_1
thf(fact_6540_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_6541_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_6542_of__int__le__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% of_int_le_iff
thf(fact_6543_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_17405671764205052669omplex @ Z2 )
= ( numera6690914467698888265omplex @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6544_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_1_of_int_real @ Z2 )
= ( numeral_numeral_real @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6545_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_1_of_int_rat @ Z2 )
= ( numeral_numeral_rat @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6546_of__int__eq__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ( ring_1_of_int_int @ Z2 )
= ( numeral_numeral_int @ N ) )
= ( Z2
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_6547_of__int__numeral,axiom,
! [K: num] :
( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
= ( numera6690914467698888265omplex @ K ) ) ).
% of_int_numeral
thf(fact_6548_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_real @ K ) ) ).
% of_int_numeral
thf(fact_6549_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_rat @ K ) ) ).
% of_int_numeral
thf(fact_6550_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ K ) ) ).
% of_int_numeral
thf(fact_6551_of__int__less__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% of_int_less_iff
thf(fact_6552_of__int__less__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% of_int_less_iff
thf(fact_6553_of__int__less__iff,axiom,
! [W2: int,Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ).
% of_int_less_iff
thf(fact_6554_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri8010041392384452111omplex @ X )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6555_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6556_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri5074537144036343181t_real @ X )
= ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6557_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6558_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ( semiri681578069525770553at_rat @ X )
= ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
= ( X
= ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_6559_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W2 )
= ( semiri8010041392384452111omplex @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6560_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6561_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 )
= ( semiri5074537144036343181t_real @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6562_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6563_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 )
= ( semiri681578069525770553at_rat @ X ) )
= ( ( power_power_nat @ B @ W2 )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_6564_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
= ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6565_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6566_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6567_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6568_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
= ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_6569_of__int__1,axiom,
( ( ring_17405671764205052669omplex @ one_one_int )
= one_one_complex ) ).
% of_int_1
thf(fact_6570_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_6571_of__int__1,axiom,
( ( ring_1_of_int_real @ one_one_int )
= one_one_real ) ).
% of_int_1
thf(fact_6572_of__int__1,axiom,
( ( ring_1_of_int_rat @ one_one_int )
= one_one_rat ) ).
% of_int_1
thf(fact_6573_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_17405671764205052669omplex @ Z2 )
= one_one_complex )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6574_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= one_one_int )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6575_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_real @ Z2 )
= one_one_real )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6576_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_rat @ Z2 )
= one_one_rat )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_6577_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_mult
thf(fact_6578_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_rat @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_mult
thf(fact_6579_of__int__mult,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z2 ) )
= ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_mult
thf(fact_6580_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_add
thf(fact_6581_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_real @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_add
thf(fact_6582_of__int__add,axiom,
! [W2: int,Z2: int] :
( ( ring_1_of_int_rat @ ( plus_plus_int @ W2 @ Z2 ) )
= ( plus_plus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_add
thf(fact_6583_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_6584_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_1_of_int_rat @ X )
= ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6585_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_1_of_int_real @ X )
= ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6586_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_1_of_int_int @ X )
= ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6587_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ( ring_17405671764205052669omplex @ X )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 ) )
= ( X
= ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_6588_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 )
= ( ring_1_of_int_rat @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6589_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 )
= ( ring_1_of_int_real @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6590_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 )
= ( ring_1_of_int_int @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6591_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 )
= ( ring_17405671764205052669omplex @ X ) )
= ( ( power_power_int @ B @ W2 )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_6592_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N ) )
= ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_6593_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N ) )
= ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_6594_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N ) )
= ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_6595_of__int__power,axiom,
! [Z2: int,N: nat] :
( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N ) )
= ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N ) ) ).
% of_int_power
thf(fact_6596_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_6597_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_6598_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
= one_one_rat ) ).
% dbl_inc_simps(2)
thf(fact_6599_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_6600_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_inc_simps(4)
thf(fact_6601_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_6602_dbl__inc__simps_I4_J,axiom,
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% dbl_inc_simps(4)
thf(fact_6603_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% dbl_inc_simps(4)
thf(fact_6604_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
= ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% dbl_inc_simps(4)
thf(fact_6605_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_6606_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_6607_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
= ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_6608_dbl__inc__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% dbl_inc_simps(5)
thf(fact_6609_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6610_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6611_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6612_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_6613_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ M ) )
= ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% of_nat_Suc
thf(fact_6614_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_6615_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ M ) )
= ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% of_nat_Suc
thf(fact_6616_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_6617_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ M ) )
= ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% of_nat_Suc
thf(fact_6618_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W2: num] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
= ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_6619_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N: nat] :
( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_6620_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_6621_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6622_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6623_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6624_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_6625_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
= ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_6626_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_6627_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
= ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_6628_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_6629_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
= ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_6630_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_6631_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_6632_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_6633_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6634_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6635_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_6636_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6637_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6638_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6639_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_6640_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6641_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6642_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6643_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_6644_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_6645_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_6646_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_6647_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6648_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6649_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_6650_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y4: nat] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
= ( semiri8010041392384452111omplex @ Y4 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y4 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6651_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y4: nat] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( semiri1314217659103216013at_int @ Y4 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y4 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6652_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y4: nat] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
= ( semiri5074537144036343181t_real @ Y4 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y4 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6653_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y4: nat] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= ( semiri1316708129612266289at_nat @ Y4 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y4 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6654_numeral__power__eq__of__nat__cancel__iff,axiom,
! [X: num,N: nat,Y4: nat] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
= ( semiri681578069525770553at_rat @ Y4 ) )
= ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= Y4 ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_6655_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y4: nat,X: num,N: nat] :
( ( ( semiri8010041392384452111omplex @ Y4 )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
= ( Y4
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6656_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y4: nat,X: num,N: nat] :
( ( ( semiri1314217659103216013at_int @ Y4 )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y4
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6657_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y4: nat,X: num,N: nat] :
( ( ( semiri5074537144036343181t_real @ Y4 )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( Y4
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6658_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y4: nat,X: num,N: nat] :
( ( ( semiri1316708129612266289at_nat @ Y4 )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( Y4
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6659_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [Y4: nat,X: num,N: nat] :
( ( ( semiri681578069525770553at_rat @ Y4 )
= ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( Y4
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_6660_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6661_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6662_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6663_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W2: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_6664_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6665_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6666_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6667_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_6668_of__int__le__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6669_of__int__le__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6670_of__int__le__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_6671_of__int__numeral__le__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_le_iff
thf(fact_6672_of__int__numeral__le__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_le_iff
thf(fact_6673_of__int__numeral__le__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_le_iff
thf(fact_6674_of__int__numeral__less__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_less_iff
thf(fact_6675_of__int__numeral__less__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_less_iff
thf(fact_6676_of__int__numeral__less__iff,axiom,
! [N: num,Z2: int] :
( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).
% of_int_numeral_less_iff
thf(fact_6677_of__int__less__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6678_of__int__less__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
= ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6679_of__int__less__numeral__iff,axiom,
! [Z2: int,N: num] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_6680_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6681_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6682_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_6683_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_6684_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_6685_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_6686_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_6687_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_6688_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_6689_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6690_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6691_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_6692_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6693_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6694_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_6695_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6696_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6697_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_6698_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y4: int] :
( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
= ( ring_17405671764205052669omplex @ Y4 ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y4 ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6699_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y4: int] :
( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
= ( ring_1_of_int_real @ Y4 ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y4 ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6700_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y4: int] :
( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
= ( ring_1_of_int_rat @ Y4 ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y4 ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6701_numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y4: int] :
( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= ( ring_1_of_int_int @ Y4 ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y4 ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_6702_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y4: int,X: num,N: nat] :
( ( ( ring_17405671764205052669omplex @ Y4 )
= ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
= ( Y4
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6703_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y4: int,X: num,N: nat] :
( ( ( ring_1_of_int_real @ Y4 )
= ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( Y4
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6704_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y4: int,X: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y4 )
= ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( Y4
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6705_of__int__eq__numeral__power__cancel__iff,axiom,
! [Y4: int,X: num,N: nat] :
( ( ( ring_1_of_int_int @ Y4 )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( Y4
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_6706_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6707_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6708_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W2: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_6709_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6710_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6711_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W2: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W2 ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_6712_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6713_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6714_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6715_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_6716_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_6717_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_6718_even__of__nat,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% even_of_nat
thf(fact_6719_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6720_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6721_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6722_of__nat__less__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
= ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_6723_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6724_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6725_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6726_numeral__power__less__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_6727_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6728_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6729_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6730_numeral__power__le__of__nat__cancel__iff,axiom,
! [I2: num,N: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_6731_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6732_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6733_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6734_of__nat__le__numeral__power__cancel__iff,axiom,
! [X: nat,I2: num,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_6735_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6736_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6737_numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_6738_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6739_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6740_of__int__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_6741_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6742_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6743_of__int__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_6744_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6745_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6746_numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_6747_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y4: int] :
( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
= ( ring_1_of_int_real @ Y4 ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y4 ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6748_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y4: int] :
( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= ( ring_1_of_int_int @ Y4 ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y4 ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6749_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y4: int] :
( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
= ( ring_17405671764205052669omplex @ Y4 ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y4 ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6750_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y4: int] :
( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
= ( ring_1_of_int_rat @ Y4 ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y4 ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6751_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [X: num,N: nat,Y4: int] :
( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
= ( ring_18347121197199848620nteger @ Y4 ) )
= ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
= Y4 ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6752_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y4: int,X: num,N: nat] :
( ( ( ring_1_of_int_real @ Y4 )
= ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( Y4
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6753_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y4: int,X: num,N: nat] :
( ( ( ring_1_of_int_int @ Y4 )
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( Y4
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6754_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y4: int,X: num,N: nat] :
( ( ( ring_17405671764205052669omplex @ Y4 )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
= ( Y4
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6755_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y4: int,X: num,N: nat] :
( ( ( ring_1_of_int_rat @ Y4 )
= ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( Y4
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6756_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [Y4: int,X: num,N: nat] :
( ( ( ring_18347121197199848620nteger @ Y4 )
= ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( Y4
= ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6757_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] :
( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_6758_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] :
( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_6759_divmod__algorithm__code_I5_J,axiom,
! [M: num,N: num] :
( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_6760_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6761_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6762_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6763_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6764_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6765_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6766_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6767_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6768_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6769_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6770_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6771_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
= ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6772_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6773_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6774_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6775_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6776_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_6777_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_6778_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_6779_mult__of__int__commute,axiom,
! [X: int,Y4: real] :
( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y4 )
= ( times_times_real @ Y4 @ ( ring_1_of_int_real @ X ) ) ) ).
% mult_of_int_commute
thf(fact_6780_mult__of__int__commute,axiom,
! [X: int,Y4: rat] :
( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y4 )
= ( times_times_rat @ Y4 @ ( ring_1_of_int_rat @ X ) ) ) ).
% mult_of_int_commute
thf(fact_6781_mult__of__int__commute,axiom,
! [X: int,Y4: int] :
( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y4 )
= ( times_times_int @ Y4 @ ( ring_1_of_int_int @ X ) ) ) ).
% mult_of_int_commute
thf(fact_6782_mult__of__nat__commute,axiom,
! [X: nat,Y4: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y4 )
= ( times_times_int @ Y4 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6783_mult__of__nat__commute,axiom,
! [X: nat,Y4: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y4 )
= ( times_times_real @ Y4 @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6784_mult__of__nat__commute,axiom,
! [X: nat,Y4: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y4 )
= ( times_times_nat @ Y4 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6785_mult__of__nat__commute,axiom,
! [X: nat,Y4: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y4 )
= ( times_times_rat @ Y4 @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_6786_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_6787_old_Oprod_Ocase,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_6788_old_Oprod_Ocase,axiom,
! [F: int > int > product_prod_int_int,X1: int,X22: int] :
( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_6789_old_Oprod_Ocase,axiom,
! [F: int > int > $o,X1: int,X22: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_6790_old_Oprod_Ocase,axiom,
! [F: int > int > int,X1: int,X22: int] :
( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_6791_case__prodE2,axiom,
! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( Q @ ( produc27273713700761075at_nat @ P @ Z2 ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( Z2
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_6792_case__prodE2,axiom,
! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z2: product_prod_nat_nat] :
( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z2 ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( Z2
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_6793_case__prodE2,axiom,
! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z2: product_prod_int_int] :
( ( Q @ ( produc4245557441103728435nt_int @ P @ Z2 ) )
=> ~ ! [X3: int,Y3: int] :
( ( Z2
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_6794_case__prodE2,axiom,
! [Q: $o > $o,P: int > int > $o,Z2: product_prod_int_int] :
( ( Q @ ( produc4947309494688390418_int_o @ P @ Z2 ) )
=> ~ ! [X3: int,Y3: int] :
( ( Z2
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_6795_case__prodE2,axiom,
! [Q: int > $o,P: int > int > int,Z2: product_prod_int_int] :
( ( Q @ ( produc8211389475949308722nt_int @ P @ Z2 ) )
=> ~ ! [X3: int,Y3: int] :
( ( Z2
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_6796_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_6797_case__prod__eta,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_6798_case__prod__eta,axiom,
! [F: product_prod_int_int > product_prod_int_int] :
( ( produc4245557441103728435nt_int
@ ^ [X2: int,Y: int] : ( F @ ( product_Pair_int_int @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_6799_case__prod__eta,axiom,
! [F: product_prod_int_int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [X2: int,Y: int] : ( F @ ( product_Pair_int_int @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_6800_case__prod__eta,axiom,
! [F: product_prod_int_int > int] :
( ( produc8211389475949308722nt_int
@ ^ [X2: int,Y: int] : ( F @ ( product_Pair_int_int @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_6801_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
( ! [X3: nat,Y3: nat] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
=> ( ( produc27273713700761075at_nat @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_6802_cond__case__prod__eta,axiom,
! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [X3: nat,Y3: nat] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
=> ( ( produc8739625826339149834_nat_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_6803_cond__case__prod__eta,axiom,
! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
( ! [X3: int,Y3: int] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
=> ( ( produc4245557441103728435nt_int @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_6804_cond__case__prod__eta,axiom,
! [F: int > int > $o,G: product_prod_int_int > $o] :
( ! [X3: int,Y3: int] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
=> ( ( produc4947309494688390418_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_6805_cond__case__prod__eta,axiom,
! [F: int > int > int,G: product_prod_int_int > int] :
( ! [X3: int,Y3: int] :
( ( F @ X3 @ Y3 )
= ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
=> ( ( produc8211389475949308722nt_int @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_6806_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6807_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6808_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6809_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_6810_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_6811_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_6812_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_6813_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% of_nat_less_0_iff
thf(fact_6814_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri8010041392384452111omplex @ ( suc @ N ) )
!= zero_zero_complex ) ).
% of_nat_neq_0
thf(fact_6815_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_6816_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_6817_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_6818_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri681578069525770553at_rat @ ( suc @ N ) )
!= zero_zero_rat ) ).
% of_nat_neq_0
thf(fact_6819_div__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% div_mult2_eq'
thf(fact_6820_div__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% div_mult2_eq'
thf(fact_6821_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6822_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6823_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6824_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_6825_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6826_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6827_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6828_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_6829_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_6830_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% of_nat_mono
thf(fact_6831_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_6832_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_6833_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_6834_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_6835_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_6836_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_6837_of__nat__dvd__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% of_nat_dvd_iff
thf(fact_6838_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_6839_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_6840_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_6841_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_6842_int__cases,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_6843_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_6844_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_6845_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% of_nat_mod
thf(fact_6846_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mod
thf(fact_6847_of__nat__mod,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
= ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mod
thf(fact_6848_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_6849_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_6850_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_6851_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_6852_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_6853_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W3: int,Z4: int] :
? [N2: nat] :
( Z4
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_6854_zdiv__int,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% zdiv_int
thf(fact_6855_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_6856_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_6857_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6858_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6859_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6860_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_6861_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_6862_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_6863_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_6864_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_6865_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_6866_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W3: int,Z4: int] :
? [N2: nat] :
( Z4
= ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_6867_real__of__nat__div,axiom,
! [D: nat,N: nat] :
( ( dvd_dvd_nat @ D @ N )
=> ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
= ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div
thf(fact_6868_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_6869_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_6870_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_6871_of__int__leD,axiom,
! [N: int,X: code_integer] :
( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% of_int_leD
thf(fact_6872_of__int__leD,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% of_int_leD
thf(fact_6873_of__int__leD,axiom,
! [N: int,X: rat] :
( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% of_int_leD
thf(fact_6874_of__int__leD,axiom,
! [N: int,X: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% of_int_leD
thf(fact_6875_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% of_int_pos
thf(fact_6876_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% of_int_pos
thf(fact_6877_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_pos
thf(fact_6878_of__int__lessD,axiom,
! [N: int,X: code_integer] :
( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% of_int_lessD
thf(fact_6879_of__int__lessD,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_real @ one_one_real @ X ) ) ) ).
% of_int_lessD
thf(fact_6880_of__int__lessD,axiom,
! [N: int,X: rat] :
( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% of_int_lessD
thf(fact_6881_of__int__lessD,axiom,
! [N: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_6882_mod__mult2__eq_H,axiom,
! [A: code_integer,M: nat,N: nat] :
( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_6883_mod__mult2__eq_H,axiom,
! [A: int,M: nat,N: nat] :
( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_6884_mod__mult2__eq_H,axiom,
! [A: nat,M: nat,N: nat] :
( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% mod_mult2_eq'
thf(fact_6885_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_6886_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_6887_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_6888_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_6889_of__int__neg__numeral,axiom,
! [K: num] :
( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% of_int_neg_numeral
thf(fact_6890_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_6891_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_6892_field__char__0__class_Oof__nat__div,axiom,
! [M: nat,N: nat] :
( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_6893_int__le__real__less,axiom,
( ord_less_eq_int
= ( ^ [N2: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% int_le_real_less
thf(fact_6894_int__less__real__le,axiom,
( ord_less_int
= ( ^ [N2: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% int_less_real_le
thf(fact_6895_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_6896_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_6897_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_6898_nat__less__real__le,axiom,
( ord_less_nat
= ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_6899_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_6900_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_6901_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_6902_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_6903_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_6904_dbl__inc__def,axiom,
( neg_nu8557863876264182079omplex
= ( ^ [X2: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% dbl_inc_def
thf(fact_6905_dbl__inc__def,axiom,
( neg_nu8295874005876285629c_real
= ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% dbl_inc_def
thf(fact_6906_dbl__inc__def,axiom,
( neg_nu5219082963157363817nc_rat
= ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% dbl_inc_def
thf(fact_6907_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_6908_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_6909_real__of__int__div__aux,axiom,
! [X: int,D: int] :
( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
= ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% real_of_int_div_aux
thf(fact_6910_real__of__nat__div__aux,axiom,
! [X: nat,D: nat] :
( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_6911_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_6912_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_6913_of__nat__less__two__power,axiom,
! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% of_nat_less_two_power
thf(fact_6914_inverse__of__nat__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_6915_inverse__of__nat__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( N != zero_zero_nat )
=> ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_6916_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_6917_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).
% real_of_int_div3
thf(fact_6918_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_6919_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y4: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y4 ) ) )
= ( ( ( ord_less_eq_nat @ Y4 @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y4 ) ) ) )
& ( ( ord_less_nat @ X @ Y4 )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_6920_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_6921_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).
% real_of_nat_div3
thf(fact_6922_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_6923_even__of__int__iff,axiom,
! [K: int] :
( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% even_of_int_iff
thf(fact_6924_even__of__int__iff,axiom,
! [K: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% even_of_int_iff
thf(fact_6925_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_6926_divmod__step__nat__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L: num] :
( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_6927_divmod__step__int__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L: num] :
( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_6928_Bernoulli__inequality__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_6929_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% double_gauss_sum
thf(fact_6930_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum
thf(fact_6931_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum
thf(fact_6932_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum
thf(fact_6933_double__gauss__sum,axiom,
! [N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% double_gauss_sum
thf(fact_6934_double__arith__series,axiom,
! [A: complex,D: complex,N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6935_double__arith__series,axiom,
! [A: int,D: int,N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6936_double__arith__series,axiom,
! [A: rat,D: rat,N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6937_double__arith__series,axiom,
! [A: nat,D: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6938_double__arith__series,axiom,
! [A: real,D: real,N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
= ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% double_arith_series
thf(fact_6939_divmod__step__def,axiom,
( unique5026877609467782581ep_nat
= ( ^ [L: num] :
( produc2626176000494625587at_nat
@ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_6940_divmod__step__def,axiom,
( unique5024387138958732305ep_int
= ( ^ [L: num] :
( produc4245557441103728435nt_int
@ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_6941_divmod__step__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L: num] :
( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_def
thf(fact_6942_gauss__sum,axiom,
! [N: nat] :
( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_6943_gauss__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% gauss_sum
thf(fact_6944_arith__series,axiom,
! [A: int,D: int,N: nat] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_6945_arith__series,axiom,
! [A: nat,D: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% arith_series
thf(fact_6946_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6947_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6948_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6949_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6950_double__gauss__sum__from__Suc__0,axiom,
! [N: nat] :
( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_6951_sum__gp__offset,axiom,
! [X: complex,M: nat,N: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6952_sum__gp__offset,axiom,
! [X: rat,M: nat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6953_sum__gp__offset,axiom,
! [X: real,M: nat,N: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
= ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp_offset
thf(fact_6954_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_6955_nat__approx__posE,axiom,
! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ~ ! [N3: nat] :
~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E ) ) ).
% nat_approx_posE
thf(fact_6956_nat__approx__posE,axiom,
! [E: rat] :
( ( ord_less_rat @ zero_zero_rat @ E )
=> ~ ! [N3: nat] :
~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E ) ) ).
% nat_approx_posE
thf(fact_6957_floor__exists,axiom,
! [X: real] :
? [Z: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_6958_floor__exists,axiom,
! [X: rat] :
? [Z: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z @ one_one_int ) ) ) ) ).
% floor_exists
thf(fact_6959_floor__exists1,axiom,
! [X: real] :
? [X3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
& ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
=> ( Y5 = X3 ) ) ) ).
% floor_exists1
thf(fact_6960_floor__exists1,axiom,
! [X: rat] :
? [X3: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
& ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
=> ( Y5 = X3 ) ) ) ).
% floor_exists1
thf(fact_6961_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( topolo6980174941875973593q_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_6962_lemma__termdiff3,axiom,
! [H2: real,Z2: real,K5: real,N: nat] :
( ( H2 != zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H2 ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_6963_lemma__termdiff3,axiom,
! [H2: complex,Z2: complex,K5: real,N: nat] :
( ( H2 != zero_zero_complex )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K5 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H2 ) ) @ K5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_6964_case__prodI2,axiom,
! [P6: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P6
= ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc127349428274296955eger_o @ C @ P6 ) ) ).
% case_prodI2
thf(fact_6965_case__prodI2,axiom,
! [P6: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P6
= ( produc8603105652947943368nteger @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc6253627499356882019eger_o @ C @ P6 ) ) ).
% case_prodI2
thf(fact_6966_case__prodI2,axiom,
! [P6: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( P6
= ( produc5700946648718959541nt_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc1573362020775583542_int_o @ C @ P6 ) ) ).
% case_prodI2
thf(fact_6967_case__prodI2,axiom,
! [P6: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( P6
= ( produc4305682042979456191nt_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc2558449545302689196_int_o @ C @ P6 ) ) ).
% case_prodI2
thf(fact_6968_case__prodI2,axiom,
! [P6: product_prod_int_int,C: int > int > $o] :
( ! [A3: int,B2: int] :
( ( P6
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc4947309494688390418_int_o @ C @ P6 ) ) ).
% case_prodI2
thf(fact_6969_case__prodI,axiom,
! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( F @ A @ B )
=> ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) ) ) ).
% case_prodI
thf(fact_6970_case__prodI,axiom,
! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( F @ A @ B )
=> ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).
% case_prodI
thf(fact_6971_case__prodI,axiom,
! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( F @ A @ B )
=> ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_6972_case__prodI,axiom,
! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( F @ A @ B )
=> ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_6973_case__prodI,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( F @ A @ B )
=> ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_6974_mem__case__prodI2,axiom,
! [P6: product_prod_int_int,Z2: nat,C: int > int > set_nat] :
( ! [A3: int,B2: int] :
( ( P6
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( member_nat @ Z2 @ ( C @ A3 @ B2 ) ) )
=> ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_6975_mem__case__prodI2,axiom,
! [P6: product_prod_int_int,Z2: int,C: int > int > set_int] :
( ! [A3: int,B2: int] :
( ( P6
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( member_int @ Z2 @ ( C @ A3 @ B2 ) ) )
=> ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_6976_mem__case__prodI2,axiom,
! [P6: product_prod_int_int,Z2: real,C: int > int > set_real] :
( ! [A3: int,B2: int] :
( ( P6
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( member_real @ Z2 @ ( C @ A3 @ B2 ) ) )
=> ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_6977_mem__case__prodI2,axiom,
! [P6: product_prod_int_int,Z2: complex,C: int > int > set_complex] :
( ! [A3: int,B2: int] :
( ( P6
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( member_complex @ Z2 @ ( C @ A3 @ B2 ) ) )
=> ( member_complex @ Z2 @ ( produc8580519160106071146omplex @ C @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_6978_mem__case__prodI2,axiom,
! [P6: product_prod_int_int,Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat] :
( ! [A3: int,B2: int] :
( ( P6
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( member8440522571783428010at_nat @ Z2 @ ( C @ A3 @ B2 ) ) )
=> ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_6979_mem__case__prodI2,axiom,
! [P6: produc8763457246119570046nteger,Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P6
= ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( member_nat @ Z2 @ ( C @ A3 @ B2 ) ) )
=> ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_6980_mem__case__prodI2,axiom,
! [P6: produc8763457246119570046nteger,Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P6
= ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( member_int @ Z2 @ ( C @ A3 @ B2 ) ) )
=> ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_6981_mem__case__prodI2,axiom,
! [P6: produc8763457246119570046nteger,Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P6
= ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( member_real @ Z2 @ ( C @ A3 @ B2 ) ) )
=> ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_6982_mem__case__prodI2,axiom,
! [P6: produc8763457246119570046nteger,Z2: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex] :
( ! [A3: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
( ( P6
= ( produc6137756002093451184nteger @ A3 @ B2 ) )
=> ( member_complex @ Z2 @ ( C @ A3 @ B2 ) ) )
=> ( member_complex @ Z2 @ ( produc2592262431452330817omplex @ C @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_6983_mem__case__prodI2,axiom,
! [P6: produc7773217078559923341nt_int,Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( P6
= ( produc4305682042979456191nt_int @ A3 @ B2 ) )
=> ( member_nat @ Z2 @ ( C @ A3 @ B2 ) ) )
=> ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P6 ) ) ) ).
% mem_case_prodI2
thf(fact_6984_mem__case__prodI,axiom,
! [Z2: nat,C: int > int > set_nat,A: int,B: int] :
( ( member_nat @ Z2 @ ( C @ A @ B ) )
=> ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_6985_mem__case__prodI,axiom,
! [Z2: int,C: int > int > set_int,A: int,B: int] :
( ( member_int @ Z2 @ ( C @ A @ B ) )
=> ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_6986_mem__case__prodI,axiom,
! [Z2: real,C: int > int > set_real,A: int,B: int] :
( ( member_real @ Z2 @ ( C @ A @ B ) )
=> ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_6987_mem__case__prodI,axiom,
! [Z2: complex,C: int > int > set_complex,A: int,B: int] :
( ( member_complex @ Z2 @ ( C @ A @ B ) )
=> ( member_complex @ Z2 @ ( produc8580519160106071146omplex @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_6988_mem__case__prodI,axiom,
! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,A: int,B: int] :
( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
=> ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_6989_mem__case__prodI,axiom,
! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_nat @ Z2 @ ( C @ A @ B ) )
=> ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_6990_mem__case__prodI,axiom,
! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_int @ Z2 @ ( C @ A @ B ) )
=> ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_6991_mem__case__prodI,axiom,
! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_real @ Z2 @ ( C @ A @ B ) )
=> ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_6992_mem__case__prodI,axiom,
! [Z2: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( member_complex @ Z2 @ ( C @ A @ B ) )
=> ( member_complex @ Z2 @ ( produc2592262431452330817omplex @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_6993_mem__case__prodI,axiom,
! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( member_nat @ Z2 @ ( C @ A @ B ) )
=> ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_6994_case__prodI2_H,axiom,
! [P6: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [A3: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B2 )
= P6 )
=> ( C @ A3 @ B2 @ X ) )
=> ( produc8739625826339149834_nat_o @ C @ P6 @ X ) ) ).
% case_prodI2'
thf(fact_6995_nat__int__comparison_I1_J,axiom,
( ( ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
= ( ^ [A4: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_6996_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_6997_mem__case__prodE,axiom,
! [Z2: nat,C: int > int > set_nat,P6: product_prod_int_int] :
( ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P6 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P6
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_6998_mem__case__prodE,axiom,
! [Z2: int,C: int > int > set_int,P6: product_prod_int_int] :
( ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P6 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P6
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member_int @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_6999_mem__case__prodE,axiom,
! [Z2: real,C: int > int > set_real,P6: product_prod_int_int] :
( ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P6 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P6
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member_real @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7000_mem__case__prodE,axiom,
! [Z2: complex,C: int > int > set_complex,P6: product_prod_int_int] :
( ( member_complex @ Z2 @ ( produc8580519160106071146omplex @ C @ P6 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P6
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member_complex @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7001_mem__case__prodE,axiom,
! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,P6: product_prod_int_int] :
( ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P6 ) )
=> ~ ! [X3: int,Y3: int] :
( ( P6
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7002_mem__case__prodE,axiom,
! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P6: produc8763457246119570046nteger] :
( ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P6 ) )
=> ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P6
= ( produc6137756002093451184nteger @ X3 @ Y3 ) )
=> ~ ( member_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7003_mem__case__prodE,axiom,
! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P6: produc8763457246119570046nteger] :
( ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P6 ) )
=> ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P6
= ( produc6137756002093451184nteger @ X3 @ Y3 ) )
=> ~ ( member_int @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7004_mem__case__prodE,axiom,
! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P6: produc8763457246119570046nteger] :
( ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P6 ) )
=> ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P6
= ( produc6137756002093451184nteger @ X3 @ Y3 ) )
=> ~ ( member_real @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7005_mem__case__prodE,axiom,
! [Z2: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,P6: produc8763457246119570046nteger] :
( ( member_complex @ Z2 @ ( produc2592262431452330817omplex @ C @ P6 ) )
=> ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P6
= ( produc6137756002093451184nteger @ X3 @ Y3 ) )
=> ~ ( member_complex @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7006_mem__case__prodE,axiom,
! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P6: produc7773217078559923341nt_int] :
( ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P6 ) )
=> ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P6
= ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
=> ~ ( member_nat @ Z2 @ ( C @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_7007_case__prodE,axiom,
! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P6: produc8763457246119570046nteger] :
( ( produc127349428274296955eger_o @ C @ P6 )
=> ~ ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P6
= ( produc6137756002093451184nteger @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_7008_case__prodE,axiom,
! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P6: produc1908205239877642774nteger] :
( ( produc6253627499356882019eger_o @ C @ P6 )
=> ~ ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger] :
( ( P6
= ( produc8603105652947943368nteger @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_7009_case__prodE,axiom,
! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P6: produc2285326912895808259nt_int] :
( ( produc1573362020775583542_int_o @ C @ P6 )
=> ~ ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P6
= ( produc5700946648718959541nt_int @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_7010_case__prodE,axiom,
! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P6: produc7773217078559923341nt_int] :
( ( produc2558449545302689196_int_o @ C @ P6 )
=> ~ ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P6
= ( produc4305682042979456191nt_int @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_7011_case__prodE,axiom,
! [C: int > int > $o,P6: product_prod_int_int] :
( ( produc4947309494688390418_int_o @ C @ P6 )
=> ~ ! [X3: int,Y3: int] :
( ( P6
= ( product_Pair_int_int @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_7012_case__prodD,axiom,
! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_7013_case__prodD,axiom,
! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_7014_case__prodD,axiom,
! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_7015_case__prodD,axiom,
! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_7016_case__prodD,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_7017_case__prodE_H,axiom,
! [C: nat > nat > product_prod_nat_nat > $o,P6: product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ C @ P6 @ Z2 )
=> ~ ! [X3: nat,Y3: nat] :
( ( P6
= ( product_Pair_nat_nat @ X3 @ Y3 ) )
=> ~ ( C @ X3 @ Y3 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_7018_case__prodD_H,axiom,
! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
=> ( R @ A @ B @ C ) ) ).
% case_prodD'
thf(fact_7019_complex__mod__triangle__ineq2,axiom,
! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% complex_mod_triangle_ineq2
thf(fact_7020_monoseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% monoseq_realpow
thf(fact_7021_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( produc8211389475949308722nt_int
@ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_7022_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_7023_real__arch__simple,axiom,
! [X: rat] :
? [N3: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% real_arch_simple
thf(fact_7024_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_7025_reals__Archimedean2,axiom,
! [X: rat] :
? [N3: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% reals_Archimedean2
thf(fact_7026_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_7027_ex__le__of__int,axiom,
! [X: real] :
? [Z: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ).
% ex_le_of_int
thf(fact_7028_ex__le__of__int,axiom,
! [X: rat] :
? [Z: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ).
% ex_le_of_int
thf(fact_7029_ex__less__of__int,axiom,
! [X: real] :
? [Z: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z ) ) ).
% ex_less_of_int
thf(fact_7030_ex__less__of__int,axiom,
! [X: rat] :
? [Z: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ).
% ex_less_of_int
thf(fact_7031_ex__of__int__less,axiom,
! [X: real] :
? [Z: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ).
% ex_of_int_less
thf(fact_7032_ex__of__int__less,axiom,
! [X: rat] :
? [Z: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ).
% ex_of_int_less
thf(fact_7033_ex__less__of__nat__mult,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_7034_ex__less__of__nat__mult,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N3: nat] : ( ord_less_rat @ Y4 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_7035_norm__divide__numeral,axiom,
! [A: real,W2: num] :
( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W2 ) ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W2 ) ) ) ).
% norm_divide_numeral
thf(fact_7036_norm__divide__numeral,axiom,
! [A: complex,W2: num] :
( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W2 ) ) ) ).
% norm_divide_numeral
thf(fact_7037_norm__mult__numeral1,axiom,
! [W2: num,A: real] :
( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ A ) )
= ( times_times_real @ ( numeral_numeral_real @ W2 ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% norm_mult_numeral1
thf(fact_7038_norm__mult__numeral1,axiom,
! [W2: num,A: complex] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ A ) )
= ( times_times_real @ ( numeral_numeral_real @ W2 ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% norm_mult_numeral1
thf(fact_7039_norm__mult__numeral2,axiom,
! [A: real,W2: num] :
( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) )
= ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W2 ) ) ) ).
% norm_mult_numeral2
thf(fact_7040_norm__mult__numeral2,axiom,
! [A: complex,W2: num] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W2 ) ) ) ).
% norm_mult_numeral2
thf(fact_7041_norm__neg__numeral,axiom,
! [W2: num] :
( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( numeral_numeral_real @ W2 ) ) ).
% norm_neg_numeral
thf(fact_7042_norm__neg__numeral,axiom,
! [W2: num] :
( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( numeral_numeral_real @ W2 ) ) ).
% norm_neg_numeral
thf(fact_7043_norm__numeral,axiom,
! [W2: num] :
( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W2 ) )
= ( numeral_numeral_real @ W2 ) ) ).
% norm_numeral
thf(fact_7044_norm__numeral,axiom,
! [W2: num] :
( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W2 ) )
= ( numeral_numeral_real @ W2 ) ) ).
% norm_numeral
thf(fact_7045_norm__one,axiom,
( ( real_V7735802525324610683m_real @ one_one_real )
= one_one_real ) ).
% norm_one
thf(fact_7046_norm__one,axiom,
( ( real_V1022390504157884413omplex @ one_one_complex )
= one_one_real ) ).
% norm_one
thf(fact_7047_norm__mult,axiom,
! [X: real,Y4: real] :
( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y4 ) )
= ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) ) ).
% norm_mult
thf(fact_7048_norm__mult,axiom,
! [X: complex,Y4: complex] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y4 ) )
= ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) ) ).
% norm_mult
thf(fact_7049_norm__divide,axiom,
! [A: real,B: real] :
( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% norm_divide
thf(fact_7050_norm__divide,axiom,
! [A: complex,B: complex] :
( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% norm_divide
thf(fact_7051_norm__power,axiom,
! [X: real,N: nat] :
( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
= ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% norm_power
thf(fact_7052_norm__power,axiom,
! [X: complex,N: nat] :
( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
= ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% norm_power
thf(fact_7053_norm__uminus__minus,axiom,
! [X: real,Y4: real] :
( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y4 ) )
= ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) ) ).
% norm_uminus_minus
thf(fact_7054_norm__uminus__minus,axiom,
! [X: complex,Y4: complex] :
( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y4 ) )
= ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) ) ).
% norm_uminus_minus
thf(fact_7055_nonzero__norm__divide,axiom,
! [B: real,A: real] :
( ( B != zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_7056_nonzero__norm__divide,axiom,
! [B: complex,A: complex] :
( ( B != zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% nonzero_norm_divide
thf(fact_7057_power__eq__imp__eq__norm,axiom,
! [W2: real,N: nat,Z2: real] :
( ( ( power_power_real @ W2 @ N )
= ( power_power_real @ Z2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V7735802525324610683m_real @ W2 )
= ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_7058_power__eq__imp__eq__norm,axiom,
! [W2: complex,N: nat,Z2: complex] :
( ( ( power_power_complex @ W2 @ N )
= ( power_power_complex @ Z2 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( real_V1022390504157884413omplex @ W2 )
= ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_7059_norm__mult__less,axiom,
! [X: real,R2: real,Y4: real,S2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y4 ) @ S2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y4 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% norm_mult_less
thf(fact_7060_norm__mult__less,axiom,
! [X: complex,R2: real,Y4: complex,S2: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y4 ) @ S2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y4 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% norm_mult_less
thf(fact_7061_norm__mult__ineq,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y4 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) ) ).
% norm_mult_ineq
thf(fact_7062_norm__mult__ineq,axiom,
! [X: complex,Y4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y4 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) ) ).
% norm_mult_ineq
thf(fact_7063_norm__triangle__lt,axiom,
! [X: real,Y4: real,E: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) @ E )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) @ E ) ) ).
% norm_triangle_lt
thf(fact_7064_norm__triangle__lt,axiom,
! [X: complex,Y4: complex,E: real] :
( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) @ E )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) @ E ) ) ).
% norm_triangle_lt
thf(fact_7065_norm__add__less,axiom,
! [X: real,R2: real,Y4: real,S2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y4 ) @ S2 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% norm_add_less
thf(fact_7066_norm__add__less,axiom,
! [X: complex,R2: real,Y4: complex,S2: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R2 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y4 ) @ S2 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% norm_add_less
thf(fact_7067_norm__power__ineq,axiom,
! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% norm_power_ineq
thf(fact_7068_norm__power__ineq,axiom,
! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% norm_power_ineq
thf(fact_7069_norm__triangle__mono,axiom,
! [A: real,R2: real,B: real,S2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S2 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% norm_triangle_mono
thf(fact_7070_norm__triangle__mono,axiom,
! [A: complex,R2: real,B: complex,S2: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S2 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% norm_triangle_mono
thf(fact_7071_norm__triangle__ineq,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) ) ).
% norm_triangle_ineq
thf(fact_7072_norm__triangle__ineq,axiom,
! [X: complex,Y4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) ) ).
% norm_triangle_ineq
thf(fact_7073_norm__triangle__le,axiom,
! [X: real,Y4: real,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) @ E )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y4 ) ) @ E ) ) ).
% norm_triangle_le
thf(fact_7074_norm__triangle__le,axiom,
! [X: complex,Y4: complex,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) @ E )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y4 ) ) @ E ) ) ).
% norm_triangle_le
thf(fact_7075_norm__add__leD,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% norm_add_leD
thf(fact_7076_norm__add__leD,axiom,
! [A: complex,B: complex,C: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% norm_add_leD
thf(fact_7077_norm__diff__triangle__less,axiom,
! [X: real,Y4: real,E1: real,Z2: real,E22: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y4 @ Z2 ) ) @ E22 )
=> ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_7078_norm__diff__triangle__less,axiom,
! [X: complex,Y4: complex,E1: real,Z2: complex,E22: real] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) @ E1 )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y4 @ Z2 ) ) @ E22 )
=> ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_7079_norm__triangle__le__diff,axiom,
! [X: real,Y4: real,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y4 ) ) @ E )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) @ E ) ) ).
% norm_triangle_le_diff
thf(fact_7080_norm__triangle__le__diff,axiom,
! [X: complex,Y4: complex,E: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y4 ) ) @ E )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) @ E ) ) ).
% norm_triangle_le_diff
thf(fact_7081_norm__diff__triangle__le,axiom,
! [X: real,Y4: real,E1: real,Z2: real,E22: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) @ E1 )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y4 @ Z2 ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_7082_norm__diff__triangle__le,axiom,
! [X: complex,Y4: complex,E1: real,Z2: complex,E22: real] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) @ E1 )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y4 @ Z2 ) ) @ E22 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_7083_norm__triangle__ineq4,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% norm_triangle_ineq4
thf(fact_7084_norm__triangle__ineq4,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% norm_triangle_ineq4
thf(fact_7085_norm__triangle__sub,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y4 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y4 ) ) ) ) ).
% norm_triangle_sub
thf(fact_7086_norm__triangle__sub,axiom,
! [X: complex,Y4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y4 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y4 ) ) ) ) ).
% norm_triangle_sub
thf(fact_7087_norm__diff__ineq,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% norm_diff_ineq
thf(fact_7088_norm__diff__ineq,axiom,
! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% norm_diff_ineq
thf(fact_7089_power__eq__1__iff,axiom,
! [W2: real,N: nat] :
( ( ( power_power_real @ W2 @ N )
= one_one_real )
=> ( ( ( real_V7735802525324610683m_real @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_7090_power__eq__1__iff,axiom,
! [W2: complex,N: nat] :
( ( ( power_power_complex @ W2 @ N )
= one_one_complex )
=> ( ( ( real_V1022390504157884413omplex @ W2 )
= one_one_real )
| ( N = zero_zero_nat ) ) ) ).
% power_eq_1_iff
thf(fact_7091_norm__diff__triangle__ineq,axiom,
! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_7092_norm__diff__triangle__ineq,axiom,
! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_7093_square__norm__one,axiom,
! [X: real] :
( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_real )
=> ( ( real_V7735802525324610683m_real @ X )
= one_one_real ) ) ).
% square_norm_one
thf(fact_7094_square__norm__one,axiom,
! [X: complex] :
( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= one_one_complex )
=> ( ( real_V1022390504157884413omplex @ X )
= one_one_real ) ) ).
% square_norm_one
thf(fact_7095_norm__power__diff,axiom,
! [Z2: real,W2: real,M: nat] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z2 @ M ) @ ( power_power_real @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z2 @ W2 ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_7096_norm__power__diff,axiom,
! [Z2: complex,W2: complex,M: nat] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z2 @ M ) @ ( power_power_complex @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z2 @ W2 ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_7097_ln__series,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ( ln_ln_real @ X )
= ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_7098_of__nat__code__if,axiom,
( semiri8010041392384452111omplex
= ( ^ [N2: nat] :
( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
@ ( produc1917071388513777916omplex
@ ^ [M2: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ one_one_complex ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7099_of__nat__code__if,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] :
( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
@ ( produc6840382203811409530at_int
@ ^ [M2: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ one_one_int ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7100_of__nat__code__if,axiom,
( semiri5074537144036343181t_real
= ( ^ [N2: nat] :
( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
@ ( produc1703576794950452218t_real
@ ^ [M2: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ one_one_real ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7101_of__nat__code__if,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
@ ( produc6842872674320459806at_nat
@ ^ [M2: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M2 ) ) @ one_one_nat ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7102_of__nat__code__if,axiom,
( semiri681578069525770553at_rat
= ( ^ [N2: nat] :
( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
@ ( produc6207742614233964070at_rat
@ ^ [M2: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ one_one_rat ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_7103_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M2: nat,N2: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N2 = zero_zero_nat )
| ( ord_less_nat @ M2 @ N2 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
@ ( produc2626176000494625587at_nat
@ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
@ ( divmod_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_7104_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( arctan @ X )
= ( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_7105_round__unique,axiom,
! [X: real,Y4: int] :
( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y4 ) )
=> ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( archim8280529875227126926d_real @ X )
= Y4 ) ) ) ).
% round_unique
thf(fact_7106_round__unique,axiom,
! [X: rat,Y4: int] :
( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y4 ) )
=> ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= Y4 ) ) ) ).
% round_unique
thf(fact_7107_lemma__termdiff2,axiom,
! [H2: complex,Z2: complex,N: nat] :
( ( H2 != zero_zero_complex )
=> ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_complex @ H2
@ ( groups2073611262835488442omplex
@ ^ [P4: nat] :
( groups2073611262835488442omplex
@ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ Q4 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_7108_lemma__termdiff2,axiom,
! [H2: rat,Z2: rat,N: nat] :
( ( H2 != zero_zero_rat )
=> ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ N ) @ ( power_power_rat @ Z2 @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_rat @ H2
@ ( groups2906978787729119204at_rat
@ ^ [P4: nat] :
( groups2906978787729119204at_rat
@ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ Q4 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_7109_lemma__termdiff2,axiom,
! [H2: real,Z2: real,N: nat] :
( ( H2 != zero_zero_real )
=> ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
= ( times_times_real @ H2
@ ( groups6591440286371151544t_real
@ ^ [P4: nat] :
( groups6591440286371151544t_real
@ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ Q4 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_7110_lessThan__iff,axiom,
! [I2: rat,K: rat] :
( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K ) )
= ( ord_less_rat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_7111_lessThan__iff,axiom,
! [I2: num,K: num] :
( ( member_num @ I2 @ ( set_ord_lessThan_num @ K ) )
= ( ord_less_num @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_7112_lessThan__iff,axiom,
! [I2: int,K: int] :
( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_7113_lessThan__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_7114_lessThan__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K ) )
= ( ord_less_real @ I2 @ K ) ) ).
% lessThan_iff
thf(fact_7115_lessThan__subset__iff,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y4 ) )
= ( ord_less_eq_rat @ X @ Y4 ) ) ).
% lessThan_subset_iff
thf(fact_7116_lessThan__subset__iff,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y4 ) )
= ( ord_less_eq_num @ X @ Y4 ) ) ).
% lessThan_subset_iff
thf(fact_7117_lessThan__subset__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y4 ) )
= ( ord_less_eq_int @ X @ Y4 ) ) ).
% lessThan_subset_iff
thf(fact_7118_lessThan__subset__iff,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y4 ) )
= ( ord_less_eq_nat @ X @ Y4 ) ) ).
% lessThan_subset_iff
thf(fact_7119_lessThan__subset__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y4 ) )
= ( ord_less_eq_real @ X @ Y4 ) ) ).
% lessThan_subset_iff
thf(fact_7120_round__numeral,axiom,
! [N: num] :
( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% round_numeral
thf(fact_7121_round__numeral,axiom,
! [N: num] :
( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% round_numeral
thf(fact_7122_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_7123_round__1,axiom,
( ( archim8280529875227126926d_real @ one_one_real )
= one_one_int ) ).
% round_1
thf(fact_7124_round__1,axiom,
( ( archim7778729529865785530nd_rat @ one_one_rat )
= one_one_int ) ).
% round_1
thf(fact_7125_sum_OlessThan__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_7126_sum_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_7127_sum_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_7128_sum_OlessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_7129_round__neg__numeral,axiom,
! [N: num] :
( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% round_neg_numeral
thf(fact_7130_round__neg__numeral,axiom,
! [N: num] :
( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% round_neg_numeral
thf(fact_7131_powser__zero,axiom,
! [F: nat > complex] :
( ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_7132_powser__zero,axiom,
! [F: nat > real] :
( ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
= ( F @ zero_zero_nat ) ) ).
% powser_zero
thf(fact_7133_lessThan__non__empty,axiom,
! [X: int] :
( ( set_ord_lessThan_int @ X )
!= bot_bot_set_int ) ).
% lessThan_non_empty
thf(fact_7134_lessThan__non__empty,axiom,
! [X: real] :
( ( set_or5984915006950818249n_real @ X )
!= bot_bot_set_real ) ).
% lessThan_non_empty
thf(fact_7135_lessThan__def,axiom,
( set_ord_lessThan_rat
= ( ^ [U3: rat] :
( collect_rat
@ ^ [X2: rat] : ( ord_less_rat @ X2 @ U3 ) ) ) ) ).
% lessThan_def
thf(fact_7136_lessThan__def,axiom,
( set_ord_lessThan_num
= ( ^ [U3: num] :
( collect_num
@ ^ [X2: num] : ( ord_less_num @ X2 @ U3 ) ) ) ) ).
% lessThan_def
thf(fact_7137_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U3: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_int @ X2 @ U3 ) ) ) ) ).
% lessThan_def
thf(fact_7138_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U3: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ U3 ) ) ) ) ).
% lessThan_def
thf(fact_7139_lessThan__def,axiom,
( set_or5984915006950818249n_real
= ( ^ [U3: real] :
( collect_real
@ ^ [X2: real] : ( ord_less_real @ X2 @ U3 ) ) ) ) ).
% lessThan_def
thf(fact_7140_Iio__eq__empty__iff,axiom,
! [N: extended_enat] :
( ( ( set_or8419480210114673929d_enat @ N )
= bot_bo7653980558646680370d_enat )
= ( N = bot_bo4199563552545308370d_enat ) ) ).
% Iio_eq_empty_iff
thf(fact_7141_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_7142_lessThan__strict__subset__iff,axiom,
! [M: rat,N: rat] :
( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
= ( ord_less_rat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_7143_lessThan__strict__subset__iff,axiom,
! [M: num,N: num] :
( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_7144_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_7145_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_7146_lessThan__strict__subset__iff,axiom,
! [M: real,N: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
= ( ord_less_real @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_7147_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_7148_round__mono,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_rat @ X @ Y4 )
=> ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y4 ) ) ) ).
% round_mono
thf(fact_7149_sum_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_7150_sum_Onat__diff__reindex,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_7151_sum__diff__distrib,axiom,
! [Q: real > nat,P: real > nat,N: real] :
( ! [X3: real] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X2: real] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_7152_sum__diff__distrib,axiom,
! [Q: nat > nat,P: nat > nat,N: nat] :
( ! [X3: nat] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_7153_sum_OlessThan__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_7154_sum_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_7155_sum_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_7156_sum_OlessThan__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_7157_sumr__diff__mult__const2,axiom,
! [F: nat > int,N: nat,R2: int] :
( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R2 ) )
= ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ R2 )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sumr_diff_mult_const2
thf(fact_7158_sumr__diff__mult__const2,axiom,
! [F: nat > rat,N: nat,R2: rat] :
( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R2 ) )
= ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ R2 )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sumr_diff_mult_const2
thf(fact_7159_sumr__diff__mult__const2,axiom,
! [F: nat > real,N: nat,R2: real] :
( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R2 ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ R2 )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sumr_diff_mult_const2
thf(fact_7160_sum__lessThan__telescope_H,axiom,
! [F: nat > rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_7161_sum__lessThan__telescope_H,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_7162_sum__lessThan__telescope_H,axiom,
! [F: nat > real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% sum_lessThan_telescope'
thf(fact_7163_sum__lessThan__telescope,axiom,
! [F: nat > rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_7164_sum__lessThan__telescope,axiom,
! [F: nat > int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_7165_sum__lessThan__telescope,axiom,
! [F: nat > real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% sum_lessThan_telescope
thf(fact_7166_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_7167_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_7168_one__diff__power__eq,axiom,
! [X: complex,N: nat] :
( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_7169_one__diff__power__eq,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_7170_one__diff__power__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_7171_one__diff__power__eq,axiom,
! [X: real,N: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq
thf(fact_7172_power__diff__1__eq,axiom,
! [X: complex,N: nat] :
( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex )
= ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_7173_power__diff__1__eq,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat )
= ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_7174_power__diff__1__eq,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
= ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_7175_power__diff__1__eq,axiom,
! [X: real,N: nat] :
( ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real )
= ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_1_eq
thf(fact_7176_geometric__sum,axiom,
! [X: complex,N: nat] :
( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).
% geometric_sum
thf(fact_7177_geometric__sum,axiom,
! [X: rat,N: nat] :
( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).
% geometric_sum
thf(fact_7178_geometric__sum,axiom,
! [X: real,N: nat] :
( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).
% geometric_sum
thf(fact_7179_round__diff__minimal,axiom,
! [Z2: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z2 ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ M ) ) ) ) ).
% round_diff_minimal
thf(fact_7180_round__diff__minimal,axiom,
! [Z2: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z2 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% round_diff_minimal
thf(fact_7181_sum__gp__strict,axiom,
! [X: complex,N: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri8010041392384452111omplex @ N ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_7182_sum__gp__strict,axiom,
! [X: rat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri681578069525770553at_rat @ N ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_7183_sum__gp__strict,axiom,
! [X: real,N: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( semiri5074537144036343181t_real @ N ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp_strict
thf(fact_7184_lemma__termdiff1,axiom,
! [Z2: complex,H2: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [P4: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z2 @ P4 ) ) @ ( power_power_complex @ Z2 @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups2073611262835488442omplex
@ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ Z2 @ P4 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ M @ P4 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_7185_lemma__termdiff1,axiom,
! [Z2: rat,H2: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [P4: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z2 @ P4 ) ) @ ( power_power_rat @ Z2 @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups2906978787729119204at_rat
@ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ Z2 @ P4 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ M @ P4 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_7186_lemma__termdiff1,axiom,
! [Z2: int,H2: int,M: nat] :
( ( groups3539618377306564664at_int
@ ^ [P4: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z2 @ P4 ) ) @ ( power_power_int @ Z2 @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups3539618377306564664at_int
@ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ Z2 @ P4 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z2 @ ( minus_minus_nat @ M @ P4 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_7187_lemma__termdiff1,axiom,
! [Z2: real,H2: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [P4: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z2 @ P4 ) ) @ ( power_power_real @ Z2 @ M ) )
@ ( set_ord_lessThan_nat @ M ) )
= ( groups6591440286371151544t_real
@ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ Z2 @ P4 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ M @ P4 ) ) ) )
@ ( set_ord_lessThan_nat @ M ) ) ) ).
% lemma_termdiff1
thf(fact_7188_diff__power__eq__sum,axiom,
! [X: complex,N: nat,Y4: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N ) ) @ ( power_power_complex @ Y4 @ ( suc @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y4 )
@ ( groups2073611262835488442omplex
@ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ X @ P4 ) @ ( power_power_complex @ Y4 @ ( minus_minus_nat @ N @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_7189_diff__power__eq__sum,axiom,
! [X: rat,N: nat,Y4: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N ) ) @ ( power_power_rat @ Y4 @ ( suc @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y4 )
@ ( groups2906978787729119204at_rat
@ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ X @ P4 ) @ ( power_power_rat @ Y4 @ ( minus_minus_nat @ N @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_7190_diff__power__eq__sum,axiom,
! [X: int,N: nat,Y4: int] :
( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y4 @ ( suc @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y4 )
@ ( groups3539618377306564664at_int
@ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ X @ P4 ) @ ( power_power_int @ Y4 @ ( minus_minus_nat @ N @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_7191_diff__power__eq__sum,axiom,
! [X: real,N: nat,Y4: real] :
( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N ) ) @ ( power_power_real @ Y4 @ ( suc @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y4 )
@ ( groups6591440286371151544t_real
@ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ X @ P4 ) @ ( power_power_real @ Y4 @ ( minus_minus_nat @ N @ P4 ) ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_7192_power__diff__sumr2,axiom,
! [X: complex,N: nat,Y4: complex] :
( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y4 @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y4 )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y4 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_7193_power__diff__sumr2,axiom,
! [X: rat,N: nat,Y4: rat] :
( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y4 @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y4 )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y4 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_7194_power__diff__sumr2,axiom,
! [X: int,N: nat,Y4: int] :
( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y4 @ N ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y4 )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y4 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_7195_power__diff__sumr2,axiom,
! [X: real,N: nat,Y4: real] :
( ( minus_minus_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ N ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y4 )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y4 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% power_diff_sumr2
thf(fact_7196_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > rat,K5: rat,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
=> ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_7197_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > int,K5: int,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K5 )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_7198_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > nat,K5: nat,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_7199_real__sum__nat__ivl__bounded2,axiom,
! [N: nat,F: nat > real,K5: real,K: nat] :
( ! [P7: nat] :
( ( ord_less_nat @ P7 @ N )
=> ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ K5 )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_7200_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M2: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M2 @ N2 ) @ ( modulo_modulo_nat @ M2 @ N2 ) ) ) ) ).
% divmod_nat_def
thf(fact_7201_one__diff__power__eq_H,axiom,
! [X: complex,N: nat] :
( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_7202_one__diff__power__eq_H,axiom,
! [X: rat,N: nat] :
( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_7203_one__diff__power__eq_H,axiom,
! [X: int,N: nat] :
( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
= ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_7204_one__diff__power__eq_H,axiom,
! [X: real,N: nat] :
( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% one_diff_power_eq'
thf(fact_7205_sum__split__even__odd,axiom,
! [F: nat > real,G: nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_7206_of__int__round__le,axiom,
! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% of_int_round_le
thf(fact_7207_of__int__round__le,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% of_int_round_le
thf(fact_7208_of__int__round__ge,axiom,
! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% of_int_round_ge
thf(fact_7209_of__int__round__ge,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% of_int_round_ge
thf(fact_7210_of__int__round__gt,axiom,
! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% of_int_round_gt
thf(fact_7211_of__int__round__gt,axiom,
! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% of_int_round_gt
thf(fact_7212_of__int__round__abs__le,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% of_int_round_abs_le
thf(fact_7213_of__int__round__abs__le,axiom,
! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% of_int_round_abs_le
thf(fact_7214_round__unique_H,axiom,
! [X: real,N: int] :
( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( archim8280529875227126926d_real @ X )
= N ) ) ).
% round_unique'
thf(fact_7215_round__unique_H,axiom,
! [X: rat,N: int] :
( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
=> ( ( archim7778729529865785530nd_rat @ X )
= N ) ) ).
% round_unique'
thf(fact_7216_suminf__geometric,axiom,
! [C: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
=> ( ( suminf_real @ ( power_power_real @ C ) )
= ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% suminf_geometric
thf(fact_7217_suminf__geometric,axiom,
! [C: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
=> ( ( suminf_complex @ ( power_power_complex @ C ) )
= ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% suminf_geometric
thf(fact_7218_sum__bounds__lt__plus1,axiom,
! [F: nat > nat,Mm: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_7219_sum__bounds__lt__plus1,axiom,
! [F: nat > real,Mm: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_7220_pi__series,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( suminf_real
@ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% pi_series
thf(fact_7221_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ zero_zero_real @ M2 ) )
@ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= one_one_real ) ).
% sumr_cos_zero_one
thf(fact_7222_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( summable_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_7223_geometric__deriv__sums,axiom,
! [Z2: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
=> ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) )
@ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_7224_geometric__deriv__sums,axiom,
! [Z2: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
=> ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) )
@ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_7225_summable__iff__shift,axiom,
! [F: nat > real,K: nat] :
( ( summable_real
@ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
= ( summable_real @ F ) ) ).
% summable_iff_shift
thf(fact_7226_cos__coeff__0,axiom,
( ( cos_coeff @ zero_zero_nat )
= one_one_real ) ).
% cos_coeff_0
thf(fact_7227_summable__cmult__iff,axiom,
! [C: complex,F: nat > complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
= ( ( C = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_cmult_iff
thf(fact_7228_summable__cmult__iff,axiom,
! [C: real,F: nat > real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_cmult_iff
thf(fact_7229_summable__divide__iff,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
= ( ( C = zero_zero_complex )
| ( summable_complex @ F ) ) ) ).
% summable_divide_iff
thf(fact_7230_summable__divide__iff,axiom,
! [F: nat > real,C: real] :
( ( summable_real
@ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
= ( ( C = zero_zero_real )
| ( summable_real @ F ) ) ) ).
% summable_divide_iff
thf(fact_7231_powser__sums__zero__iff,axiom,
! [A: nat > complex,X: complex] :
( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
@ X )
= ( ( A @ zero_zero_nat )
= X ) ) ).
% powser_sums_zero_iff
thf(fact_7232_powser__sums__zero__iff,axiom,
! [A: nat > real,X: real] :
( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
@ X )
= ( ( A @ zero_zero_nat )
= X ) ) ).
% powser_sums_zero_iff
thf(fact_7233_summable__geometric__iff,axiom,
! [C: real] :
( ( summable_real @ ( power_power_real @ C ) )
= ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% summable_geometric_iff
thf(fact_7234_summable__geometric__iff,axiom,
! [C: complex] :
( ( summable_complex @ ( power_power_complex @ C ) )
= ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% summable_geometric_iff
thf(fact_7235_summable__divide,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( summable_complex
@ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) ) ) ).
% summable_divide
thf(fact_7236_summable__divide,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) ) ) ).
% summable_divide
thf(fact_7237_summable__Suc__iff,axiom,
! [F: nat > real] :
( ( summable_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
= ( summable_real @ F ) ) ).
% summable_Suc_iff
thf(fact_7238_sums__mult2,axiom,
! [F: nat > real,A: real,C: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
@ ( times_times_real @ A @ C ) ) ) ).
% sums_mult2
thf(fact_7239_sums__mult,axiom,
! [F: nat > real,A: real,C: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
@ ( times_times_real @ C @ A ) ) ) ).
% sums_mult
thf(fact_7240_sums__divide,axiom,
! [F: nat > complex,A: complex,C: complex] :
( ( sums_complex @ F @ A )
=> ( sums_complex
@ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C )
@ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% sums_divide
thf(fact_7241_sums__divide,axiom,
! [F: nat > real,A: real,C: real] :
( ( sums_real @ F @ A )
=> ( sums_real
@ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C )
@ ( divide_divide_real @ A @ C ) ) ) ).
% sums_divide
thf(fact_7242_sums__le,axiom,
! [F: nat > real,G: nat > real,S2: real,T: real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_real @ F @ S2 )
=> ( ( sums_real @ G @ T )
=> ( ord_less_eq_real @ S2 @ T ) ) ) ) ).
% sums_le
thf(fact_7243_sums__le,axiom,
! [F: nat > nat,G: nat > nat,S2: nat,T: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_nat @ F @ S2 )
=> ( ( sums_nat @ G @ T )
=> ( ord_less_eq_nat @ S2 @ T ) ) ) ) ).
% sums_le
thf(fact_7244_sums__le,axiom,
! [F: nat > int,G: nat > int,S2: int,T: int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( sums_int @ F @ S2 )
=> ( ( sums_int @ G @ T )
=> ( ord_less_eq_int @ S2 @ T ) ) ) ) ).
% sums_le
thf(fact_7245_summable__mult2,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ).
% summable_mult2
thf(fact_7246_summable__mult,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) ) ) ).
% summable_mult
thf(fact_7247_summable__add,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% summable_add
thf(fact_7248_summable__add,axiom,
! [F: nat > nat,G: nat > nat] :
( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( summable_nat
@ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% summable_add
thf(fact_7249_summable__add,axiom,
! [F: nat > int,G: nat > int] :
( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( summable_int
@ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% summable_add
thf(fact_7250_sums__add,axiom,
! [F: nat > real,A: real,G: nat > real,B: real] :
( ( sums_real @ F @ A )
=> ( ( sums_real @ G @ B )
=> ( sums_real
@ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus_real @ A @ B ) ) ) ) ).
% sums_add
thf(fact_7251_sums__add,axiom,
! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
( ( sums_nat @ F @ A )
=> ( ( sums_nat @ G @ B )
=> ( sums_nat
@ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus_nat @ A @ B ) ) ) ) ).
% sums_add
thf(fact_7252_sums__add,axiom,
! [F: nat > int,A: int,G: nat > int,B: int] :
( ( sums_int @ F @ A )
=> ( ( sums_int @ G @ B )
=> ( sums_int
@ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus_int @ A @ B ) ) ) ) ).
% sums_add
thf(fact_7253_summable__ignore__initial__segment,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( summable_real
@ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_7254_summable__comparison__test_H,axiom,
! [G: nat > real,N4: nat,F: nat > real] :
( ( summable_real @ G )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N4 @ N3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test'
thf(fact_7255_summable__comparison__test_H,axiom,
! [G: nat > real,N4: nat,F: nat > complex] :
( ( summable_real @ G )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N4 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test'
thf(fact_7256_summable__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N7 @ N3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real @ F ) ) ) ).
% summable_comparison_test
thf(fact_7257_summable__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N7 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_complex @ F ) ) ) ).
% summable_comparison_test
thf(fact_7258_suminf__le,axiom,
! [F: nat > real,G: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% suminf_le
thf(fact_7259_suminf__le,axiom,
! [F: nat > nat,G: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% suminf_le
thf(fact_7260_suminf__le,axiom,
! [F: nat > int,G: nat > int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% suminf_le
thf(fact_7261_sums__mult__iff,axiom,
! [C: complex,F: nat > complex,D: complex] :
( ( C != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
@ ( times_times_complex @ C @ D ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_7262_sums__mult__iff,axiom,
! [C: real,F: nat > real,D: real] :
( ( C != zero_zero_real )
=> ( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
@ ( times_times_real @ C @ D ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult_iff
thf(fact_7263_sums__mult2__iff,axiom,
! [C: complex,F: nat > complex,D: complex] :
( ( C != zero_zero_complex )
=> ( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
@ ( times_times_complex @ D @ C ) )
= ( sums_complex @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_7264_sums__mult2__iff,axiom,
! [C: real,F: nat > real,D: real] :
( ( C != zero_zero_real )
=> ( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
@ ( times_times_real @ D @ C ) )
= ( sums_real @ F @ D ) ) ) ).
% sums_mult2_iff
thf(fact_7265_summable__mult__D,axiom,
! [C: complex,F: nat > complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
=> ( ( C != zero_zero_complex )
=> ( summable_complex @ F ) ) ) ).
% summable_mult_D
thf(fact_7266_summable__mult__D,axiom,
! [C: real,F: nat > real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
=> ( ( C != zero_zero_real )
=> ( summable_real @ F ) ) ) ).
% summable_mult_D
thf(fact_7267_summable__zero__power,axiom,
summable_real @ ( power_power_real @ zero_zero_real ) ).
% summable_zero_power
thf(fact_7268_summable__zero__power,axiom,
summable_int @ ( power_power_int @ zero_zero_int ) ).
% summable_zero_power
thf(fact_7269_summable__zero__power,axiom,
summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% summable_zero_power
thf(fact_7270_suminf__mult2,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( ( times_times_real @ ( suminf_real @ F ) @ C )
= ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ) ).
% suminf_mult2
thf(fact_7271_suminf__mult,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
= ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% suminf_mult
thf(fact_7272_suminf__add,axiom,
! [F: nat > real,G: nat > real] :
( ( summable_real @ F )
=> ( ( summable_real @ G )
=> ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
= ( suminf_real
@ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% suminf_add
thf(fact_7273_suminf__add,axiom,
! [F: nat > nat,G: nat > nat] :
( ( summable_nat @ F )
=> ( ( summable_nat @ G )
=> ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
= ( suminf_nat
@ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% suminf_add
thf(fact_7274_suminf__add,axiom,
! [F: nat > int,G: nat > int] :
( ( summable_int @ F )
=> ( ( summable_int @ G )
=> ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
= ( suminf_int
@ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% suminf_add
thf(fact_7275_suminf__divide,axiom,
! [F: nat > complex,C: complex] :
( ( summable_complex @ F )
=> ( ( suminf_complex
@ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
= ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% suminf_divide
thf(fact_7276_suminf__divide,axiom,
! [F: nat > real,C: real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
= ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% suminf_divide
thf(fact_7277_powser__insidea,axiom,
! [F: nat > real,X: real,Z2: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X ) )
=> ( summable_real
@ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) ) ) ) ) ).
% powser_insidea
thf(fact_7278_powser__insidea,axiom,
! [F: nat > complex,X: complex,Z2: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X ) )
=> ( summable_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) ) ) ) ) ).
% powser_insidea
thf(fact_7279_suminf__eq__zero__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ( ( suminf_real @ F )
= zero_zero_real )
= ( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_real ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_7280_suminf__eq__zero__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ( ( suminf_nat @ F )
= zero_zero_nat )
= ( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_nat ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_7281_suminf__eq__zero__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ( ( suminf_int @ F )
= zero_zero_int )
= ( ! [N2: nat] :
( ( F @ N2 )
= zero_zero_int ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_7282_suminf__nonneg,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_7283_suminf__nonneg,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_7284_suminf__nonneg,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_nonneg
thf(fact_7285_suminf__pos,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% suminf_pos
thf(fact_7286_suminf__pos,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% suminf_pos
thf(fact_7287_suminf__pos,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% suminf_pos
thf(fact_7288_sums__mult__D,axiom,
! [C: complex,F: nat > complex,A: complex] :
( ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
@ A )
=> ( ( C != zero_zero_complex )
=> ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% sums_mult_D
thf(fact_7289_sums__mult__D,axiom,
! [C: real,F: nat > real,A: real] :
( ( sums_real
@ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
@ A )
=> ( ( C != zero_zero_real )
=> ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% sums_mult_D
thf(fact_7290_sums__Suc__imp,axiom,
! [F: nat > complex,S2: complex] :
( ( ( F @ zero_zero_nat )
= zero_zero_complex )
=> ( ( sums_complex
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ S2 )
=> ( sums_complex @ F @ S2 ) ) ) ).
% sums_Suc_imp
thf(fact_7291_sums__Suc__imp,axiom,
! [F: nat > real,S2: real] :
( ( ( F @ zero_zero_nat )
= zero_zero_real )
=> ( ( sums_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ S2 )
=> ( sums_real @ F @ S2 ) ) ) ).
% sums_Suc_imp
thf(fact_7292_sums__Suc__iff,axiom,
! [F: nat > real,S2: real] :
( ( sums_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ S2 )
= ( sums_real @ F @ ( plus_plus_real @ S2 @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc_iff
thf(fact_7293_sums__Suc,axiom,
! [F: nat > real,L2: real] :
( ( sums_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ L2 )
=> ( sums_real @ F @ ( plus_plus_real @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_7294_sums__Suc,axiom,
! [F: nat > nat,L2: nat] :
( ( sums_nat
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ L2 )
=> ( sums_nat @ F @ ( plus_plus_nat @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_7295_sums__Suc,axiom,
! [F: nat > int,L2: int] :
( ( sums_int
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
@ L2 )
=> ( sums_int @ F @ ( plus_plus_int @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% sums_Suc
thf(fact_7296_summable__0__powser,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% summable_0_powser
thf(fact_7297_summable__0__powser,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% summable_0_powser
thf(fact_7298_summable__zero__power_H,axiom,
! [F: nat > complex] :
( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% summable_zero_power'
thf(fact_7299_summable__zero__power_H,axiom,
! [F: nat > real] :
( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% summable_zero_power'
thf(fact_7300_summable__zero__power_H,axiom,
! [F: nat > int] :
( summable_int
@ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).
% summable_zero_power'
thf(fact_7301_sums__zero__iff__shift,axiom,
! [N: nat,F: nat > complex,S2: complex] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( F @ I3 )
= zero_zero_complex ) )
=> ( ( sums_complex
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
@ S2 )
= ( sums_complex @ F @ S2 ) ) ) ).
% sums_zero_iff_shift
thf(fact_7302_sums__zero__iff__shift,axiom,
! [N: nat,F: nat > real,S2: real] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ( F @ I3 )
= zero_zero_real ) )
=> ( ( sums_real
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
@ S2 )
= ( sums_real @ F @ S2 ) ) ) ).
% sums_zero_iff_shift
thf(fact_7303_summable__powser__split__head,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) ) )
= ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) ) ) ).
% summable_powser_split_head
thf(fact_7304_summable__powser__split__head,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) ) )
= ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) ) ) ).
% summable_powser_split_head
thf(fact_7305_powser__split__head_I3_J,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
=> ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) ) ) ) ).
% powser_split_head(3)
thf(fact_7306_powser__split__head_I3_J,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) ) ) ) ).
% powser_split_head(3)
thf(fact_7307_summable__powser__ignore__initial__segment,axiom,
! [F: nat > complex,M: nat,Z2: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_complex @ Z2 @ N2 ) ) )
= ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_7308_summable__powser__ignore__initial__segment,axiom,
! [F: nat > real,M: nat,Z2: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_real @ Z2 @ N2 ) ) )
= ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_7309_summable__norm__comparison__test,axiom,
! [F: nat > complex,G: nat > real] :
( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N7 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_7310_summable__rabs__comparison__test,axiom,
! [F: nat > real,G: nat > real] :
( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N7 @ N3 )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable_real @ G )
=> ( summable_real
@ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_7311_suminf__pos2,axiom,
! [F: nat > real,I2: nat] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_7312_suminf__pos2,axiom,
! [F: nat > nat,I2: nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_7313_suminf__pos2,axiom,
! [F: nat > int,I2: nat] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
=> ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% suminf_pos2
thf(fact_7314_suminf__pos__iff,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
= ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_7315_suminf__pos__iff,axiom,
! [F: nat > nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
= ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_7316_suminf__pos__iff,axiom,
! [F: nat > int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
= ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_7317_suminf__le__const,axiom,
! [F: nat > int,X: int] :
( ( summable_int @ F )
=> ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_7318_suminf__le__const,axiom,
! [F: nat > nat,X: nat] :
( ( summable_nat @ F )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_7319_suminf__le__const,axiom,
! [F: nat > real,X: real] :
( ( summable_real @ F )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
% suminf_le_const
thf(fact_7320_powser__sums__if,axiom,
! [M: nat,Z2: complex] :
( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z2 @ N2 ) )
@ ( power_power_complex @ Z2 @ M ) ) ).
% powser_sums_if
thf(fact_7321_powser__sums__if,axiom,
! [M: nat,Z2: real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z2 @ N2 ) )
@ ( power_power_real @ Z2 @ M ) ) ).
% powser_sums_if
thf(fact_7322_powser__sums__if,axiom,
! [M: nat,Z2: int] :
( sums_int
@ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z2 @ N2 ) )
@ ( power_power_int @ Z2 @ M ) ) ).
% powser_sums_if
thf(fact_7323_summableI__nonneg__bounded,axiom,
! [F: nat > int,X: int] :
( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( summable_int @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_7324_summableI__nonneg__bounded,axiom,
! [F: nat > nat,X: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( summable_nat @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_7325_summableI__nonneg__bounded,axiom,
! [F: nat > real,X: real] :
( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X )
=> ( summable_real @ F ) ) ) ).
% summableI_nonneg_bounded
thf(fact_7326_complete__algebra__summable__geometric,axiom,
! [X: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
=> ( summable_real @ ( power_power_real @ X ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_7327_complete__algebra__summable__geometric,axiom,
! [X: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
=> ( summable_complex @ ( power_power_complex @ X ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_7328_summable__geometric,axiom,
! [C: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
=> ( summable_real @ ( power_power_real @ C ) ) ) ).
% summable_geometric
thf(fact_7329_summable__geometric,axiom,
! [C: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
=> ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% summable_geometric
thf(fact_7330_sums__iff__shift,axiom,
! [F: nat > real,N: nat,S2: real] :
( ( sums_real
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
@ S2 )
= ( sums_real @ F @ ( plus_plus_real @ S2 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% sums_iff_shift
thf(fact_7331_sums__split__initial__segment,axiom,
! [F: nat > real,S2: real,N: nat] :
( ( sums_real @ F @ S2 )
=> ( sums_real
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
@ ( minus_minus_real @ S2 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_7332_sums__iff__shift_H,axiom,
! [F: nat > real,N: nat,S2: real] :
( ( sums_real
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
@ ( minus_minus_real @ S2 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
= ( sums_real @ F @ S2 ) ) ).
% sums_iff_shift'
thf(fact_7333_suminf__split__head,axiom,
! [F: nat > real] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
= ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% suminf_split_head
thf(fact_7334_sums__If__finite__set_H,axiom,
! [G: nat > real,S3: real,A2: set_nat,S4: real,F: nat > real] :
( ( sums_real @ G @ S3 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( S4
= ( plus_plus_real @ S3
@ ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
@ A2 ) ) )
=> ( sums_real
@ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A2 ) @ ( F @ N2 ) @ ( G @ N2 ) )
@ S4 ) ) ) ) ).
% sums_If_finite_set'
thf(fact_7335_sum__le__suminf,axiom,
! [F: nat > int,I5: set_nat] :
( ( summable_int @ F )
=> ( ( finite_finite_nat @ I5 )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
=> ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_7336_sum__le__suminf,axiom,
! [F: nat > nat,I5: set_nat] :
( ( summable_nat @ F )
=> ( ( finite_finite_nat @ I5 )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_7337_sum__le__suminf,axiom,
! [F: nat > real,I5: set_nat] :
( ( summable_real @ F )
=> ( ( finite_finite_nat @ I5 )
=> ( ! [N3: nat] :
( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% sum_le_suminf
thf(fact_7338_suminf__split__initial__segment,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( ( suminf_real @ F )
= ( plus_plus_real
@ ( suminf_real
@ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
@ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_7339_suminf__minus__initial__segment,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( ( suminf_real
@ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
= ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_7340_powser__sums__zero,axiom,
! [A: nat > complex] :
( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_7341_powser__sums__zero,axiom,
! [A: nat > real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
@ ( A @ zero_zero_nat ) ) ).
% powser_sums_zero
thf(fact_7342_powser__inside,axiom,
! [F: nat > real,X: real,Z2: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
=> ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) ) ) ) ).
% powser_inside
thf(fact_7343_powser__inside,axiom,
! [F: nat > complex,X: complex,Z2: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
=> ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X ) )
=> ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) ) ) ) ).
% powser_inside
thf(fact_7344_sum__less__suminf,axiom,
! [F: nat > int,N: nat] :
( ( summable_int @ F )
=> ( ! [M4: nat] :
( ( ord_less_eq_nat @ N @ M4 )
=> ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_7345_sum__less__suminf,axiom,
! [F: nat > nat,N: nat] :
( ( summable_nat @ F )
=> ( ! [M4: nat] :
( ( ord_less_eq_nat @ N @ M4 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_7346_sum__less__suminf,axiom,
! [F: nat > real,N: nat] :
( ( summable_real @ F )
=> ( ! [M4: nat] :
( ( ord_less_eq_nat @ N @ M4 )
=> ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_less_suminf
thf(fact_7347_pi__less__4,axiom,
ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% pi_less_4
thf(fact_7348_pi__ge__two,axiom,
ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% pi_ge_two
thf(fact_7349_pi__half__neq__two,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
!= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% pi_half_neq_two
thf(fact_7350_powser__split__head_I1_J,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
=> ( ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
= ( plus_plus_complex @ ( F @ zero_zero_nat )
@ ( times_times_complex
@ ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) ) )
@ Z2 ) ) ) ) ).
% powser_split_head(1)
thf(fact_7351_powser__split__head_I1_J,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
=> ( ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
= ( plus_plus_real @ ( F @ zero_zero_nat )
@ ( times_times_real
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) ) )
@ Z2 ) ) ) ) ).
% powser_split_head(1)
thf(fact_7352_powser__split__head_I2_J,axiom,
! [F: nat > complex,Z2: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
=> ( ( times_times_complex
@ ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z2 @ N2 ) ) )
@ Z2 )
= ( minus_minus_complex
@ ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_7353_powser__split__head_I2_J,axiom,
! [F: nat > real,Z2: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
=> ( ( times_times_real
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z2 @ N2 ) ) )
@ Z2 )
= ( minus_minus_real
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) )
@ ( F @ zero_zero_nat ) ) ) ) ).
% powser_split_head(2)
thf(fact_7354_summable__partial__sum__bound,axiom,
! [F: nat > complex,E: real] :
( ( summable_complex @ F )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ~ ! [N8: nat] :
~ ! [M3: nat] :
( ( ord_less_eq_nat @ N8 @ M3 )
=> ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M3 @ N9 ) ) ) @ E ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_7355_summable__partial__sum__bound,axiom,
! [F: nat > real,E: real] :
( ( summable_real @ F )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ~ ! [N8: nat] :
~ ! [M3: nat] :
( ( ord_less_eq_nat @ N8 @ M3 )
=> ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M3 @ N9 ) ) ) @ E ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_7356_suminf__exist__split,axiom,
! [R2: real,F: nat > real] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( summable_real @ F )
=> ? [N8: nat] :
! [N9: nat] :
( ( ord_less_eq_nat @ N8 @ N9 )
=> ( ord_less_real
@ ( real_V7735802525324610683m_real
@ ( suminf_real
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
@ R2 ) ) ) ) ).
% suminf_exist_split
thf(fact_7357_suminf__exist__split,axiom,
! [R2: real,F: nat > complex] :
( ( ord_less_real @ zero_zero_real @ R2 )
=> ( ( summable_complex @ F )
=> ? [N8: nat] :
! [N9: nat] :
( ( ord_less_eq_nat @ N8 @ N9 )
=> ( ord_less_real
@ ( real_V1022390504157884413omplex
@ ( suminf_complex
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
@ R2 ) ) ) ) ).
% suminf_exist_split
thf(fact_7358_summable__power__series,axiom,
! [F: nat > real,Z2: real] :
( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
=> ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( summable_real
@ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z2 @ I4 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_7359_Abel__lemma,axiom,
! [R2: real,R0: real,A: nat > complex,M7: real] :
( ( ord_less_eq_real @ zero_zero_real @ R2 )
=> ( ( ord_less_real @ R2 @ R0 )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M7 )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R2 @ N2 ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_7360_summable__ratio__test,axiom,
! [C: real,N4: nat,F: nat > real] :
( ( ord_less_real @ C @ one_one_real )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N4 @ N3 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
=> ( summable_real @ F ) ) ) ).
% summable_ratio_test
thf(fact_7361_summable__ratio__test,axiom,
! [C: real,N4: nat,F: nat > complex] :
( ( ord_less_real @ C @ one_one_real )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ N4 @ N3 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
=> ( summable_complex @ F ) ) ) ).
% summable_ratio_test
thf(fact_7362_sum__less__suminf2,axiom,
! [F: nat > int,N: nat,I2: nat] :
( ( summable_int @ F )
=> ( ! [M4: nat] :
( ( ord_less_eq_nat @ N @ M4 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
=> ( ( ord_less_eq_nat @ N @ I2 )
=> ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
=> ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_7363_sum__less__suminf2,axiom,
! [F: nat > nat,N: nat,I2: nat] :
( ( summable_nat @ F )
=> ( ! [M4: nat] :
( ( ord_less_eq_nat @ N @ M4 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
=> ( ( ord_less_eq_nat @ N @ I2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_7364_sum__less__suminf2,axiom,
! [F: nat > real,N: nat,I2: nat] :
( ( summable_real @ F )
=> ( ! [M4: nat] :
( ( ord_less_eq_nat @ N @ M4 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
=> ( ( ord_less_eq_nat @ N @ I2 )
=> ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_7365_geometric__sums,axiom,
! [C: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
=> ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% geometric_sums
thf(fact_7366_geometric__sums,axiom,
! [C: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
=> ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% geometric_sums
thf(fact_7367_power__half__series,axiom,
( sums_real
@ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
@ one_one_real ) ).
% power_half_series
thf(fact_7368_pi__half__neq__zero,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
!= zero_zero_real ) ).
% pi_half_neq_zero
thf(fact_7369_pi__half__less__two,axiom,
ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% pi_half_less_two
thf(fact_7370_pi__half__le__two,axiom,
ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% pi_half_le_two
thf(fact_7371_pi__half__gt__zero,axiom,
ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% pi_half_gt_zero
thf(fact_7372_pi__half__ge__zero,axiom,
ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% pi_half_ge_zero
thf(fact_7373_m2pi__less__pi,axiom,
ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% m2pi_less_pi
thf(fact_7374_arctan__ubound,axiom,
! [Y4: real] : ( ord_less_real @ ( arctan @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arctan_ubound
thf(fact_7375_arctan__one,axiom,
( ( arctan @ one_one_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% arctan_one
thf(fact_7376_minus__pi__half__less__zero,axiom,
ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% minus_pi_half_less_zero
thf(fact_7377_arctan__lbound,axiom,
! [Y4: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y4 ) ) ).
% arctan_lbound
thf(fact_7378_arctan__bounded,axiom,
! [Y4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y4 ) )
& ( ord_less_real @ ( arctan @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% arctan_bounded
thf(fact_7379_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums_real @ G @ X )
=> ( sums_real
@ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ X ) ) ).
% sums_if'
thf(fact_7380_sums__if,axiom,
! [G: nat > real,X: real,F: nat > real,Y4: real] :
( ( sums_real @ G @ X )
=> ( ( sums_real @ F @ Y4 )
=> ( sums_real
@ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( plus_plus_real @ X @ Y4 ) ) ) ) ).
% sums_if
thf(fact_7381_sum__pos__lt__pair,axiom,
! [F: nat > real,K: nat] :
( ( summable_real @ F )
=> ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
=> ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_7382_machin__Euler,axiom,
( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% machin_Euler
thf(fact_7383_machin,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% machin
thf(fact_7384_sin__cos__npi,axiom,
! [N: nat] :
( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% sin_cos_npi
thf(fact_7385_diffs__equiv,axiom,
! [C: nat > complex,X: complex] :
( ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
=> ( sums_complex
@ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
@ ( suminf_complex
@ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% diffs_equiv
thf(fact_7386_diffs__equiv,axiom,
! [C: nat > real,X: real] :
( ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
=> ( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% diffs_equiv
thf(fact_7387_cos__pi__eq__zero,axiom,
! [M: nat] :
( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ).
% cos_pi_eq_zero
thf(fact_7388_monoI1,axiom,
! [X7: nat > real] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_real @ ( X7 @ M4 ) @ ( X7 @ N3 ) ) )
=> ( topolo6980174941875973593q_real @ X7 ) ) ).
% monoI1
thf(fact_7389_monoI1,axiom,
! [X7: nat > set_nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_set_nat @ ( X7 @ M4 ) @ ( X7 @ N3 ) ) )
=> ( topolo7278393974255667507et_nat @ X7 ) ) ).
% monoI1
thf(fact_7390_monoI1,axiom,
! [X7: nat > rat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_rat @ ( X7 @ M4 ) @ ( X7 @ N3 ) ) )
=> ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% monoI1
thf(fact_7391_monoI1,axiom,
! [X7: nat > num] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_num @ ( X7 @ M4 ) @ ( X7 @ N3 ) ) )
=> ( topolo1459490580787246023eq_num @ X7 ) ) ).
% monoI1
thf(fact_7392_monoI1,axiom,
! [X7: nat > nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ ( X7 @ M4 ) @ ( X7 @ N3 ) ) )
=> ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% monoI1
thf(fact_7393_monoI1,axiom,
! [X7: nat > int] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_int @ ( X7 @ M4 ) @ ( X7 @ N3 ) ) )
=> ( topolo4899668324122417113eq_int @ X7 ) ) ).
% monoI1
thf(fact_7394_monoI2,axiom,
! [X7: nat > real] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ M4 ) ) )
=> ( topolo6980174941875973593q_real @ X7 ) ) ).
% monoI2
thf(fact_7395_monoI2,axiom,
! [X7: nat > set_nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_set_nat @ ( X7 @ N3 ) @ ( X7 @ M4 ) ) )
=> ( topolo7278393974255667507et_nat @ X7 ) ) ).
% monoI2
thf(fact_7396_monoI2,axiom,
! [X7: nat > rat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( X7 @ M4 ) ) )
=> ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% monoI2
thf(fact_7397_monoI2,axiom,
! [X7: nat > num] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_num @ ( X7 @ N3 ) @ ( X7 @ M4 ) ) )
=> ( topolo1459490580787246023eq_num @ X7 ) ) ).
% monoI2
thf(fact_7398_monoI2,axiom,
! [X7: nat > nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ M4 ) ) )
=> ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% monoI2
thf(fact_7399_monoI2,axiom,
! [X7: nat > int] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
=> ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ M4 ) ) )
=> ( topolo4899668324122417113eq_int @ X7 ) ) ).
% monoI2
thf(fact_7400_monoseq__def,axiom,
( topolo6980174941875973593q_real
= ( ^ [X6: nat > real] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) )
| ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7401_monoseq__def,axiom,
( topolo7278393974255667507et_nat
= ( ^ [X6: nat > set_nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_set_nat @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) )
| ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_set_nat @ ( X6 @ N2 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7402_monoseq__def,axiom,
( topolo4267028734544971653eq_rat
= ( ^ [X6: nat > rat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_rat @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) )
| ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7403_monoseq__def,axiom,
( topolo1459490580787246023eq_num
= ( ^ [X6: nat > num] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_num @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) )
| ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_num @ ( X6 @ N2 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7404_monoseq__def,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X6: nat > nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) )
| ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7405_monoseq__def,axiom,
( topolo4899668324122417113eq_int
= ( ^ [X6: nat > int] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_int @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) )
| ! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% monoseq_def
thf(fact_7406_cos__zero,axiom,
( ( cos_complex @ zero_zero_complex )
= one_one_complex ) ).
% cos_zero
thf(fact_7407_cos__zero,axiom,
( ( cos_real @ zero_zero_real )
= one_one_real ) ).
% cos_zero
thf(fact_7408_cos__pi,axiom,
( ( cos_real @ pi )
= ( uminus_uminus_real @ one_one_real ) ) ).
% cos_pi
thf(fact_7409_cos__periodic__pi2,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_periodic_pi2
thf(fact_7410_cos__periodic__pi,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% cos_periodic_pi
thf(fact_7411_sin__periodic__pi2,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ pi @ X ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_periodic_pi2
thf(fact_7412_sin__periodic__pi,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ X @ pi ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_periodic_pi
thf(fact_7413_sin__cos__squared__add3,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
= one_one_complex ) ).
% sin_cos_squared_add3
thf(fact_7414_sin__cos__squared__add3,axiom,
! [X: real] :
( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
= one_one_real ) ).
% sin_cos_squared_add3
thf(fact_7415_sin__npi,axiom,
! [N: nat] :
( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= zero_zero_real ) ).
% sin_npi
thf(fact_7416_sin__npi2,axiom,
! [N: nat] :
( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
= zero_zero_real ) ).
% sin_npi2
thf(fact_7417_sin__npi__int,axiom,
! [N: int] :
( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
= zero_zero_real ) ).
% sin_npi_int
thf(fact_7418_cos__pi__half,axiom,
( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ).
% cos_pi_half
thf(fact_7419_sin__two__pi,axiom,
( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= zero_zero_real ) ).
% sin_two_pi
thf(fact_7420_sin__pi__half,axiom,
( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_pi_half
thf(fact_7421_cos__two__pi,axiom,
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= one_one_real ) ).
% cos_two_pi
thf(fact_7422_cos__periodic,axiom,
! [X: real] :
( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( cos_real @ X ) ) ).
% cos_periodic
thf(fact_7423_sin__periodic,axiom,
! [X: real] :
( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( sin_real @ X ) ) ).
% sin_periodic
thf(fact_7424_cos__2pi__minus,axiom,
! [X: real] :
( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
= ( cos_real @ X ) ) ).
% cos_2pi_minus
thf(fact_7425_cos__npi2,axiom,
! [N: nat] :
( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% cos_npi2
thf(fact_7426_cos__npi,axiom,
! [N: nat] :
( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% cos_npi
thf(fact_7427_sin__cos__squared__add2,axiom,
! [X: real] :
( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_cos_squared_add2
thf(fact_7428_sin__cos__squared__add2,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_complex ) ).
% sin_cos_squared_add2
thf(fact_7429_sin__cos__squared__add,axiom,
! [X: real] :
( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% sin_cos_squared_add
thf(fact_7430_sin__cos__squared__add,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_complex ) ).
% sin_cos_squared_add
thf(fact_7431_sin__2npi,axiom,
! [N: nat] :
( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
= zero_zero_real ) ).
% sin_2npi
thf(fact_7432_cos__2npi,axiom,
! [N: nat] :
( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
= one_one_real ) ).
% cos_2npi
thf(fact_7433_sin__2pi__minus,axiom,
! [X: real] :
( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
= ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% sin_2pi_minus
thf(fact_7434_sin__int__2pin,axiom,
! [N: int] :
( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
= zero_zero_real ) ).
% sin_int_2pin
thf(fact_7435_cos__int__2pin,axiom,
! [N: int] :
( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
= one_one_real ) ).
% cos_int_2pin
thf(fact_7436_cos__3over2__pi,axiom,
( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
= zero_zero_real ) ).
% cos_3over2_pi
thf(fact_7437_sin__3over2__pi,axiom,
( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% sin_3over2_pi
thf(fact_7438_cos__npi__int,axiom,
! [N: int] :
( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
= one_one_real ) )
& ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
=> ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% cos_npi_int
thf(fact_7439_sin__add,axiom,
! [X: real,Y4: real] :
( ( sin_real @ ( plus_plus_real @ X @ Y4 ) )
= ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% sin_add
thf(fact_7440_sin__diff,axiom,
! [X: real,Y4: real] :
( ( sin_real @ ( minus_minus_real @ X @ Y4 ) )
= ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% sin_diff
thf(fact_7441_polar__Ex,axiom,
! [X: real,Y4: real] :
? [R3: real,A3: real] :
( ( X
= ( times_times_real @ R3 @ ( cos_real @ A3 ) ) )
& ( Y4
= ( times_times_real @ R3 @ ( sin_real @ A3 ) ) ) ) ).
% polar_Ex
thf(fact_7442_cos__one__sin__zero,axiom,
! [X: complex] :
( ( ( cos_complex @ X )
= one_one_complex )
=> ( ( sin_complex @ X )
= zero_zero_complex ) ) ).
% cos_one_sin_zero
thf(fact_7443_cos__one__sin__zero,axiom,
! [X: real] :
( ( ( cos_real @ X )
= one_one_real )
=> ( ( sin_real @ X )
= zero_zero_real ) ) ).
% cos_one_sin_zero
thf(fact_7444_cos__add,axiom,
! [X: real,Y4: real] :
( ( cos_real @ ( plus_plus_real @ X @ Y4 ) )
= ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% cos_add
thf(fact_7445_cos__diff,axiom,
! [X: real,Y4: real] :
( ( cos_real @ ( minus_minus_real @ X @ Y4 ) )
= ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) ) ) ) ).
% cos_diff
thf(fact_7446_sin__zero__norm__cos__one,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
=> ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
= one_one_real ) ) ).
% sin_zero_norm_cos_one
thf(fact_7447_sin__zero__norm__cos__one,axiom,
! [X: complex] :
( ( ( sin_complex @ X )
= zero_zero_complex )
=> ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
= one_one_real ) ) ).
% sin_zero_norm_cos_one
thf(fact_7448_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
=> ( ( abs_abs_real @ ( cos_real @ X ) )
= one_one_real ) ) ).
% sin_zero_abs_cos_one
thf(fact_7449_sin__double,axiom,
! [X: complex] :
( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% sin_double
thf(fact_7450_sin__double,axiom,
! [X: real] :
( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% sin_double
thf(fact_7451_sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% sin_le_one
thf(fact_7452_cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% cos_le_one
thf(fact_7453_sin__cos__le1,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) @ one_one_real ) ).
% sin_cos_le1
thf(fact_7454_sin__squared__eq,axiom,
! [X: complex] :
( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sin_squared_eq
thf(fact_7455_sin__squared__eq,axiom,
! [X: real] :
( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sin_squared_eq
thf(fact_7456_cos__squared__eq,axiom,
! [X: complex] :
( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cos_squared_eq
thf(fact_7457_cos__squared__eq,axiom,
! [X: real] :
( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cos_squared_eq
thf(fact_7458_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% sin_ge_minus_one
thf(fact_7459_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% cos_ge_minus_one
thf(fact_7460_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% abs_sin_le_one
thf(fact_7461_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% abs_cos_le_one
thf(fact_7462_sin__times__sin,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% sin_times_sin
thf(fact_7463_sin__times__sin,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_times_sin
thf(fact_7464_sin__times__cos,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( sin_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% sin_times_cos
thf(fact_7465_sin__times__cos,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( sin_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_times_cos
thf(fact_7466_cos__times__sin,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( cos_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% cos_times_sin
thf(fact_7467_cos__times__sin,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( cos_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_times_sin
thf(fact_7468_sin__plus__sin,axiom,
! [W2: complex,Z2: complex] :
( ( plus_plus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_7469_sin__plus__sin,axiom,
! [W2: real,Z2: real] :
( ( plus_plus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_7470_sin__diff__sin,axiom,
! [W2: complex,Z2: complex] :
( ( minus_minus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_7471_sin__diff__sin,axiom,
! [W2: real,Z2: real] :
( ( minus_minus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_7472_cos__diff__cos,axiom,
! [W2: complex,Z2: complex] :
( ( minus_minus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z2 @ W2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_7473_cos__diff__cos,axiom,
! [W2: real,Z2: real] :
( ( minus_minus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z2 @ W2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_7474_cos__double,axiom,
! [X: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
= ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cos_double
thf(fact_7475_cos__double,axiom,
! [X: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
= ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cos_double
thf(fact_7476_cos__double__sin,axiom,
! [W2: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_7477_cos__double__sin,axiom,
! [W2: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_7478_cos__two__neq__zero,axiom,
( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
!= zero_zero_real ) ).
% cos_two_neq_zero
thf(fact_7479_sin__zero__iff__int2,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
= ( ? [I4: int] :
( X
= ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_7480_diffs__def,axiom,
( diffs_int
= ( ^ [C2: nat > int,N2: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( C2 @ ( suc @ N2 ) ) ) ) ) ).
% diffs_def
thf(fact_7481_diffs__def,axiom,
( diffs_real
= ( ^ [C2: nat > real,N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( C2 @ ( suc @ N2 ) ) ) ) ) ).
% diffs_def
thf(fact_7482_diffs__def,axiom,
( diffs_rat
= ( ^ [C2: nat > rat,N2: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( C2 @ ( suc @ N2 ) ) ) ) ) ).
% diffs_def
thf(fact_7483_sincos__total__pi,axiom,
! [Y4: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T3: real] :
( ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ pi )
& ( X
= ( cos_real @ T3 ) )
& ( Y4
= ( sin_real @ T3 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_7484_sin__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
= ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_7485_cos__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
= ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_7486_termdiff__converges__all,axiom,
! [C: nat > complex,X: complex] :
( ! [X3: complex] :
( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
=> ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% termdiff_converges_all
thf(fact_7487_termdiff__converges__all,axiom,
! [C: nat > real,X: real] :
( ! [X3: real] :
( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% termdiff_converges_all
thf(fact_7488_sin__gt__zero__02,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_gt_zero_02
thf(fact_7489_cos__two__less__zero,axiom,
ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% cos_two_less_zero
thf(fact_7490_cos__is__zero,axiom,
? [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ X3 )
= zero_zero_real )
& ! [Y5: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
& ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ Y5 )
= zero_zero_real ) )
=> ( Y5 = X3 ) ) ) ).
% cos_is_zero
thf(fact_7491_cos__two__le__zero,axiom,
ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% cos_two_le_zero
thf(fact_7492_cos__total,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ? [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ pi )
& ( ( cos_real @ X3 )
= Y4 )
& ! [Y5: real] :
( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
& ( ord_less_eq_real @ Y5 @ pi )
& ( ( cos_real @ Y5 )
= Y4 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% cos_total
thf(fact_7493_sincos__total__pi__half,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T3: real] :
( ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( X
= ( cos_real @ T3 ) )
& ( Y4
= ( sin_real @ T3 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_7494_sincos__total__2pi__le,axiom,
! [X: real,Y4: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ? [T3: real] :
( ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
& ( X
= ( cos_real @ T3 ) )
& ( Y4
= ( sin_real @ T3 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_7495_sincos__total__2pi,axiom,
! [X: real,Y4: real] :
( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real )
=> ~ ! [T3: real] :
( ( ord_less_eq_real @ zero_zero_real @ T3 )
=> ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ( X
= ( cos_real @ T3 ) )
=> ( Y4
!= ( sin_real @ T3 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_7496_cos__times__cos,axiom,
! [W2: complex,Z2: complex] :
( ( times_times_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% cos_times_cos
thf(fact_7497_cos__times__cos,axiom,
! [W2: real,Z2: real] :
( ( times_times_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_times_cos
thf(fact_7498_cos__plus__cos,axiom,
! [W2: complex,Z2: complex] :
( ( plus_plus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_7499_cos__plus__cos,axiom,
! [W2: real,Z2: real] :
( ( plus_plus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_7500_sin__gt__zero2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% sin_gt_zero2
thf(fact_7501_sin__lt__zero,axiom,
! [X: real] :
( ( ord_less_real @ pi @ X )
=> ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% sin_lt_zero
thf(fact_7502_cos__double__less__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
=> ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% cos_double_less_one
thf(fact_7503_sin__30,axiom,
( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_30
thf(fact_7504_cos__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% cos_gt_zero
thf(fact_7505_sin__monotone__2pi__le,axiom,
! [Y4: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( sin_real @ Y4 ) @ ( sin_real @ X ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_7506_sin__mono__le__eq,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) )
= ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_7507_sin__inj__pi,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ( sin_real @ X )
= ( sin_real @ Y4 ) )
=> ( X = Y4 ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_7508_cos__60,axiom,
( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_60
thf(fact_7509_cos__one__2pi__int,axiom,
! [X: real] :
( ( ( cos_real @ X )
= one_one_real )
= ( ? [X2: int] :
( X
= ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_7510_cos__double__cos,axiom,
! [W2: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% cos_double_cos
thf(fact_7511_cos__double__cos,axiom,
! [W2: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W2 ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% cos_double_cos
thf(fact_7512_cos__treble__cos,axiom,
! [X: complex] :
( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
= ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).
% cos_treble_cos
thf(fact_7513_cos__treble__cos,axiom,
! [X: real] :
( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
= ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).
% cos_treble_cos
thf(fact_7514_termdiff__converges,axiom,
! [X: real,K5: real,C: nat > real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K5 )
=> ( ! [X3: real] :
( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% termdiff_converges
thf(fact_7515_termdiff__converges,axiom,
! [X: complex,K5: real,C: nat > complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K5 )
=> ( ! [X3: complex] :
( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
=> ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) )
=> ( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% termdiff_converges
thf(fact_7516_sin__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ pi @ X )
=> ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% sin_le_zero
thf(fact_7517_sin__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% sin_less_zero
thf(fact_7518_sin__monotone__2pi,axiom,
! [Y4: real,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
=> ( ( ord_less_real @ Y4 @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sin_real @ Y4 ) @ ( sin_real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_7519_sin__mono__less__eq,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) )
= ( ord_less_real @ X @ Y4 ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_7520_sin__total,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ? [X3: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
& ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ X3 )
= Y4 )
& ! [Y5: real] :
( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
& ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ Y5 )
= Y4 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% sin_total
thf(fact_7521_cos__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_7522_cos__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% cos_ge_zero
thf(fact_7523_cos__one__2pi,axiom,
! [X: real] :
( ( ( cos_real @ X )
= one_one_real )
= ( ? [X2: nat] :
( X
= ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
| ? [X2: nat] :
( X
= ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_7524_sin__pi__divide__n__gt__0,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_7525_sin__zero__iff__int,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
= ( ? [I4: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
& ( X
= ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_7526_cos__zero__iff__int,axiom,
! [X: real] :
( ( ( cos_real @ X )
= zero_zero_real )
= ( ? [I4: int] :
( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
& ( X
= ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_7527_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( sin_real @ X )
= zero_zero_real )
=> ? [N3: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_7528_sin__zero__iff,axiom,
! [X: real] :
( ( ( sin_real @ X )
= zero_zero_real )
= ( ? [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
| ? [N2: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( X
= ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_7529_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ( cos_real @ X )
= zero_zero_real )
=> ? [N3: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_7530_cos__zero__iff,axiom,
! [X: real] :
( ( ( cos_real @ X )
= zero_zero_real )
= ( ? [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( X
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
| ? [N2: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
& ( X
= ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_7531_mono__SucI1,axiom,
! [X7: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
=> ( topolo6980174941875973593q_real @ X7 ) ) ).
% mono_SucI1
thf(fact_7532_mono__SucI1,axiom,
! [X7: nat > set_nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
=> ( topolo7278393974255667507et_nat @ X7 ) ) ).
% mono_SucI1
thf(fact_7533_mono__SucI1,axiom,
! [X7: nat > rat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
=> ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% mono_SucI1
thf(fact_7534_mono__SucI1,axiom,
! [X7: nat > num] :
( ! [N3: nat] : ( ord_less_eq_num @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
=> ( topolo1459490580787246023eq_num @ X7 ) ) ).
% mono_SucI1
thf(fact_7535_mono__SucI1,axiom,
! [X7: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
=> ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% mono_SucI1
thf(fact_7536_mono__SucI1,axiom,
! [X7: nat > int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
=> ( topolo4899668324122417113eq_int @ X7 ) ) ).
% mono_SucI1
thf(fact_7537_mono__SucI2,axiom,
! [X7: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
=> ( topolo6980174941875973593q_real @ X7 ) ) ).
% mono_SucI2
thf(fact_7538_mono__SucI2,axiom,
! [X7: nat > set_nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
=> ( topolo7278393974255667507et_nat @ X7 ) ) ).
% mono_SucI2
thf(fact_7539_mono__SucI2,axiom,
! [X7: nat > rat] :
( ! [N3: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
=> ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% mono_SucI2
thf(fact_7540_mono__SucI2,axiom,
! [X7: nat > num] :
( ! [N3: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
=> ( topolo1459490580787246023eq_num @ X7 ) ) ).
% mono_SucI2
thf(fact_7541_mono__SucI2,axiom,
! [X7: nat > nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
=> ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% mono_SucI2
thf(fact_7542_mono__SucI2,axiom,
! [X7: nat > int] :
( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
=> ( topolo4899668324122417113eq_int @ X7 ) ) ).
% mono_SucI2
thf(fact_7543_monoseq__Suc,axiom,
( topolo6980174941875973593q_real
= ( ^ [X6: nat > real] :
( ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7544_monoseq__Suc,axiom,
( topolo7278393974255667507et_nat
= ( ^ [X6: nat > set_nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_set_nat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7545_monoseq__Suc,axiom,
( topolo4267028734544971653eq_rat
= ( ^ [X6: nat > rat] :
( ! [N2: nat] : ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7546_monoseq__Suc,axiom,
( topolo1459490580787246023eq_num
= ( ^ [X6: nat > num] :
( ! [N2: nat] : ( ord_less_eq_num @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7547_monoseq__Suc,axiom,
( topolo4902158794631467389eq_nat
= ( ^ [X6: nat > nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7548_monoseq__Suc,axiom,
( topolo4899668324122417113eq_int
= ( ^ [X6: nat > int] :
( ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% monoseq_Suc
thf(fact_7549_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ T3 )
& ( ord_less_real @ T3 @ X )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_7550_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ? [T3: real] :
( ( ord_less_real @ X @ T3 )
& ( ord_less_real @ T3 @ zero_zero_real )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_7551_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( cos_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_7552_sin__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( sin_real @ X ) ) ).
% sin_paired
thf(fact_7553_tan__double,axiom,
! [X: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
= ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_7554_tan__double,axiom,
! [X: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
!= zero_zero_real )
=> ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
= ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_7555_complex__unimodular__polar,axiom,
! [Z2: complex] :
( ( ( real_V1022390504157884413omplex @ Z2 )
= one_one_real )
=> ~ ! [T3: real] :
( ( ord_less_eq_real @ zero_zero_real @ T3 )
=> ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
=> ( Z2
!= ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_7556_tan__periodic__pi,axiom,
! [X: real] :
( ( tan_real @ ( plus_plus_real @ X @ pi ) )
= ( tan_real @ X ) ) ).
% tan_periodic_pi
thf(fact_7557_fact__0,axiom,
( ( semiri5044797733671781792omplex @ zero_zero_nat )
= one_one_complex ) ).
% fact_0
thf(fact_7558_fact__0,axiom,
( ( semiri773545260158071498ct_rat @ zero_zero_nat )
= one_one_rat ) ).
% fact_0
thf(fact_7559_fact__0,axiom,
( ( semiri1406184849735516958ct_int @ zero_zero_nat )
= one_one_int ) ).
% fact_0
thf(fact_7560_fact__0,axiom,
( ( semiri2265585572941072030t_real @ zero_zero_nat )
= one_one_real ) ).
% fact_0
thf(fact_7561_fact__0,axiom,
( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
= one_one_nat ) ).
% fact_0
thf(fact_7562_fact__1,axiom,
( ( semiri5044797733671781792omplex @ one_one_nat )
= one_one_complex ) ).
% fact_1
thf(fact_7563_fact__1,axiom,
( ( semiri773545260158071498ct_rat @ one_one_nat )
= one_one_rat ) ).
% fact_1
thf(fact_7564_fact__1,axiom,
( ( semiri1406184849735516958ct_int @ one_one_nat )
= one_one_int ) ).
% fact_1
thf(fact_7565_fact__1,axiom,
( ( semiri2265585572941072030t_real @ one_one_nat )
= one_one_real ) ).
% fact_1
thf(fact_7566_fact__1,axiom,
( ( semiri1408675320244567234ct_nat @ one_one_nat )
= one_one_nat ) ).
% fact_1
thf(fact_7567_fact__Suc__0,axiom,
( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
= one_one_complex ) ).
% fact_Suc_0
thf(fact_7568_fact__Suc__0,axiom,
( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
= one_one_rat ) ).
% fact_Suc_0
thf(fact_7569_fact__Suc__0,axiom,
( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
= one_one_int ) ).
% fact_Suc_0
thf(fact_7570_fact__Suc__0,axiom,
( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
= one_one_real ) ).
% fact_Suc_0
thf(fact_7571_fact__Suc__0,axiom,
( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% fact_Suc_0
thf(fact_7572_fact__Suc,axiom,
! [N: nat] :
( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_Suc
thf(fact_7573_fact__Suc,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_Suc
thf(fact_7574_fact__Suc,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ ( suc @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_Suc
thf(fact_7575_fact__Suc,axiom,
! [N: nat] :
( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_Suc
thf(fact_7576_tan__npi,axiom,
! [N: nat] :
( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= zero_zero_real ) ).
% tan_npi
thf(fact_7577_tan__periodic__n,axiom,
! [X: real,N: num] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic_n
thf(fact_7578_tan__periodic__nat,axiom,
! [X: real,N: nat] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic_nat
thf(fact_7579_tan__periodic__int,axiom,
! [X: real,I2: int] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic_int
thf(fact_7580_fact__2,axiom,
( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_7581_fact__2,axiom,
( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_7582_fact__2,axiom,
( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_7583_fact__2,axiom,
( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_7584_fact__2,axiom,
( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% fact_2
thf(fact_7585_norm__cos__sin,axiom,
! [T: real] :
( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
= one_one_real ) ).
% norm_cos_sin
thf(fact_7586_tan__periodic,axiom,
! [X: real] :
( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( tan_real @ X ) ) ).
% tan_periodic
thf(fact_7587_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_zero
thf(fact_7588_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_zero
thf(fact_7589_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_ge_zero
thf(fact_7590_fact__ge__zero,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_zero
thf(fact_7591_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_gt_zero
thf(fact_7592_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_gt_zero
thf(fact_7593_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_gt_zero
thf(fact_7594_fact__gt__zero,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_gt_zero
thf(fact_7595_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% fact_not_neg
thf(fact_7596_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% fact_not_neg
thf(fact_7597_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% fact_not_neg
thf(fact_7598_fact__not__neg,axiom,
! [N: nat] :
~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% fact_not_neg
thf(fact_7599_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% fact_ge_1
thf(fact_7600_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% fact_ge_1
thf(fact_7601_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% fact_ge_1
thf(fact_7602_fact__ge__1,axiom,
! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_1
thf(fact_7603_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_mono
thf(fact_7604_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% fact_mono
thf(fact_7605_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_mono
thf(fact_7606_fact__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_mono
thf(fact_7607_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% fact_dvd
thf(fact_7608_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% fact_dvd
thf(fact_7609_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% fact_dvd
thf(fact_7610_fact__dvd,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% fact_dvd
thf(fact_7611_complex__add,axiom,
! [A: real,B: real,C: real,D: real] :
( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
= ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% complex_add
thf(fact_7612_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7613_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7614_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7615_fact__less__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono
thf(fact_7616_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
= zero_zero_int ) ) ).
% fact_mod
thf(fact_7617_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
= zero_z3403309356797280102nteger ) ) ).
% fact_mod
thf(fact_7618_fact__mod,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
= zero_zero_nat ) ) ).
% fact_mod
thf(fact_7619_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7620_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7621_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7622_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7623_fact__fact__dvd__fact,axiom,
! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% fact_fact_dvd_fact
thf(fact_7624_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7625_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7626_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7627_fact__le__power,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% fact_le_power
thf(fact_7628_complex__mult,axiom,
! [A: real,B: real,C: real,D: real] :
( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
= ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% complex_mult
thf(fact_7629_one__complex_Ocode,axiom,
( one_one_complex
= ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% one_complex.code
thf(fact_7630_Complex__eq__1,axiom,
! [A: real,B: real] :
( ( ( complex2 @ A @ B )
= one_one_complex )
= ( ( A = one_one_real )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_1
thf(fact_7631_tan__def,axiom,
( tan_complex
= ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X2 ) @ ( cos_complex @ X2 ) ) ) ) ).
% tan_def
thf(fact_7632_tan__def,axiom,
( tan_real
= ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ X2 ) @ ( cos_real @ X2 ) ) ) ) ).
% tan_def
thf(fact_7633_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% choose_dvd
thf(fact_7634_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% choose_dvd
thf(fact_7635_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% choose_dvd
thf(fact_7636_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% choose_dvd
thf(fact_7637_choose__dvd,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% choose_dvd
thf(fact_7638_fact__numeral,axiom,
! [K: num] :
( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
= ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7639_fact__numeral,axiom,
! [K: num] :
( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
= ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7640_fact__numeral,axiom,
! [K: num] :
( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
= ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7641_fact__numeral,axiom,
! [K: num] :
( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
= ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7642_fact__numeral,axiom,
! [K: num] :
( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
= ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% fact_numeral
thf(fact_7643_Complex__eq__neg__1,axiom,
! [A: real,B: real] :
( ( ( complex2 @ A @ B )
= ( uminus1482373934393186551omplex @ one_one_complex ) )
= ( ( A
= ( uminus_uminus_real @ one_one_real ) )
& ( B = zero_zero_real ) ) ) ).
% Complex_eq_neg_1
thf(fact_7644_square__fact__le__2__fact,axiom,
! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% square_fact_le_2_fact
thf(fact_7645_tan__45,axiom,
( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
= one_one_real ) ).
% tan_45
thf(fact_7646_fact__num__eq__if,axiom,
( semiri5044797733671781792omplex
= ( ^ [M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7647_fact__num__eq__if,axiom,
( semiri1406184849735516958ct_int
= ( ^ [M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7648_fact__num__eq__if,axiom,
( semiri773545260158071498ct_rat
= ( ^ [M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7649_fact__num__eq__if,axiom,
( semiri2265585572941072030t_real
= ( ^ [M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7650_fact__num__eq__if,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_7651_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1406184849735516958ct_int @ N )
= ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7652_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri773545260158071498ct_rat @ N )
= ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7653_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri2265585572941072030t_real @ N )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7654_fact__reduce,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( semiri1408675320244567234ct_nat @ N )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% fact_reduce
thf(fact_7655_tan__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% tan_gt_zero
thf(fact_7656_lemma__tan__total,axiom,
! [Y4: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ord_less_real @ Y4 @ ( tan_real @ X3 ) ) ) ) ).
% lemma_tan_total
thf(fact_7657_tan__total,axiom,
! [Y4: real] :
? [X3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
& ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X3 )
= Y4 )
& ! [Y5: real] :
( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
& ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ Y5 )
= Y4 ) )
=> ( Y5 = X3 ) ) ) ).
% tan_total
thf(fact_7658_tan__monotone,axiom,
! [Y4: real,X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
=> ( ( ord_less_real @ Y4 @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( tan_real @ Y4 ) @ ( tan_real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_7659_tan__monotone_H,axiom,
! [Y4: real,X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
=> ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ Y4 @ X )
= ( ord_less_real @ ( tan_real @ Y4 ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_7660_tan__mono__lt__eq,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
=> ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) )
= ( ord_less_real @ X @ Y4 ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_7661_lemma__tan__total1,axiom,
! [Y4: real] :
? [X3: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
& ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X3 )
= Y4 ) ) ).
% lemma_tan_total1
thf(fact_7662_tan__minus__45,axiom,
( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% tan_minus_45
thf(fact_7663_tan__inverse,axiom,
! [Y4: real] :
( ( divide_divide_real @ one_one_real @ ( tan_real @ Y4 ) )
= ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y4 ) ) ) ).
% tan_inverse
thf(fact_7664_add__tan__eq,axiom,
! [X: complex,Y4: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y4 )
!= zero_zero_complex )
=> ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) )
= ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y4 ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y4 ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_7665_add__tan__eq,axiom,
! [X: real,Y4: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y4 )
!= zero_zero_real )
=> ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) )
= ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_7666_tan__total__pos,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ? [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X3 )
= Y4 ) ) ) ).
% tan_total_pos
thf(fact_7667_tan__pos__pi2__le,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_7668_tan__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% tan_less_zero
thf(fact_7669_tan__mono__le,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ Y4 )
=> ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) ) ) ) ).
% tan_mono_le
thf(fact_7670_tan__mono__le__eq,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
=> ( ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) )
= ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_7671_tan__bound__pi2,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% tan_bound_pi2
thf(fact_7672_arctan__unique,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ( tan_real @ X )
= Y4 )
=> ( ( arctan @ Y4 )
= X ) ) ) ) ).
% arctan_unique
thf(fact_7673_arctan__tan,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( arctan @ ( tan_real @ X ) )
= X ) ) ) ).
% arctan_tan
thf(fact_7674_arctan,axiom,
! [Y4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y4 ) )
& ( ord_less_real @ ( arctan @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ ( arctan @ Y4 ) )
= Y4 ) ) ).
% arctan
thf(fact_7675_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > complex > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% Maclaurin_zero
thf(fact_7676_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > real > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% Maclaurin_zero
thf(fact_7677_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > rat > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% Maclaurin_zero
thf(fact_7678_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > nat > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% Maclaurin_zero
thf(fact_7679_Maclaurin__zero,axiom,
! [X: real,N: nat,Diff: nat > int > real] :
( ( X = zero_zero_real )
=> ( ( N != zero_zero_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% Maclaurin_zero
thf(fact_7680_Maclaurin__lemma,axiom,
! [H2: real,F: real > real,J: nat > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ? [B7: real] :
( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ B7 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_7681_lemma__tan__add1,axiom,
! [X: complex,Y4: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y4 )
!= zero_zero_complex )
=> ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) )
= ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y4 ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y4 ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_7682_lemma__tan__add1,axiom,
! [X: real,Y4: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y4 )
!= zero_zero_real )
=> ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) )
= ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y4 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_7683_tan__diff,axiom,
! [X: complex,Y4: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y4 )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y4 ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( minus_minus_complex @ X @ Y4 ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_7684_tan__diff,axiom,
! [X: real,Y4: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y4 )
!= zero_zero_real )
=> ( ( ( cos_real @ ( minus_minus_real @ X @ Y4 ) )
!= zero_zero_real )
=> ( ( tan_real @ ( minus_minus_real @ X @ Y4 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_7685_tan__add,axiom,
! [X: complex,Y4: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( ( cos_complex @ Y4 )
!= zero_zero_complex )
=> ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y4 ) )
!= zero_zero_complex )
=> ( ( tan_complex @ ( plus_plus_complex @ X @ Y4 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y4 ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_7686_tan__add,axiom,
! [X: real,Y4: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( ( cos_real @ Y4 )
!= zero_zero_real )
=> ( ( ( cos_real @ ( plus_plus_real @ X @ Y4 ) )
!= zero_zero_real )
=> ( ( tan_real @ ( plus_plus_real @ X @ Y4 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y4 ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_7687_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ? [Z: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z )
& ( ord_less_real @ Z @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
& ( ( tan_real @ Z )
= X ) ) ) ).
% tan_total_pi4
thf(fact_7688_tan__half,axiom,
( tan_complex
= ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_complex ) ) ) ) ).
% tan_half
thf(fact_7689_tan__half,axiom,
( tan_real
= ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ) ).
% tan_half
thf(fact_7690_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).
% cos_coeff_def
thf(fact_7691_cos__paired,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( cos_real @ X ) ) ).
% cos_paired
thf(fact_7692_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ T3 )
& ( ord_less_real @ T3 @ X )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_7693_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ X )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_7694_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_7695_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( sin_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_7696_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_7697_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_self
thf(fact_7698_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_7699_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_7700_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% fact_ge_Suc_0_nat
thf(fact_7701_dvd__fact,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_7702_fact__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
= ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_7703_fact__div__fact__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq_nat @ R2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_7704_sin__coeff__Suc,axiom,
! [N: nat] :
( ( sin_coeff @ ( suc @ N ) )
= ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% sin_coeff_Suc
thf(fact_7705_cos__coeff__Suc,axiom,
! [N: nat] :
( ( cos_coeff @ ( suc @ N ) )
= ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% cos_coeff_Suc
thf(fact_7706_sin__tan,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( sin_real @ X )
= ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_7707_cos__tan,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( cos_real @ X )
= ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_7708_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X != zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
& ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( exp_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_7709_pochhammer__double,axiom,
! [Z2: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_7710_pochhammer__double,axiom,
! [Z2: real,N: nat] :
( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_7711_pochhammer__double,axiom,
! [Z2: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% pochhammer_double
thf(fact_7712_of__nat__code,axiom,
( semiri8010041392384452111omplex
= ( ^ [N2: nat] :
( semiri2816024913162550771omplex
@ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
@ N2
@ zero_zero_complex ) ) ) ).
% of_nat_code
thf(fact_7713_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] :
( semiri8420488043553186161ux_int
@ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
@ N2
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_7714_of__nat__code,axiom,
( semiri5074537144036343181t_real
= ( ^ [N2: nat] :
( semiri7260567687927622513x_real
@ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
@ N2
@ zero_zero_real ) ) ) ).
% of_nat_code
thf(fact_7715_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
@ N2
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_7716_of__nat__code,axiom,
( semiri681578069525770553at_rat
= ( ^ [N2: nat] :
( semiri7787848453975740701ux_rat
@ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
@ N2
@ zero_zero_rat ) ) ) ).
% of_nat_code
thf(fact_7717_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% real_sqrt_eq_1_iff
thf(fact_7718_real__sqrt__one,axiom,
( ( sqrt @ one_one_real )
= one_one_real ) ).
% real_sqrt_one
thf(fact_7719_exp__zero,axiom,
( ( exp_complex @ zero_zero_complex )
= one_one_complex ) ).
% exp_zero
thf(fact_7720_exp__zero,axiom,
( ( exp_real @ zero_zero_real )
= one_one_real ) ).
% exp_zero
thf(fact_7721_real__sqrt__gt__1__iff,axiom,
! [Y4: real] :
( ( ord_less_real @ one_one_real @ ( sqrt @ Y4 ) )
= ( ord_less_real @ one_one_real @ Y4 ) ) ).
% real_sqrt_gt_1_iff
thf(fact_7722_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ).
% real_sqrt_lt_1_iff
thf(fact_7723_real__sqrt__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ).
% real_sqrt_le_1_iff
thf(fact_7724_real__sqrt__ge__1__iff,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y4 ) )
= ( ord_less_eq_real @ one_one_real @ Y4 ) ) ).
% real_sqrt_ge_1_iff
thf(fact_7725_pochhammer__0,axiom,
! [A: complex] :
( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
= one_one_complex ) ).
% pochhammer_0
thf(fact_7726_pochhammer__0,axiom,
! [A: real] :
( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
= one_one_real ) ).
% pochhammer_0
thf(fact_7727_pochhammer__0,axiom,
! [A: rat] :
( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% pochhammer_0
thf(fact_7728_pochhammer__0,axiom,
! [A: nat] :
( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% pochhammer_0
thf(fact_7729_pochhammer__0,axiom,
! [A: int] :
( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
= one_one_int ) ).
% pochhammer_0
thf(fact_7730_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp_real @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ).
% exp_eq_one_iff
thf(fact_7731_real__sqrt__mult__self,axiom,
! [A: real] :
( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
= ( abs_abs_real @ A ) ) ).
% real_sqrt_mult_self
thf(fact_7732_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times_real @ X @ X ) )
= ( abs_abs_real @ X ) ) ).
% real_sqrt_abs2
thf(fact_7733_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% real_sqrt_four
thf(fact_7734_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_less_exp_iff
thf(fact_7735_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ).
% exp_less_one_iff
thf(fact_7736_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% one_le_exp_iff
thf(fact_7737_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% exp_le_one_iff
thf(fact_7738_real__sqrt__abs,axiom,
! [X: real] :
( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( abs_abs_real @ X ) ) ).
% real_sqrt_abs
thf(fact_7739_real__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X ) ) ).
% real_sqrt_pow2
thf(fact_7740_real__sqrt__pow2__iff,axiom,
! [X: real] :
( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% real_sqrt_pow2_iff
thf(fact_7741_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X: real,Y4: real,Xa2: real,Ya: real] :
( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_squared_eq
thf(fact_7742_real__sqrt__mult,axiom,
! [X: real,Y4: real] :
( ( sqrt @ ( times_times_real @ X @ Y4 ) )
= ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ).
% real_sqrt_mult
thf(fact_7743_exp__times__arg__commute,axiom,
! [A2: complex] :
( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
= ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).
% exp_times_arg_commute
thf(fact_7744_exp__times__arg__commute,axiom,
! [A2: real] :
( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
= ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).
% exp_times_arg_commute
thf(fact_7745_real__sqrt__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_one
thf(fact_7746_mult__exp__exp,axiom,
! [X: complex,Y4: complex] :
( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y4 ) )
= ( exp_complex @ ( plus_plus_complex @ X @ Y4 ) ) ) ).
% mult_exp_exp
thf(fact_7747_mult__exp__exp,axiom,
! [X: real,Y4: real] :
( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) )
= ( exp_real @ ( plus_plus_real @ X @ Y4 ) ) ) ).
% mult_exp_exp
thf(fact_7748_exp__add__commuting,axiom,
! [X: complex,Y4: complex] :
( ( ( times_times_complex @ X @ Y4 )
= ( times_times_complex @ Y4 @ X ) )
=> ( ( exp_complex @ ( plus_plus_complex @ X @ Y4 ) )
= ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y4 ) ) ) ) ).
% exp_add_commuting
thf(fact_7749_exp__add__commuting,axiom,
! [X: real,Y4: real] :
( ( ( times_times_real @ X @ Y4 )
= ( times_times_real @ Y4 @ X ) )
=> ( ( exp_real @ ( plus_plus_real @ X @ Y4 ) )
= ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) ) ) ) ).
% exp_add_commuting
thf(fact_7750_exp__diff,axiom,
! [X: complex,Y4: complex] :
( ( exp_complex @ ( minus_minus_complex @ X @ Y4 ) )
= ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y4 ) ) ) ).
% exp_diff
thf(fact_7751_exp__diff,axiom,
! [X: real,Y4: real] :
( ( exp_real @ ( minus_minus_real @ X @ Y4 ) )
= ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y4 ) ) ) ).
% exp_diff
thf(fact_7752_pochhammer__pos,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_7753_pochhammer__pos,axiom,
! [X: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_7754_pochhammer__pos,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_7755_pochhammer__pos,axiom,
! [X: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_pos
thf(fact_7756_pochhammer__eq__0__mono,axiom,
! [A: complex,N: nat,M: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s2602460028002588243omplex @ A @ M )
= zero_zero_complex ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_7757_pochhammer__eq__0__mono,axiom,
! [A: real,N: nat,M: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s7457072308508201937r_real @ A @ M )
= zero_zero_real ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_7758_pochhammer__eq__0__mono,axiom,
! [A: rat,N: nat,M: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s4028243227959126397er_rat @ A @ M )
= zero_zero_rat ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_7759_pochhammer__neq__0__mono,axiom,
! [A: complex,M: nat,N: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ M )
!= zero_zero_complex )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s2602460028002588243omplex @ A @ N )
!= zero_zero_complex ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_7760_pochhammer__neq__0__mono,axiom,
! [A: real,M: nat,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ M )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s7457072308508201937r_real @ A @ N )
!= zero_zero_real ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_7761_pochhammer__neq__0__mono,axiom,
! [A: rat,M: nat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ M )
!= zero_zero_rat )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ( comm_s4028243227959126397er_rat @ A @ N )
!= zero_zero_rat ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_7762_pochhammer__fact,axiom,
( semiri5044797733671781792omplex
= ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% pochhammer_fact
thf(fact_7763_pochhammer__fact,axiom,
( semiri773545260158071498ct_rat
= ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% pochhammer_fact
thf(fact_7764_pochhammer__fact,axiom,
( semiri1406184849735516958ct_int
= ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% pochhammer_fact
thf(fact_7765_pochhammer__fact,axiom,
( semiri2265585572941072030t_real
= ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% pochhammer_fact
thf(fact_7766_pochhammer__fact,axiom,
( semiri1408675320244567234ct_nat
= ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% pochhammer_fact
thf(fact_7767_exp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% exp_gt_one
thf(fact_7768_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y4 ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y4 ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_7769_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% exp_ge_add_one_self
thf(fact_7770_le__real__sqrt__sumsq,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y4 @ Y4 ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_7771_exp__minus__inverse,axiom,
! [X: real] :
( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
= one_one_real ) ).
% exp_minus_inverse
thf(fact_7772_exp__minus__inverse,axiom,
! [X: complex] :
( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
= one_one_complex ) ).
% exp_minus_inverse
thf(fact_7773_exp__of__nat__mult,axiom,
! [N: nat,X: complex] :
( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
= ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% exp_of_nat_mult
thf(fact_7774_exp__of__nat__mult,axiom,
! [N: nat,X: real] :
( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
= ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% exp_of_nat_mult
thf(fact_7775_exp__of__nat2__mult,axiom,
! [X: complex,N: nat] :
( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
= ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).
% exp_of_nat2_mult
thf(fact_7776_exp__of__nat2__mult,axiom,
! [X: real,N: nat] :
( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
= ( power_power_real @ ( exp_real @ X ) @ N ) ) ).
% exp_of_nat2_mult
thf(fact_7777_pochhammer__nonneg,axiom,
! [X: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_7778_pochhammer__nonneg,axiom,
! [X: rat,N: nat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_7779_pochhammer__nonneg,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_7780_pochhammer__nonneg,axiom,
! [X: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% pochhammer_nonneg
thf(fact_7781_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
= one_one_complex ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
= zero_zero_complex ) ) ) ).
% pochhammer_0_left
thf(fact_7782_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= one_one_real ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
= zero_zero_real ) ) ) ).
% pochhammer_0_left
thf(fact_7783_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= one_one_rat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
= zero_zero_rat ) ) ) ).
% pochhammer_0_left
thf(fact_7784_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% pochhammer_0_left
thf(fact_7785_pochhammer__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% pochhammer_0_left
thf(fact_7786_sqrt2__less__2,axiom,
ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% sqrt2_less_2
thf(fact_7787_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_7788_lemma__exp__total,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ one_one_real @ Y4 )
=> ? [X3: real] :
( ( ord_less_eq_real @ zero_zero_real @ X3 )
& ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y4 @ one_one_real ) )
& ( ( exp_real @ X3 )
= Y4 ) ) ) ).
% lemma_exp_total
thf(fact_7789_ln__x__over__x__mono,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y4 ) @ Y4 ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_7790_pochhammer__rec,axiom,
! [A: complex,N: nat] :
( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
= ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7791_pochhammer__rec,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7792_pochhammer__rec,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7793_pochhammer__rec,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7794_pochhammer__rec,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% pochhammer_rec
thf(fact_7795_pochhammer__Suc,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7796_pochhammer__Suc,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7797_pochhammer__Suc,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7798_pochhammer__Suc,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% pochhammer_Suc
thf(fact_7799_pochhammer__rec_H,axiom,
! [Z2: int,N: nat] :
( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N ) )
= ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7800_pochhammer__rec_H,axiom,
! [Z2: real,N: nat] :
( ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) )
= ( times_times_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7801_pochhammer__rec_H,axiom,
! [Z2: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N ) )
= ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7802_pochhammer__rec_H,axiom,
! [Z2: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) )
= ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) ) ).
% pochhammer_rec'
thf(fact_7803_pochhammer__eq__0__iff,axiom,
! [A: complex,N: nat] :
( ( ( comm_s2602460028002588243omplex @ A @ N )
= zero_zero_complex )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_7804_pochhammer__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( comm_s7457072308508201937r_real @ A @ N )
= zero_zero_real )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_7805_pochhammer__eq__0__iff,axiom,
! [A: rat,N: nat] :
( ( ( comm_s4028243227959126397er_rat @ A @ N )
= zero_zero_rat )
= ( ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ( A
= ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_7806_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
= zero_zero_complex )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7807_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
= zero_z3403309356797280102nteger )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7808_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
= zero_zero_int )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7809_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
= zero_zero_real )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7810_pochhammer__of__nat__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
= zero_zero_rat )
= ( ord_less_nat @ N @ K ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_7811_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7812_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7813_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7814_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7815_pochhammer__of__nat__eq__0__lemma,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_7816_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
!= zero_zero_complex ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7817_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
!= zero_z3403309356797280102nteger ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7818_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
!= zero_zero_int ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7819_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
!= zero_zero_real ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7820_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
!= zero_zero_rat ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_7821_pochhammer__product_H,axiom,
! [Z2: int,N: nat,M: nat] :
( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N @ M ) )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_7822_pochhammer__product_H,axiom,
! [Z2: real,N: nat,M: nat] :
( ( comm_s7457072308508201937r_real @ Z2 @ ( plus_plus_nat @ N @ M ) )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_7823_pochhammer__product_H,axiom,
! [Z2: nat,N: nat,M: nat] :
( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N @ M ) )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_7824_pochhammer__product_H,axiom,
! [Z2: rat,N: nat,M: nat] :
( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N @ M ) )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% pochhammer_product'
thf(fact_7825_real__less__rsqrt,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 )
=> ( ord_less_real @ X @ ( sqrt @ Y4 ) ) ) ).
% real_less_rsqrt
thf(fact_7826_sqrt__le__D,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( sqrt @ X ) @ Y4 )
=> ( ord_less_eq_real @ X @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% sqrt_le_D
thf(fact_7827_real__le__rsqrt,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 )
=> ( ord_less_eq_real @ X @ ( sqrt @ Y4 ) ) ) ).
% real_le_rsqrt
thf(fact_7828_exp__le,axiom,
ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% exp_le
thf(fact_7829_exp__divide__power__eq,axiom,
! [N: nat,X: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
= ( exp_complex @ X ) ) ) ).
% exp_divide_power_eq
thf(fact_7830_exp__divide__power__eq,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
= ( exp_real @ X ) ) ) ).
% exp_divide_power_eq
thf(fact_7831_tanh__altdef,axiom,
( tanh_real
= ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_7832_tanh__altdef,axiom,
( tanh_complex
= ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_7833_pochhammer__product,axiom,
! [M: nat,N: nat,Z2: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4660882817536571857er_int @ Z2 @ N )
= ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_7834_pochhammer__product,axiom,
! [M: nat,N: nat,Z2: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s7457072308508201937r_real @ Z2 @ N )
= ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_7835_pochhammer__product,axiom,
! [M: nat,N: nat,Z2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4663373288045622133er_nat @ Z2 @ N )
= ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_7836_pochhammer__product,axiom,
! [M: nat,N: nat,Z2: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( comm_s4028243227959126397er_rat @ Z2 @ N )
= ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% pochhammer_product
thf(fact_7837_real__le__lsqrt,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ X @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( sqrt @ X ) @ Y4 ) ) ) ) ).
% real_le_lsqrt
thf(fact_7838_real__sqrt__unique,axiom,
! [Y4: real,X: real] :
( ( ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( sqrt @ X )
= Y4 ) ) ) ).
% real_sqrt_unique
thf(fact_7839_lemma__real__divide__sqrt__less,axiom,
! [U2: real] :
( ( ord_less_real @ zero_zero_real @ U2 )
=> ( ord_less_real @ ( divide_divide_real @ U2 @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U2 ) ) ).
% lemma_real_divide_sqrt_less
thf(fact_7840_real__sqrt__sum__squares__eq__cancel,axiom,
! [X: real,Y4: real] :
( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= X )
=> ( Y4 = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_7841_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X: real,Y4: real] :
( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= Y4 )
=> ( X = zero_zero_real ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_7842_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_7843_real__sqrt__sum__squares__ge2,axiom,
! [Y4: real,X: real] : ( ord_less_eq_real @ Y4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_7844_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_7845_exp__half__le2,axiom,
ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% exp_half_le2
thf(fact_7846_sqrt__ge__absD,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y4 ) )
=> ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y4 ) ) ).
% sqrt_ge_absD
thf(fact_7847_cos__45,axiom,
( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
= ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_45
thf(fact_7848_sin__45,axiom,
( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
= ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_45
thf(fact_7849_tan__60,axiom,
( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
= ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% tan_60
thf(fact_7850_exp__double,axiom,
! [Z2: complex] :
( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) )
= ( power_power_complex @ ( exp_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% exp_double
thf(fact_7851_exp__double,axiom,
! [Z2: real] :
( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) )
= ( power_power_real @ ( exp_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% exp_double
thf(fact_7852_pochhammer__absorb__comp,axiom,
! [R2: complex,K: nat] :
( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
= ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7853_pochhammer__absorb__comp,axiom,
! [R2: code_integer,K: nat] :
( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
= ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7854_pochhammer__absorb__comp,axiom,
! [R2: int,K: nat] :
( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
= ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7855_pochhammer__absorb__comp,axiom,
! [R2: real,K: nat] :
( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
= ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7856_pochhammer__absorb__comp,axiom,
! [R2: rat,K: nat] :
( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
= ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% pochhammer_absorb_comp
thf(fact_7857_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% pochhammer_same
thf(fact_7858_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% pochhammer_same
thf(fact_7859_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% pochhammer_same
thf(fact_7860_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% pochhammer_same
thf(fact_7861_pochhammer__same,axiom,
! [N: nat] :
( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% pochhammer_same
thf(fact_7862_real__less__lsqrt,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_real @ X @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ ( sqrt @ X ) @ Y4 ) ) ) ) ).
% real_less_lsqrt
thf(fact_7863_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y4 ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_7864_sqrt__even__pow2,axiom,
! [N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% sqrt_even_pow2
thf(fact_7865_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y4 ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_7866_real__sqrt__ge__abs2,axiom,
! [Y4: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_7867_real__sqrt__ge__abs1,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_7868_ln__sqrt,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( sqrt @ X ) )
= ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% ln_sqrt
thf(fact_7869_cos__30,axiom,
( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
= ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% cos_30
thf(fact_7870_sin__60,axiom,
( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
= ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% sin_60
thf(fact_7871_complex__norm,axiom,
! [X: real,Y4: real] :
( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y4 ) )
= ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_norm
thf(fact_7872_exp__bound__half,axiom,
! [Z2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% exp_bound_half
thf(fact_7873_exp__bound__half,axiom,
! [Z2: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% exp_bound_half
thf(fact_7874_pochhammer__minus_H,axiom,
! [B: complex,K: nat] :
( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7875_pochhammer__minus_H,axiom,
! [B: code_integer,K: nat] :
( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7876_pochhammer__minus_H,axiom,
! [B: int,K: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7877_pochhammer__minus_H,axiom,
! [B: real,K: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7878_pochhammer__minus_H,axiom,
! [B: rat,K: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% pochhammer_minus'
thf(fact_7879_pochhammer__minus,axiom,
! [B: complex,K: nat] :
( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7880_pochhammer__minus,axiom,
! [B: code_integer,K: nat] :
( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
= ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7881_pochhammer__minus,axiom,
! [B: int,K: nat] :
( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
= ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7882_pochhammer__minus,axiom,
! [B: real,K: nat] :
( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7883_pochhammer__minus,axiom,
! [B: rat,K: nat] :
( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% pochhammer_minus
thf(fact_7884_arsinh__real__aux,axiom,
! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% arsinh_real_aux
thf(fact_7885_exp__bound,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_7886_real__sqrt__power__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( sqrt @ X ) @ N )
= ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_7887_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y4: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_7888_arith__geo__mean__sqrt,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y4 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_7889_tan__30,axiom,
( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
= ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% tan_30
thf(fact_7890_real__exp__bound__lemma,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_7891_cos__x__y__le__one,axiom,
! [X: real,Y4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% cos_x_y_le_one
thf(fact_7892_real__sqrt__sum__squares__less,axiom,
! [X: real,U2: real,Y4: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U2 @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ord_less_real @ ( abs_abs_real @ Y4 ) @ ( divide_divide_real @ U2 @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U2 ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_7893_arcosh__real__def,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( arcosh_real @ X )
= ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_7894_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_7895_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_7896_cos__arctan,axiom,
! [X: real] :
( ( cos_real @ ( arctan @ X ) )
= ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_7897_sin__arctan,axiom,
! [X: real] :
( ( sin_real @ ( arctan @ X ) )
= ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_7898_exp__bound__lemma,axiom,
! [Z2: real] :
( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_7899_exp__bound__lemma,axiom,
! [Z2: complex] :
( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_7900_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( exp_real @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_7901_sqrt__sum__squares__half__less,axiom,
! [X: real,U2: real,Y4: real] :
( ( ord_less_real @ X @ ( divide_divide_real @ U2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_real @ Y4 @ ( divide_divide_real @ U2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U2 ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_7902_exp__lower__Taylor__quadratic,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_7903_sin__cos__sqrt,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
=> ( ( sin_real @ X )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_7904_arctan__half,axiom,
( arctan
= ( ^ [X2: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X2 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_7905_tanh__real__altdef,axiom,
( tanh_real
= ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_7906_fact__double,axiom,
! [N: nat] :
( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% fact_double
thf(fact_7907_fact__double,axiom,
! [N: nat] :
( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% fact_double
thf(fact_7908_fact__double,axiom,
! [N: nat] :
( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
= ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% fact_double
thf(fact_7909_pochhammer__times__pochhammer__half,axiom,
! [Z2: complex,N: nat] :
( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_7910_pochhammer__times__pochhammer__half,axiom,
! [Z2: real,N: nat] :
( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups129246275422532515t_real
@ ^ [K3: nat] : ( plus_plus_real @ Z2 @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_7911_pochhammer__times__pochhammer__half,axiom,
! [Z2: rat,N: nat] :
( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
= ( groups73079841787564623at_rat
@ ^ [K3: nat] : ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_7912_pochhammer__code,axiom,
( comm_s2602460028002588243omplex
= ( ^ [A4: complex,N2: nat] :
( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
@ ( set_fo1517530859248394432omplex
@ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A4 @ ( semiri8010041392384452111omplex @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_complex ) ) ) ) ).
% pochhammer_code
thf(fact_7913_pochhammer__code,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A4: int,N2: nat] :
( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
@ ( set_fo2581907887559384638at_int
@ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_int ) ) ) ) ).
% pochhammer_code
thf(fact_7914_pochhammer__code,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A4: real,N2: nat] :
( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
@ ( set_fo3111899725591712190t_real
@ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_real ) ) ) ) ).
% pochhammer_code
thf(fact_7915_pochhammer__code,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A4: rat,N2: nat] :
( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_rat ) ) ) ) ).
% pochhammer_code
thf(fact_7916_pochhammer__code,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A4: nat,N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
@ ( set_fo2584398358068434914at_nat
@ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ O ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ N2 @ one_one_nat )
@ one_one_nat ) ) ) ) ).
% pochhammer_code
thf(fact_7917_arsinh__real__def,axiom,
( arsinh_real
= ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_7918_cos__arcsin,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( cos_real @ ( arcsin @ X ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_7919_sin__arccos__abs,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
=> ( ( sin_real @ ( arccos @ Y4 ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_7920_prod_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups708209901874060359at_nat
@ ^ [Uu2: nat] : one_one_nat
@ A2 )
= one_one_nat ) ).
% prod.neutral_const
thf(fact_7921_prod_Oneutral__const,axiom,
! [A2: set_nat] :
( ( groups705719431365010083at_int
@ ^ [Uu2: nat] : one_one_int
@ A2 )
= one_one_int ) ).
% prod.neutral_const
thf(fact_7922_prod_Oneutral__const,axiom,
! [A2: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [Uu2: int] : one_one_int
@ A2 )
= one_one_int ) ).
% prod.neutral_const
thf(fact_7923_prod_Oempty,axiom,
! [G: nat > complex] :
( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
= one_one_complex ) ).
% prod.empty
thf(fact_7924_prod_Oempty,axiom,
! [G: nat > real] :
( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
= one_one_real ) ).
% prod.empty
thf(fact_7925_prod_Oempty,axiom,
! [G: nat > rat] :
( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
= one_one_rat ) ).
% prod.empty
thf(fact_7926_prod_Oempty,axiom,
! [G: int > complex] :
( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
= one_one_complex ) ).
% prod.empty
thf(fact_7927_prod_Oempty,axiom,
! [G: int > real] :
( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
= one_one_real ) ).
% prod.empty
thf(fact_7928_prod_Oempty,axiom,
! [G: int > rat] :
( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
= one_one_rat ) ).
% prod.empty
thf(fact_7929_prod_Oempty,axiom,
! [G: int > nat] :
( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
= one_one_nat ) ).
% prod.empty
thf(fact_7930_prod_Oempty,axiom,
! [G: real > complex] :
( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
= one_one_complex ) ).
% prod.empty
thf(fact_7931_prod_Oempty,axiom,
! [G: real > real] :
( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
= one_one_real ) ).
% prod.empty
thf(fact_7932_prod_Oempty,axiom,
! [G: real > rat] :
( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
= one_one_rat ) ).
% prod.empty
thf(fact_7933_prod_Oinfinite,axiom,
! [A2: set_nat,G: nat > complex] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups6464643781859351333omplex @ G @ A2 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_7934_prod_Oinfinite,axiom,
! [A2: set_complex,G: complex > complex] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ( ( groups3708469109370488835omplex @ G @ A2 )
= one_one_complex ) ) ).
% prod.infinite
thf(fact_7935_prod_Oinfinite,axiom,
! [A2: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups129246275422532515t_real @ G @ A2 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_7936_prod_Oinfinite,axiom,
! [A2: set_complex,G: complex > real] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ( ( groups766887009212190081x_real @ G @ A2 )
= one_one_real ) ) ).
% prod.infinite
thf(fact_7937_prod_Oinfinite,axiom,
! [A2: set_nat,G: nat > rat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat @ G @ A2 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_7938_prod_Oinfinite,axiom,
! [A2: set_complex,G: complex > rat] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ( ( groups225925009352817453ex_rat @ G @ A2 )
= one_one_rat ) ) ).
% prod.infinite
thf(fact_7939_prod_Oinfinite,axiom,
! [A2: set_complex,G: complex > nat] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ( ( groups861055069439313189ex_nat @ G @ A2 )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_7940_prod_Oinfinite,axiom,
! [A2: set_complex,G: complex > int] :
( ~ ( finite3207457112153483333omplex @ A2 )
=> ( ( groups858564598930262913ex_int @ G @ A2 )
= one_one_int ) ) ).
% prod.infinite
thf(fact_7941_prod_Oinfinite,axiom,
! [A2: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups708209901874060359at_nat @ G @ A2 )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_7942_prod_Oinfinite,axiom,
! [A2: set_nat,G: nat > int] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups705719431365010083at_int @ G @ A2 )
= one_one_int ) ) ).
% prod.infinite
thf(fact_7943_arccos__1,axiom,
( ( arccos @ one_one_real )
= zero_zero_real ) ).
% arccos_1
thf(fact_7944_prod_Odelta_H,axiom,
! [S3: set_int,A: int,B: int > complex] :
( ( finite_finite_int @ S3 )
=> ( ( ( member_int @ A @ S3 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S3 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7945_prod_Odelta_H,axiom,
! [S3: set_real,A: real,B: real > complex] :
( ( finite_finite_real @ S3 )
=> ( ( ( member_real @ A @ S3 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S3 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7946_prod_Odelta_H,axiom,
! [S3: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S3 )
=> ( ( ( member_nat @ A @ S3 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S3 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7947_prod_Odelta_H,axiom,
! [S3: set_complex,A: complex,B: complex > complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( ( member_complex @ A @ S3 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S3 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= one_one_complex ) ) ) ) ).
% prod.delta'
thf(fact_7948_prod_Odelta_H,axiom,
! [S3: set_int,A: int,B: int > real] :
( ( finite_finite_int @ S3 )
=> ( ( ( member_int @ A @ S3 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S3 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7949_prod_Odelta_H,axiom,
! [S3: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S3 )
=> ( ( ( member_real @ A @ S3 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S3 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7950_prod_Odelta_H,axiom,
! [S3: set_nat,A: nat,B: nat > real] :
( ( finite_finite_nat @ S3 )
=> ( ( ( member_nat @ A @ S3 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S3 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7951_prod_Odelta_H,axiom,
! [S3: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( ( member_complex @ A @ S3 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S3 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= one_one_real ) ) ) ) ).
% prod.delta'
thf(fact_7952_prod_Odelta_H,axiom,
! [S3: set_int,A: int,B: int > rat] :
( ( finite_finite_int @ S3 )
=> ( ( ( member_int @ A @ S3 )
=> ( ( groups1072433553688619179nt_rat
@ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S3 )
=> ( ( groups1072433553688619179nt_rat
@ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
@ S3 )
= one_one_rat ) ) ) ) ).
% prod.delta'
thf(fact_7953_prod_Odelta_H,axiom,
! [S3: set_real,A: real,B: real > rat] :
( ( finite_finite_real @ S3 )
=> ( ( ( member_real @ A @ S3 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S3 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
@ S3 )
= one_one_rat ) ) ) ) ).
% prod.delta'
thf(fact_7954_prod_Odelta,axiom,
! [S3: set_int,A: int,B: int > complex] :
( ( finite_finite_int @ S3 )
=> ( ( ( member_int @ A @ S3 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S3 )
=> ( ( groups7440179247065528705omplex
@ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7955_prod_Odelta,axiom,
! [S3: set_real,A: real,B: real > complex] :
( ( finite_finite_real @ S3 )
=> ( ( ( member_real @ A @ S3 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S3 )
=> ( ( groups713298508707869441omplex
@ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7956_prod_Odelta,axiom,
! [S3: set_nat,A: nat,B: nat > complex] :
( ( finite_finite_nat @ S3 )
=> ( ( ( member_nat @ A @ S3 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S3 )
=> ( ( groups6464643781859351333omplex
@ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7957_prod_Odelta,axiom,
! [S3: set_complex,A: complex,B: complex > complex] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( ( member_complex @ A @ S3 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S3 )
=> ( ( groups3708469109370488835omplex
@ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
@ S3 )
= one_one_complex ) ) ) ) ).
% prod.delta
thf(fact_7958_prod_Odelta,axiom,
! [S3: set_int,A: int,B: int > real] :
( ( finite_finite_int @ S3 )
=> ( ( ( member_int @ A @ S3 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S3 )
=> ( ( groups2316167850115554303t_real
@ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7959_prod_Odelta,axiom,
! [S3: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S3 )
=> ( ( ( member_real @ A @ S3 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S3 )
=> ( ( groups1681761925125756287l_real
@ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7960_prod_Odelta,axiom,
! [S3: set_nat,A: nat,B: nat > real] :
( ( finite_finite_nat @ S3 )
=> ( ( ( member_nat @ A @ S3 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S3 )
=> ( ( groups129246275422532515t_real
@ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7961_prod_Odelta,axiom,
! [S3: set_complex,A: complex,B: complex > real] :
( ( finite3207457112153483333omplex @ S3 )
=> ( ( ( member_complex @ A @ S3 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_complex @ A @ S3 )
=> ( ( groups766887009212190081x_real
@ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
@ S3 )
= one_one_real ) ) ) ) ).
% prod.delta
thf(fact_7962_prod_Odelta,axiom,
! [S3: set_int,A: int,B: int > rat] :
( ( finite_finite_int @ S3 )
=> ( ( ( member_int @ A @ S3 )
=> ( ( groups1072433553688619179nt_rat
@ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_int @ A @ S3 )
=> ( ( groups1072433553688619179nt_rat
@ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
@ S3 )
= one_one_rat ) ) ) ) ).
% prod.delta
thf(fact_7963_prod_Odelta,axiom,
! [S3: set_real,A: real,B: real > rat] :
( ( finite_finite_real @ S3 )
=> ( ( ( member_real @ A @ S3 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
@ S3 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S3 )
=> ( ( groups4061424788464935467al_rat
@ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
@ S3 )
= one_one_rat ) ) ) ) ).
% prod.delta
thf(fact_7964_prod_OlessThan__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7965_prod_OlessThan__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7966_prod_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7967_prod_OlessThan__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% prod.lessThan_Suc
thf(fact_7968_arccos__minus__1,axiom,
( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
= pi ) ).
% arccos_minus_1
thf(fact_7969_cos__arccos,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ( cos_real @ ( arccos @ Y4 ) )
= Y4 ) ) ) ).
% cos_arccos
thf(fact_7970_sin__arcsin,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ( sin_real @ ( arcsin @ Y4 ) )
= Y4 ) ) ) ).
% sin_arcsin
thf(fact_7971_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > complex] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_complex ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7972_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > real] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_real ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7973_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > rat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_rat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7974_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7975_prod_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > int] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= one_one_int ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_7976_arccos__0,axiom,
( ( arccos @ zero_zero_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arccos_0
thf(fact_7977_arcsin__1,axiom,
( ( arcsin @ one_one_real )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% arcsin_1
thf(fact_7978_arcsin__minus__1,axiom,
( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% arcsin_minus_1
thf(fact_7979_prod_Oneutral,axiom,
! [A2: set_nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ A2 )
= one_one_nat ) ) ).
% prod.neutral
thf(fact_7980_prod_Oneutral,axiom,
! [A2: set_nat,G: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G @ A2 )
= one_one_int ) ) ).
% prod.neutral
thf(fact_7981_prod_Oneutral,axiom,
! [A2: set_int,G: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups1705073143266064639nt_int @ G @ A2 )
= one_one_int ) ) ).
% prod.neutral
thf(fact_7982_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > complex,A2: set_nat] :
( ( ( groups6464643781859351333omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7983_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > complex,A2: set_int] :
( ( ( groups7440179247065528705omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( ( G @ A3 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7984_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > complex,A2: set_real] :
( ( ( groups713298508707869441omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7985_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > complex,A2: set_complex] :
( ( ( groups3708469109370488835omplex @ G @ A2 )
!= one_one_complex )
=> ~ ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ( G @ A3 )
= one_one_complex ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7986_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A2: set_nat] :
( ( ( groups129246275422532515t_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7987_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > real,A2: set_int] :
( ( ( groups2316167850115554303t_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( ( G @ A3 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7988_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups1681761925125756287l_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ( G @ A3 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7989_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: complex > real,A2: set_complex] :
( ( ( groups766887009212190081x_real @ G @ A2 )
!= one_one_real )
=> ~ ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ( G @ A3 )
= one_one_real ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7990_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > rat,A2: set_nat] :
( ( ( groups73079841787564623at_rat @ G @ A2 )
!= one_one_rat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( G @ A3 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7991_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: int > rat,A2: set_int] :
( ( ( groups1072433553688619179nt_rat @ G @ A2 )
!= one_one_rat )
=> ~ ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( ( G @ A3 )
= one_one_rat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_7992_prod_Odistrib,axiom,
! [G: nat > nat,H2: nat > nat,A2: set_nat] :
( ( groups708209901874060359at_nat
@ ^ [X2: nat] : ( times_times_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A2 ) @ ( groups708209901874060359at_nat @ H2 @ A2 ) ) ) ).
% prod.distrib
thf(fact_7993_prod_Odistrib,axiom,
! [G: nat > int,H2: nat > int,A2: set_nat] :
( ( groups705719431365010083at_int
@ ^ [X2: nat] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ A2 ) @ ( groups705719431365010083at_int @ H2 @ A2 ) ) ) ).
% prod.distrib
thf(fact_7994_prod_Odistrib,axiom,
! [G: int > int,H2: int > int,A2: set_int] :
( ( groups1705073143266064639nt_int
@ ^ [X2: int] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A2 )
= ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A2 ) @ ( groups1705073143266064639nt_int @ H2 @ A2 ) ) ) ).
% prod.distrib
thf(fact_7995_prod__power__distrib,axiom,
! [F: nat > nat,A2: set_nat,N: nat] :
( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N )
= ( groups708209901874060359at_nat
@ ^ [X2: nat] : ( power_power_nat @ ( F @ X2 ) @ N )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_7996_prod__power__distrib,axiom,
! [F: nat > int,A2: set_nat,N: nat] :
( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N )
= ( groups705719431365010083at_int
@ ^ [X2: nat] : ( power_power_int @ ( F @ X2 ) @ N )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_7997_prod__power__distrib,axiom,
! [F: int > int,A2: set_int,N: nat] :
( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N )
= ( groups1705073143266064639nt_int
@ ^ [X2: int] : ( power_power_int @ ( F @ X2 ) @ N )
@ A2 ) ) ).
% prod_power_distrib
thf(fact_7998_mod__prod__eq,axiom,
! [F: nat > nat,A: nat,A2: set_nat] :
( ( modulo_modulo_nat
@ ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_7999_mod__prod__eq,axiom,
! [F: nat > int,A: int,A2: set_nat] :
( ( modulo_modulo_int
@ ( groups705719431365010083at_int
@ ^ [I4: nat] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_8000_mod__prod__eq,axiom,
! [F: int > int,A: int,A2: set_int] :
( ( modulo_modulo_int
@ ( groups1705073143266064639nt_int
@ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
@ A2 )
@ A )
= ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).
% mod_prod_eq
thf(fact_8001_prod__nonneg,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_8002_prod__nonneg,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_8003_prod__nonneg,axiom,
! [A2: set_int,F: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% prod_nonneg
thf(fact_8004_prod__mono,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8005_prod__mono,axiom,
! [A2: set_int,F: int > real,G: int > real] :
( ! [I3: int] :
( ( member_int @ I3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8006_prod__mono,axiom,
! [A2: set_real,F: real > real,G: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8007_prod__mono,axiom,
! [A2: set_complex,F: complex > real,G: complex > real] :
( ! [I3: complex] :
( ( member_complex @ I3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8008_prod__mono,axiom,
! [A2: set_nat,F: nat > rat,G: nat > rat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8009_prod__mono,axiom,
! [A2: set_int,F: int > rat,G: int > rat] :
( ! [I3: int] :
( ( member_int @ I3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8010_prod__mono,axiom,
! [A2: set_real,F: real > rat,G: real > rat] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8011_prod__mono,axiom,
! [A2: set_complex,F: complex > rat,G: complex > rat] :
( ! [I3: complex] :
( ( member_complex @ I3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8012_prod__mono,axiom,
! [A2: set_int,F: int > nat,G: int > nat] :
( ! [I3: int] :
( ( member_int @ I3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8013_prod__mono,axiom,
! [A2: set_real,F: real > nat,G: real > nat] :
( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% prod_mono
thf(fact_8014_prod__pos,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_8015_prod__pos,axiom,
! [A2: set_nat,F: nat > int] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_8016_prod__pos,axiom,
! [A2: set_int,F: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
=> ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% prod_pos
thf(fact_8017_prod__ge__1,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8018_prod__ge__1,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8019_prod__ge__1,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8020_prod__ge__1,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8021_prod__ge__1,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8022_prod__ge__1,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8023_prod__ge__1,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8024_prod__ge__1,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8025_prod__ge__1,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8026_prod__ge__1,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% prod_ge_1
thf(fact_8027_prod__atLeastAtMost__code,axiom,
! [F: nat > complex,A: nat,B: nat] :
( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1517530859248394432omplex
@ ^ [A4: nat] : ( times_times_complex @ ( F @ A4 ) )
@ A
@ B
@ one_one_complex ) ) ).
% prod_atLeastAtMost_code
thf(fact_8028_prod__atLeastAtMost__code,axiom,
! [F: nat > real,A: nat,B: nat] :
( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo3111899725591712190t_real
@ ^ [A4: nat] : ( times_times_real @ ( F @ A4 ) )
@ A
@ B
@ one_one_real ) ) ).
% prod_atLeastAtMost_code
thf(fact_8029_prod__atLeastAtMost__code,axiom,
! [F: nat > rat,A: nat,B: nat] :
( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A4: nat] : ( times_times_rat @ ( F @ A4 ) )
@ A
@ B
@ one_one_rat ) ) ).
% prod_atLeastAtMost_code
thf(fact_8030_prod__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A4: nat] : ( times_times_nat @ ( F @ A4 ) )
@ A
@ B
@ one_one_nat ) ) ).
% prod_atLeastAtMost_code
thf(fact_8031_prod__atLeastAtMost__code,axiom,
! [F: nat > int,A: nat,B: nat] :
( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A4: nat] : ( times_times_int @ ( F @ A4 ) )
@ A
@ B
@ one_one_int ) ) ).
% prod_atLeastAtMost_code
thf(fact_8032_prod_Ointer__filter,axiom,
! [A2: set_int,G: int > complex,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups7440179247065528705omplex @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups7440179247065528705omplex
@ ^ [X2: int] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8033_prod_Ointer__filter,axiom,
! [A2: set_real,G: real > complex,P: real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( groups713298508707869441omplex @ G
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups713298508707869441omplex
@ ^ [X2: real] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8034_prod_Ointer__filter,axiom,
! [A2: set_nat,G: nat > complex,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups6464643781859351333omplex @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups6464643781859351333omplex
@ ^ [X2: nat] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8035_prod_Ointer__filter,axiom,
! [A2: set_complex,G: complex > complex,P: complex > $o] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups3708469109370488835omplex @ G
@ ( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups3708469109370488835omplex
@ ^ [X2: complex] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8036_prod_Ointer__filter,axiom,
! [A2: set_int,G: int > real,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups2316167850115554303t_real @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups2316167850115554303t_real
@ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8037_prod_Ointer__filter,axiom,
! [A2: set_real,G: real > real,P: real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( groups1681761925125756287l_real @ G
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups1681761925125756287l_real
@ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8038_prod_Ointer__filter,axiom,
! [A2: set_nat,G: nat > real,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups129246275422532515t_real @ G
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups129246275422532515t_real
@ ^ [X2: nat] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8039_prod_Ointer__filter,axiom,
! [A2: set_complex,G: complex > real,P: complex > $o] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups766887009212190081x_real @ G
@ ( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups766887009212190081x_real
@ ^ [X2: complex] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8040_prod_Ointer__filter,axiom,
! [A2: set_int,G: int > rat,P: int > $o] :
( ( finite_finite_int @ A2 )
=> ( ( groups1072433553688619179nt_rat @ G
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups1072433553688619179nt_rat
@ ^ [X2: int] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8041_prod_Ointer__filter,axiom,
! [A2: set_real,G: real > rat,P: real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( groups4061424788464935467al_rat @ G
@ ( collect_real
@ ^ [X2: real] :
( ( member_real @ X2 @ A2 )
& ( P @ X2 ) ) ) )
= ( groups4061424788464935467al_rat
@ ^ [X2: real] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
@ A2 ) ) ) ).
% prod.inter_filter
thf(fact_8042_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_8043_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > int,M: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_8044_power__sum,axiom,
! [C: real,F: nat > nat,A2: set_nat] :
( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups129246275422532515t_real
@ ^ [A4: nat] : ( power_power_real @ C @ ( F @ A4 ) )
@ A2 ) ) ).
% power_sum
thf(fact_8045_power__sum,axiom,
! [C: complex,F: nat > nat,A2: set_nat] :
( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups6464643781859351333omplex
@ ^ [A4: nat] : ( power_power_complex @ C @ ( F @ A4 ) )
@ A2 ) ) ).
% power_sum
thf(fact_8046_power__sum,axiom,
! [C: nat,F: nat > nat,A2: set_nat] :
( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups708209901874060359at_nat
@ ^ [A4: nat] : ( power_power_nat @ C @ ( F @ A4 ) )
@ A2 ) ) ).
% power_sum
thf(fact_8047_power__sum,axiom,
! [C: int,F: nat > nat,A2: set_nat] :
( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
= ( groups705719431365010083at_int
@ ^ [A4: nat] : ( power_power_int @ C @ ( F @ A4 ) )
@ A2 ) ) ).
% power_sum
thf(fact_8048_power__sum,axiom,
! [C: int,F: int > nat,A2: set_int] :
( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
= ( groups1705073143266064639nt_int
@ ^ [A4: int] : ( power_power_int @ C @ ( F @ A4 ) )
@ A2 ) ) ).
% power_sum
thf(fact_8049_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_8050_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > int,M: nat,K: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_8051_prod__le__1,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8052_prod__le__1,axiom,
! [A2: set_int,F: int > real] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8053_prod__le__1,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8054_prod__le__1,axiom,
! [A2: set_complex,F: complex > real] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
& ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ one_one_real ) ) ).
% prod_le_1
thf(fact_8055_prod__le__1,axiom,
! [A2: set_nat,F: nat > rat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8056_prod__le__1,axiom,
! [A2: set_int,F: int > rat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8057_prod__le__1,axiom,
! [A2: set_real,F: real > rat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8058_prod__le__1,axiom,
! [A2: set_complex,F: complex > rat] :
( ! [X3: complex] :
( ( member_complex @ X3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
& ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ one_one_rat ) ) ).
% prod_le_1
thf(fact_8059_prod__le__1,axiom,
! [A2: set_int,F: int > nat] :
( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
& ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_8060_prod__le__1,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
& ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_8061_prod_Orelated,axiom,
! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups6464643781859351333omplex @ H2 @ S3 ) @ ( groups6464643781859351333omplex @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8062_prod_Orelated,axiom,
! [R: complex > complex > $o,S3: set_complex,H2: complex > complex,G: complex > complex] :
( ( R @ one_one_complex @ one_one_complex )
=> ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups3708469109370488835omplex @ H2 @ S3 ) @ ( groups3708469109370488835omplex @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8063_prod_Orelated,axiom,
! [R: real > real > $o,S3: set_nat,H2: nat > real,G: nat > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X15: real,Y15: real,X23: real,Y23: real] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups129246275422532515t_real @ H2 @ S3 ) @ ( groups129246275422532515t_real @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8064_prod_Orelated,axiom,
! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
( ( R @ one_one_real @ one_one_real )
=> ( ! [X15: real,Y15: real,X23: real,Y23: real] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups766887009212190081x_real @ H2 @ S3 ) @ ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8065_prod_Orelated,axiom,
! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups73079841787564623at_rat @ H2 @ S3 ) @ ( groups73079841787564623at_rat @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8066_prod_Orelated,axiom,
! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
( ( R @ one_one_rat @ one_one_rat )
=> ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups225925009352817453ex_rat @ H2 @ S3 ) @ ( groups225925009352817453ex_rat @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8067_prod_Orelated,axiom,
! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
( ( R @ one_one_nat @ one_one_nat )
=> ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups861055069439313189ex_nat @ H2 @ S3 ) @ ( groups861055069439313189ex_nat @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8068_prod_Orelated,axiom,
! [R: int > int > $o,S3: set_complex,H2: complex > int,G: complex > int] :
( ( R @ one_one_int @ one_one_int )
=> ( ! [X15: int,Y15: int,X23: int,Y23: int] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_int @ X15 @ Y15 ) @ ( times_times_int @ X23 @ Y23 ) ) )
=> ( ( finite3207457112153483333omplex @ S3 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups858564598930262913ex_int @ H2 @ S3 ) @ ( groups858564598930262913ex_int @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8069_prod_Orelated,axiom,
! [R: nat > nat > $o,S3: set_nat,H2: nat > nat,G: nat > nat] :
( ( R @ one_one_nat @ one_one_nat )
=> ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups708209901874060359at_nat @ H2 @ S3 ) @ ( groups708209901874060359at_nat @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8070_prod_Orelated,axiom,
! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
( ( R @ one_one_int @ one_one_int )
=> ( ! [X15: int,Y15: int,X23: int,Y23: int] :
( ( ( R @ X15 @ X23 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times_int @ X15 @ Y15 ) @ ( times_times_int @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S3 )
=> ( R @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R @ ( groups705719431365010083at_int @ H2 @ S3 ) @ ( groups705719431365010083at_int @ G @ S3 ) ) ) ) ) ) ).
% prod.related
thf(fact_8071_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_int,S3: set_int,I2: int > int,J: int > int,T5: set_int,G: int > complex,H2: int > complex] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
=> ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T4 )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S3 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ S3 )
= ( groups7440179247065528705omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8072_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_real,S3: set_int,I2: real > int,J: int > real,T5: set_real,G: int > complex,H2: real > complex] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T4 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T4 ) )
=> ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S3 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ S3 )
= ( groups713298508707869441omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8073_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_int,S3: set_real,I2: int > real,J: real > int,T5: set_int,G: real > complex,H2: int > complex] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
=> ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T4 )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S3 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S3 )
= ( groups7440179247065528705omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8074_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_real,S3: set_real,I2: real > real,J: real > real,T5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T4 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T4 ) )
=> ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S3 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S3 )
= ( groups713298508707869441omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8075_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_complex,S3: set_int,I2: complex > int,J: int > complex,T5: set_complex,G: int > complex,H2: complex > complex] :
( ( finite_finite_int @ S4 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T5 @ T4 ) ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
=> ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T4 )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S3 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ S3 )
= ( groups3708469109370488835omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8076_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_real,T4: set_complex,S3: set_real,I2: complex > real,J: real > complex,T5: set_complex,G: real > complex,H2: complex > complex] :
( ( finite_finite_real @ S4 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S4 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T5 @ T4 ) ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
=> ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S3 @ S4 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T4 )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ S3 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S3 )
= ( groups3708469109370488835omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8077_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_complex,T4: set_int,S3: set_complex,I2: int > complex,J: complex > int,T5: set_int,G: complex > complex,H2: int > complex] :
( ( finite3207457112153483333omplex @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
=> ( member_complex @ ( I2 @ B2 ) @ ( minus_811609699411566653omplex @ S3 @ S4 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T4 )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S3 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S3 )
= ( groups7440179247065528705omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8078_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_complex,T4: set_real,S3: set_complex,I2: real > complex,J: complex > real,T5: set_real,G: complex > complex,H2: real > complex] :
( ( finite3207457112153483333omplex @ S4 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
=> ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T4 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ T5 @ T4 ) )
=> ( member_complex @ ( I2 @ B2 ) @ ( minus_811609699411566653omplex @ S3 @ S4 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ T4 )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S3 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S3 )
= ( groups713298508707869441omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8079_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_complex,T4: set_complex,S3: set_complex,I2: complex > complex,J: complex > complex,T5: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ S4 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S4 ) )
=> ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T5 @ T4 ) ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
=> ( member_complex @ ( I2 @ B2 ) @ ( minus_811609699411566653omplex @ S3 @ S4 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ T4 )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ S3 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S3 )
= ( groups3708469109370488835omplex @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8080_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_int,T4: set_int,S3: set_int,I2: int > int,J: int > int,T5: set_int,G: int > real,H2: int > real] :
( ( finite_finite_int @ S4 )
=> ( ( finite_finite_int @ T4 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( ( I2 @ ( J @ A3 ) )
= A3 ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S4 ) )
=> ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
=> ( ( J @ ( I2 @ B2 ) )
= B2 ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ T5 @ T4 ) )
=> ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S3 @ S4 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ T4 )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ S3 )
=> ( ( H2 @ ( J @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2316167850115554303t_real @ G @ S3 )
= ( groups2316167850115554303t_real @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_8081_prod_Osetdiff__irrelevant,axiom,
! [A2: set_real,G: real > complex] :
( ( finite_finite_real @ A2 )
=> ( ( groups713298508707869441omplex @ G
@ ( minus_minus_set_real @ A2
@ ( collect_real
@ ^ [X2: real] :
( ( G @ X2 )
= one_one_complex ) ) ) )
= ( groups713298508707869441omplex @ G @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8082_prod_Osetdiff__irrelevant,axiom,
! [A2: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups3708469109370488835omplex @ G
@ ( minus_811609699411566653omplex @ A2
@ ( collect_complex
@ ^ [X2: complex] :
( ( G @ X2 )
= one_one_complex ) ) ) )
= ( groups3708469109370488835omplex @ G @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8083_prod_Osetdiff__irrelevant,axiom,
! [A2: set_real,G: real > real] :
( ( finite_finite_real @ A2 )
=> ( ( groups1681761925125756287l_real @ G
@ ( minus_minus_set_real @ A2
@ ( collect_real
@ ^ [X2: real] :
( ( G @ X2 )
= one_one_real ) ) ) )
= ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8084_prod_Osetdiff__irrelevant,axiom,
! [A2: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups766887009212190081x_real @ G
@ ( minus_811609699411566653omplex @ A2
@ ( collect_complex
@ ^ [X2: complex] :
( ( G @ X2 )
= one_one_real ) ) ) )
= ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8085_prod_Osetdiff__irrelevant,axiom,
! [A2: set_real,G: real > rat] :
( ( finite_finite_real @ A2 )
=> ( ( groups4061424788464935467al_rat @ G
@ ( minus_minus_set_real @ A2
@ ( collect_real
@ ^ [X2: real] :
( ( G @ X2 )
= one_one_rat ) ) ) )
= ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8086_prod_Osetdiff__irrelevant,axiom,
! [A2: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups225925009352817453ex_rat @ G
@ ( minus_811609699411566653omplex @ A2
@ ( collect_complex
@ ^ [X2: complex] :
( ( G @ X2 )
= one_one_rat ) ) ) )
= ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8087_prod_Osetdiff__irrelevant,axiom,
! [A2: set_real,G: real > nat] :
( ( finite_finite_real @ A2 )
=> ( ( groups4696554848551431203al_nat @ G
@ ( minus_minus_set_real @ A2
@ ( collect_real
@ ^ [X2: real] :
( ( G @ X2 )
= one_one_nat ) ) ) )
= ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8088_prod_Osetdiff__irrelevant,axiom,
! [A2: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups861055069439313189ex_nat @ G
@ ( minus_811609699411566653omplex @ A2
@ ( collect_complex
@ ^ [X2: complex] :
( ( G @ X2 )
= one_one_nat ) ) ) )
= ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8089_prod_Osetdiff__irrelevant,axiom,
! [A2: set_real,G: real > int] :
( ( finite_finite_real @ A2 )
=> ( ( groups4694064378042380927al_int @ G
@ ( minus_minus_set_real @ A2
@ ( collect_real
@ ^ [X2: real] :
( ( G @ X2 )
= one_one_int ) ) ) )
= ( groups4694064378042380927al_int @ G @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8090_prod_Osetdiff__irrelevant,axiom,
! [A2: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups858564598930262913ex_int @ G
@ ( minus_811609699411566653omplex @ A2
@ ( collect_complex
@ ^ [X2: complex] :
( ( G @ X2 )
= one_one_int ) ) ) )
= ( groups858564598930262913ex_int @ G @ A2 ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_8091_prod_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nat_diff_reindex
thf(fact_8092_prod_Onat__diff__reindex,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nat_diff_reindex
thf(fact_8093_prod_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_8094_prod_OatLeastAtMost__rev,axiom,
! [G: nat > int,N: nat,M: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_8095_less__1__prod2,axiom,
! [I5: set_int,I2: int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I2 @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8096_less__1__prod2,axiom,
! [I5: set_real,I2: real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8097_less__1__prod2,axiom,
! [I5: set_nat,I2: nat,F: nat > real] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I2 @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8098_less__1__prod2,axiom,
! [I5: set_complex,I2: complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8099_less__1__prod2,axiom,
! [I5: set_int,I2: int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( member_int @ I2 @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8100_less__1__prod2,axiom,
! [I5: set_real,I2: real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8101_less__1__prod2,axiom,
! [I5: set_nat,I2: nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I2 @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8102_less__1__prod2,axiom,
! [I5: set_complex,I2: complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8103_less__1__prod2,axiom,
! [I5: set_real,I2: real,F: real > int] :
( ( finite_finite_real @ I5 )
=> ( ( member_real @ I2 @ I5 )
=> ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8104_less__1__prod2,axiom,
! [I5: set_complex,I2: complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( member_complex @ I2 @ I5 )
=> ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_8105_arccos__le__arccos,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y4 ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_8106_less__1__prod,axiom,
! [I5: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8107_less__1__prod,axiom,
! [I5: set_nat,F: nat > real] :
( ( finite_finite_nat @ I5 )
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8108_less__1__prod,axiom,
! [I5: set_int,F: int > real] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8109_less__1__prod,axiom,
! [I5: set_real,F: real > real] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
=> ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8110_less__1__prod,axiom,
! [I5: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8111_less__1__prod,axiom,
! [I5: set_nat,F: nat > rat] :
( ( finite_finite_nat @ I5 )
=> ( ( I5 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8112_less__1__prod,axiom,
! [I5: set_int,F: int > rat] :
( ( finite_finite_int @ I5 )
=> ( ( I5 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8113_less__1__prod,axiom,
! [I5: set_real,F: real > rat] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
=> ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8114_less__1__prod,axiom,
! [I5: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ I5 )
=> ( ( I5 != bot_bot_set_complex )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8115_less__1__prod,axiom,
! [I5: set_real,F: real > int] :
( ( finite_finite_real @ I5 )
=> ( ( I5 != bot_bot_set_real )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
=> ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ).
% less_1_prod
thf(fact_8116_arccos__eq__iff,axiom,
! [X: real,Y4: real] :
( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
& ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real ) )
=> ( ( ( arccos @ X )
= ( arccos @ Y4 ) )
= ( X = Y4 ) ) ) ).
% arccos_eq_iff
thf(fact_8117_arccos__le__mono,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y4 ) )
= ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).
% arccos_le_mono
thf(fact_8118_arcsin__minus,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( arcsin @ ( uminus_uminus_real @ X ) )
= ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% arcsin_minus
thf(fact_8119_arcsin__le__arcsin,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_8120_prod_Osubset__diff,axiom,
! [B5: set_complex,A2: set_complex,G: complex > real] :
( ( ord_le211207098394363844omplex @ B5 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups766887009212190081x_real @ G @ A2 )
= ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups766887009212190081x_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8121_prod_Osubset__diff,axiom,
! [B5: set_complex,A2: set_complex,G: complex > rat] :
( ( ord_le211207098394363844omplex @ B5 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups225925009352817453ex_rat @ G @ A2 )
= ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8122_prod_Osubset__diff,axiom,
! [B5: set_complex,A2: set_complex,G: complex > nat] :
( ( ord_le211207098394363844omplex @ B5 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups861055069439313189ex_nat @ G @ A2 )
= ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8123_prod_Osubset__diff,axiom,
! [B5: set_complex,A2: set_complex,G: complex > int] :
( ( ord_le211207098394363844omplex @ B5 @ A2 )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ( ( groups858564598930262913ex_int @ G @ A2 )
= ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8124_prod_Osubset__diff,axiom,
! [B5: set_nat,A2: set_nat,G: nat > real] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups129246275422532515t_real @ G @ A2 )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups129246275422532515t_real @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8125_prod_Osubset__diff,axiom,
! [B5: set_nat,A2: set_nat,G: nat > rat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups73079841787564623at_rat @ G @ A2 )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8126_prod_Osubset__diff,axiom,
! [B5: set_nat,A2: set_nat,G: nat > nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups708209901874060359at_nat @ G @ A2 )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8127_prod_Osubset__diff,axiom,
! [B5: set_nat,A2: set_nat,G: nat > int] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( groups705719431365010083at_int @ G @ A2 )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups705719431365010083at_int @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8128_prod_Osubset__diff,axiom,
! [B5: set_int,A2: set_int,G: int > int] :
( ( ord_less_eq_set_int @ B5 @ A2 )
=> ( ( finite_finite_int @ A2 )
=> ( ( groups1705073143266064639nt_int @ G @ A2 )
= ( times_times_int @ ( groups1705073143266064639nt_int @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups1705073143266064639nt_int @ G @ B5 ) ) ) ) ) ).
% prod.subset_diff
thf(fact_8129_prod_Osame__carrier,axiom,
! [C5: set_int,A2: set_int,B5: set_int,G: int > complex,H2: int > complex] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A2 @ C5 )
=> ( ( ord_less_eq_set_int @ B5 @ C5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ( ( groups7440179247065528705omplex @ G @ A2 )
= ( groups7440179247065528705omplex @ H2 @ B5 ) )
= ( ( groups7440179247065528705omplex @ G @ C5 )
= ( groups7440179247065528705omplex @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8130_prod_Osame__carrier,axiom,
! [C5: set_real,A2: set_real,B5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C5 )
=> ( ( ord_less_eq_set_real @ A2 @ C5 )
=> ( ( ord_less_eq_set_real @ B5 @ C5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ( ( groups713298508707869441omplex @ G @ A2 )
= ( groups713298508707869441omplex @ H2 @ B5 ) )
= ( ( groups713298508707869441omplex @ G @ C5 )
= ( groups713298508707869441omplex @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8131_prod_Osame__carrier,axiom,
! [C5: set_complex,A2: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ C5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C5 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ( ( groups3708469109370488835omplex @ G @ A2 )
= ( groups3708469109370488835omplex @ H2 @ B5 ) )
= ( ( groups3708469109370488835omplex @ G @ C5 )
= ( groups3708469109370488835omplex @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8132_prod_Osame__carrier,axiom,
! [C5: set_int,A2: set_int,B5: set_int,G: int > real,H2: int > real] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A2 @ C5 )
=> ( ( ord_less_eq_set_int @ B5 @ C5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ( ( groups2316167850115554303t_real @ G @ A2 )
= ( groups2316167850115554303t_real @ H2 @ B5 ) )
= ( ( groups2316167850115554303t_real @ G @ C5 )
= ( groups2316167850115554303t_real @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8133_prod_Osame__carrier,axiom,
! [C5: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C5 )
=> ( ( ord_less_eq_set_real @ A2 @ C5 )
=> ( ( ord_less_eq_set_real @ B5 @ C5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ( ( groups1681761925125756287l_real @ G @ A2 )
= ( groups1681761925125756287l_real @ H2 @ B5 ) )
= ( ( groups1681761925125756287l_real @ G @ C5 )
= ( groups1681761925125756287l_real @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8134_prod_Osame__carrier,axiom,
! [C5: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C5 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ( ( groups766887009212190081x_real @ G @ A2 )
= ( groups766887009212190081x_real @ H2 @ B5 ) )
= ( ( groups766887009212190081x_real @ G @ C5 )
= ( groups766887009212190081x_real @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8135_prod_Osame__carrier,axiom,
! [C5: set_int,A2: set_int,B5: set_int,G: int > rat,H2: int > rat] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A2 @ C5 )
=> ( ( ord_less_eq_set_int @ B5 @ C5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_rat ) )
=> ( ( ( groups1072433553688619179nt_rat @ G @ A2 )
= ( groups1072433553688619179nt_rat @ H2 @ B5 ) )
= ( ( groups1072433553688619179nt_rat @ G @ C5 )
= ( groups1072433553688619179nt_rat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8136_prod_Osame__carrier,axiom,
! [C5: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C5 )
=> ( ( ord_less_eq_set_real @ A2 @ C5 )
=> ( ( ord_less_eq_set_real @ B5 @ C5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_rat ) )
=> ( ( ( groups4061424788464935467al_rat @ G @ A2 )
= ( groups4061424788464935467al_rat @ H2 @ B5 ) )
= ( ( groups4061424788464935467al_rat @ G @ C5 )
= ( groups4061424788464935467al_rat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8137_prod_Osame__carrier,axiom,
! [C5: set_complex,A2: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C5 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_rat ) )
=> ( ( ( groups225925009352817453ex_rat @ G @ A2 )
= ( groups225925009352817453ex_rat @ H2 @ B5 ) )
= ( ( groups225925009352817453ex_rat @ G @ C5 )
= ( groups225925009352817453ex_rat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8138_prod_Osame__carrier,axiom,
! [C5: set_int,A2: set_int,B5: set_int,G: int > nat,H2: int > nat] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A2 @ C5 )
=> ( ( ord_less_eq_set_int @ B5 @ C5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups1707563613775114915nt_nat @ G @ A2 )
= ( groups1707563613775114915nt_nat @ H2 @ B5 ) )
= ( ( groups1707563613775114915nt_nat @ G @ C5 )
= ( groups1707563613775114915nt_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_8139_prod_Osame__carrierI,axiom,
! [C5: set_int,A2: set_int,B5: set_int,G: int > complex,H2: int > complex] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A2 @ C5 )
=> ( ( ord_less_eq_set_int @ B5 @ C5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ( ( groups7440179247065528705omplex @ G @ C5 )
= ( groups7440179247065528705omplex @ H2 @ C5 ) )
=> ( ( groups7440179247065528705omplex @ G @ A2 )
= ( groups7440179247065528705omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8140_prod_Osame__carrierI,axiom,
! [C5: set_real,A2: set_real,B5: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ C5 )
=> ( ( ord_less_eq_set_real @ A2 @ C5 )
=> ( ( ord_less_eq_set_real @ B5 @ C5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ( ( groups713298508707869441omplex @ G @ C5 )
= ( groups713298508707869441omplex @ H2 @ C5 ) )
=> ( ( groups713298508707869441omplex @ G @ A2 )
= ( groups713298508707869441omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8141_prod_Osame__carrierI,axiom,
! [C5: set_complex,A2: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ C5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C5 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_complex ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_complex ) )
=> ( ( ( groups3708469109370488835omplex @ G @ C5 )
= ( groups3708469109370488835omplex @ H2 @ C5 ) )
=> ( ( groups3708469109370488835omplex @ G @ A2 )
= ( groups3708469109370488835omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8142_prod_Osame__carrierI,axiom,
! [C5: set_int,A2: set_int,B5: set_int,G: int > real,H2: int > real] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A2 @ C5 )
=> ( ( ord_less_eq_set_int @ B5 @ C5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ( ( groups2316167850115554303t_real @ G @ C5 )
= ( groups2316167850115554303t_real @ H2 @ C5 ) )
=> ( ( groups2316167850115554303t_real @ G @ A2 )
= ( groups2316167850115554303t_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8143_prod_Osame__carrierI,axiom,
! [C5: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ C5 )
=> ( ( ord_less_eq_set_real @ A2 @ C5 )
=> ( ( ord_less_eq_set_real @ B5 @ C5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ( ( groups1681761925125756287l_real @ G @ C5 )
= ( groups1681761925125756287l_real @ H2 @ C5 ) )
=> ( ( groups1681761925125756287l_real @ G @ A2 )
= ( groups1681761925125756287l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8144_prod_Osame__carrierI,axiom,
! [C5: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ C5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C5 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_real ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_real ) )
=> ( ( ( groups766887009212190081x_real @ G @ C5 )
= ( groups766887009212190081x_real @ H2 @ C5 ) )
=> ( ( groups766887009212190081x_real @ G @ A2 )
= ( groups766887009212190081x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8145_prod_Osame__carrierI,axiom,
! [C5: set_int,A2: set_int,B5: set_int,G: int > rat,H2: int > rat] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A2 @ C5 )
=> ( ( ord_less_eq_set_int @ B5 @ C5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_rat ) )
=> ( ( ( groups1072433553688619179nt_rat @ G @ C5 )
= ( groups1072433553688619179nt_rat @ H2 @ C5 ) )
=> ( ( groups1072433553688619179nt_rat @ G @ A2 )
= ( groups1072433553688619179nt_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8146_prod_Osame__carrierI,axiom,
! [C5: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ C5 )
=> ( ( ord_less_eq_set_real @ A2 @ C5 )
=> ( ( ord_less_eq_set_real @ B5 @ C5 )
=> ( ! [A3: real] :
( ( member_real @ A3 @ ( minus_minus_set_real @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_rat ) )
=> ( ( ( groups4061424788464935467al_rat @ G @ C5 )
= ( groups4061424788464935467al_rat @ H2 @ C5 ) )
=> ( ( groups4061424788464935467al_rat @ G @ A2 )
= ( groups4061424788464935467al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8147_prod_Osame__carrierI,axiom,
! [C5: set_complex,A2: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ C5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ C5 )
=> ( ( ord_le211207098394363844omplex @ B5 @ C5 )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_rat ) )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_rat ) )
=> ( ( ( groups225925009352817453ex_rat @ G @ C5 )
= ( groups225925009352817453ex_rat @ H2 @ C5 ) )
=> ( ( groups225925009352817453ex_rat @ G @ A2 )
= ( groups225925009352817453ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8148_prod_Osame__carrierI,axiom,
! [C5: set_int,A2: set_int,B5: set_int,G: int > nat,H2: int > nat] :
( ( finite_finite_int @ C5 )
=> ( ( ord_less_eq_set_int @ A2 @ C5 )
=> ( ( ord_less_eq_set_int @ B5 @ C5 )
=> ( ! [A3: int] :
( ( member_int @ A3 @ ( minus_minus_set_int @ C5 @ A2 ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ C5 @ B5 ) )
=> ( ( H2 @ B2 )
= one_one_nat ) )
=> ( ( ( groups1707563613775114915nt_nat @ G @ C5 )
= ( groups1707563613775114915nt_nat @ H2 @ C5 ) )
=> ( ( groups1707563613775114915nt_nat @ G @ A2 )
= ( groups1707563613775114915nt_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_8149_prod_Omono__neutral__left,axiom,
! [T5: set_complex,S3: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ( groups3708469109370488835omplex @ G @ S3 )
= ( groups3708469109370488835omplex @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8150_prod_Omono__neutral__left,axiom,
! [T5: set_complex,S3: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ( groups766887009212190081x_real @ G @ S3 )
= ( groups766887009212190081x_real @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8151_prod_Omono__neutral__left,axiom,
! [T5: set_complex,S3: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ( groups225925009352817453ex_rat @ G @ S3 )
= ( groups225925009352817453ex_rat @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8152_prod_Omono__neutral__left,axiom,
! [T5: set_complex,S3: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups861055069439313189ex_nat @ G @ S3 )
= ( groups861055069439313189ex_nat @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8153_prod_Omono__neutral__left,axiom,
! [T5: set_complex,S3: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups858564598930262913ex_int @ G @ S3 )
= ( groups858564598930262913ex_int @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8154_prod_Omono__neutral__left,axiom,
! [T5: set_nat,S3: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T5 )
=> ( ( ord_less_eq_set_nat @ S3 @ T5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ( groups6464643781859351333omplex @ G @ S3 )
= ( groups6464643781859351333omplex @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8155_prod_Omono__neutral__left,axiom,
! [T5: set_nat,S3: set_nat,G: nat > real] :
( ( finite_finite_nat @ T5 )
=> ( ( ord_less_eq_set_nat @ S3 @ T5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ( groups129246275422532515t_real @ G @ S3 )
= ( groups129246275422532515t_real @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8156_prod_Omono__neutral__left,axiom,
! [T5: set_nat,S3: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T5 )
=> ( ( ord_less_eq_set_nat @ S3 @ T5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ S3 )
= ( groups73079841787564623at_rat @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8157_prod_Omono__neutral__left,axiom,
! [T5: set_nat,S3: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T5 )
=> ( ( ord_less_eq_set_nat @ S3 @ T5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ S3 )
= ( groups708209901874060359at_nat @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8158_prod_Omono__neutral__left,axiom,
! [T5: set_nat,S3: set_nat,G: nat > int] :
( ( finite_finite_nat @ T5 )
=> ( ( ord_less_eq_set_nat @ S3 @ T5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G @ S3 )
= ( groups705719431365010083at_int @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_8159_prod_Omono__neutral__right,axiom,
! [T5: set_complex,S3: set_complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ( groups3708469109370488835omplex @ G @ T5 )
= ( groups3708469109370488835omplex @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8160_prod_Omono__neutral__right,axiom,
! [T5: set_complex,S3: set_complex,G: complex > real] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ( groups766887009212190081x_real @ G @ T5 )
= ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8161_prod_Omono__neutral__right,axiom,
! [T5: set_complex,S3: set_complex,G: complex > rat] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ( groups225925009352817453ex_rat @ G @ T5 )
= ( groups225925009352817453ex_rat @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8162_prod_Omono__neutral__right,axiom,
! [T5: set_complex,S3: set_complex,G: complex > nat] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups861055069439313189ex_nat @ G @ T5 )
= ( groups861055069439313189ex_nat @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8163_prod_Omono__neutral__right,axiom,
! [T5: set_complex,S3: set_complex,G: complex > int] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups858564598930262913ex_int @ G @ T5 )
= ( groups858564598930262913ex_int @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8164_prod_Omono__neutral__right,axiom,
! [T5: set_nat,S3: set_nat,G: nat > complex] :
( ( finite_finite_nat @ T5 )
=> ( ( ord_less_eq_set_nat @ S3 @ T5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ( groups6464643781859351333omplex @ G @ T5 )
= ( groups6464643781859351333omplex @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8165_prod_Omono__neutral__right,axiom,
! [T5: set_nat,S3: set_nat,G: nat > real] :
( ( finite_finite_nat @ T5 )
=> ( ( ord_less_eq_set_nat @ S3 @ T5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ( groups129246275422532515t_real @ G @ T5 )
= ( groups129246275422532515t_real @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8166_prod_Omono__neutral__right,axiom,
! [T5: set_nat,S3: set_nat,G: nat > rat] :
( ( finite_finite_nat @ T5 )
=> ( ( ord_less_eq_set_nat @ S3 @ T5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ( groups73079841787564623at_rat @ G @ T5 )
= ( groups73079841787564623at_rat @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8167_prod_Omono__neutral__right,axiom,
! [T5: set_nat,S3: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T5 )
=> ( ( ord_less_eq_set_nat @ S3 @ T5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ T5 )
= ( groups708209901874060359at_nat @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8168_prod_Omono__neutral__right,axiom,
! [T5: set_nat,S3: set_nat,G: nat > int] :
( ( finite_finite_nat @ T5 )
=> ( ( ord_less_eq_set_nat @ S3 @ T5 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_int ) )
=> ( ( groups705719431365010083at_int @ G @ T5 )
= ( groups705719431365010083at_int @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_8169_prod_Omono__neutral__cong__left,axiom,
! [T5: set_int,S3: set_int,H2: int > complex,G: int > complex] :
( ( finite_finite_int @ T5 )
=> ( ( ord_less_eq_set_int @ S3 @ T5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_complex ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ S3 )
= ( groups7440179247065528705omplex @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8170_prod_Omono__neutral__cong__left,axiom,
! [T5: set_real,S3: set_real,H2: real > complex,G: real > complex] :
( ( finite_finite_real @ T5 )
=> ( ( ord_less_eq_set_real @ S3 @ T5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_complex ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups713298508707869441omplex @ G @ S3 )
= ( groups713298508707869441omplex @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8171_prod_Omono__neutral__cong__left,axiom,
! [T5: set_complex,S3: set_complex,H2: complex > complex,G: complex > complex] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_complex ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ S3 )
= ( groups3708469109370488835omplex @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8172_prod_Omono__neutral__cong__left,axiom,
! [T5: set_int,S3: set_int,H2: int > real,G: int > real] :
( ( finite_finite_int @ T5 )
=> ( ( ord_less_eq_set_int @ S3 @ T5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_real ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups2316167850115554303t_real @ G @ S3 )
= ( groups2316167850115554303t_real @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8173_prod_Omono__neutral__cong__left,axiom,
! [T5: set_real,S3: set_real,H2: real > real,G: real > real] :
( ( finite_finite_real @ T5 )
=> ( ( ord_less_eq_set_real @ S3 @ T5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_real ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ S3 )
= ( groups1681761925125756287l_real @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8174_prod_Omono__neutral__cong__left,axiom,
! [T5: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_real ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups766887009212190081x_real @ G @ S3 )
= ( groups766887009212190081x_real @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8175_prod_Omono__neutral__cong__left,axiom,
! [T5: set_int,S3: set_int,H2: int > rat,G: int > rat] :
( ( finite_finite_int @ T5 )
=> ( ( ord_less_eq_set_int @ S3 @ T5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_rat ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1072433553688619179nt_rat @ G @ S3 )
= ( groups1072433553688619179nt_rat @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8176_prod_Omono__neutral__cong__left,axiom,
! [T5: set_real,S3: set_real,H2: real > rat,G: real > rat] :
( ( finite_finite_real @ T5 )
=> ( ( ord_less_eq_set_real @ S3 @ T5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_rat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups4061424788464935467al_rat @ G @ S3 )
= ( groups4061424788464935467al_rat @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8177_prod_Omono__neutral__cong__left,axiom,
! [T5: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_rat ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups225925009352817453ex_rat @ G @ S3 )
= ( groups225925009352817453ex_rat @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8178_prod_Omono__neutral__cong__left,axiom,
! [T5: set_int,S3: set_int,H2: int > nat,G: int > nat] :
( ( finite_finite_int @ T5 )
=> ( ( ord_less_eq_set_int @ S3 @ T5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
=> ( ( H2 @ X3 )
= one_one_nat ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G @ S3 )
= ( groups1707563613775114915nt_nat @ H2 @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_8179_prod_Omono__neutral__cong__right,axiom,
! [T5: set_int,S3: set_int,G: int > complex,H2: int > complex] :
( ( finite_finite_int @ T5 )
=> ( ( ord_less_eq_set_int @ S3 @ T5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7440179247065528705omplex @ G @ T5 )
= ( groups7440179247065528705omplex @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8180_prod_Omono__neutral__cong__right,axiom,
! [T5: set_real,S3: set_real,G: real > complex,H2: real > complex] :
( ( finite_finite_real @ T5 )
=> ( ( ord_less_eq_set_real @ S3 @ T5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups713298508707869441omplex @ G @ T5 )
= ( groups713298508707869441omplex @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8181_prod_Omono__neutral__cong__right,axiom,
! [T5: set_complex,S3: set_complex,G: complex > complex,H2: complex > complex] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_complex ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups3708469109370488835omplex @ G @ T5 )
= ( groups3708469109370488835omplex @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8182_prod_Omono__neutral__cong__right,axiom,
! [T5: set_int,S3: set_int,G: int > real,H2: int > real] :
( ( finite_finite_int @ T5 )
=> ( ( ord_less_eq_set_int @ S3 @ T5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups2316167850115554303t_real @ G @ T5 )
= ( groups2316167850115554303t_real @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8183_prod_Omono__neutral__cong__right,axiom,
! [T5: set_real,S3: set_real,G: real > real,H2: real > real] :
( ( finite_finite_real @ T5 )
=> ( ( ord_less_eq_set_real @ S3 @ T5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1681761925125756287l_real @ G @ T5 )
= ( groups1681761925125756287l_real @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8184_prod_Omono__neutral__cong__right,axiom,
! [T5: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_real ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups766887009212190081x_real @ G @ T5 )
= ( groups766887009212190081x_real @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8185_prod_Omono__neutral__cong__right,axiom,
! [T5: set_int,S3: set_int,G: int > rat,H2: int > rat] :
( ( finite_finite_int @ T5 )
=> ( ( ord_less_eq_set_int @ S3 @ T5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1072433553688619179nt_rat @ G @ T5 )
= ( groups1072433553688619179nt_rat @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8186_prod_Omono__neutral__cong__right,axiom,
! [T5: set_real,S3: set_real,G: real > rat,H2: real > rat] :
( ( finite_finite_real @ T5 )
=> ( ( ord_less_eq_set_real @ S3 @ T5 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups4061424788464935467al_rat @ G @ T5 )
= ( groups4061424788464935467al_rat @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8187_prod_Omono__neutral__cong__right,axiom,
! [T5: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
( ( finite3207457112153483333omplex @ T5 )
=> ( ( ord_le211207098394363844omplex @ S3 @ T5 )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_rat ) )
=> ( ! [X3: complex] :
( ( member_complex @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups225925009352817453ex_rat @ G @ T5 )
= ( groups225925009352817453ex_rat @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8188_prod_Omono__neutral__cong__right,axiom,
! [T5: set_int,S3: set_int,G: int > nat,H2: int > nat] :
( ( finite_finite_int @ T5 )
=> ( ( ord_less_eq_set_int @ S3 @ T5 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
=> ( ( G @ X3 )
= one_one_nat ) )
=> ( ! [X3: int] :
( ( member_int @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1707563613775114915nt_nat @ G @ T5 )
= ( groups1707563613775114915nt_nat @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_8189_arcsin__eq__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y4 ) )
= ( X = Y4 ) ) ) ) ).
% arcsin_eq_iff
thf(fact_8190_arcsin__le__mono,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) )
= ( ord_less_eq_real @ X @ Y4 ) ) ) ) ).
% arcsin_le_mono
thf(fact_8191_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8192_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8193_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8194_prod_OatLeast0__atMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_8195_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8196_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8197_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8198_prod_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_8199_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8200_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8201_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8202_prod_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_8203_prod_OlessThan__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8204_prod_OlessThan__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8205_prod_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8206_prod_OlessThan__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_8207_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ M )
@ ( groups129246275422532515t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8208_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ M )
@ ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8209_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ M )
@ ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8210_prod_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ M )
@ ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_8211_prod_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_8212_prod_OatLeast1__atMost__eq,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_8213_fact__prod,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N2: nat] :
( semiri1314217659103216013at_int
@ ( groups708209901874060359at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% fact_prod
thf(fact_8214_fact__prod,axiom,
( semiri773545260158071498ct_rat
= ( ^ [N2: nat] :
( semiri681578069525770553at_rat
@ ( groups708209901874060359at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% fact_prod
thf(fact_8215_fact__prod,axiom,
( semiri2265585572941072030t_real
= ( ^ [N2: nat] :
( semiri5074537144036343181t_real
@ ( groups708209901874060359at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% fact_prod
thf(fact_8216_fact__prod,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N2: nat] :
( semiri1316708129612266289at_nat
@ ( groups708209901874060359at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% fact_prod
thf(fact_8217_prod__mono__strict,axiom,
! [A2: set_complex,F: complex > real,G: complex > real] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A2 != bot_bot_set_complex )
=> ( ord_less_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8218_prod__mono__strict,axiom,
! [A2: set_nat,F: nat > real,G: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8219_prod__mono__strict,axiom,
! [A2: set_int,F: int > real,G: int > real] :
( ( finite_finite_int @ A2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A2 != bot_bot_set_int )
=> ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8220_prod__mono__strict,axiom,
! [A2: set_real,F: real > real,G: real > real] :
( ( finite_finite_real @ A2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
& ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A2 != bot_bot_set_real )
=> ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8221_prod__mono__strict,axiom,
! [A2: set_complex,F: complex > rat,G: complex > rat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A2 != bot_bot_set_complex )
=> ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8222_prod__mono__strict,axiom,
! [A2: set_nat,F: nat > rat,G: nat > rat] :
( ( finite_finite_nat @ A2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8223_prod__mono__strict,axiom,
! [A2: set_int,F: int > rat,G: int > rat] :
( ( finite_finite_int @ A2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A2 != bot_bot_set_int )
=> ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8224_prod__mono__strict,axiom,
! [A2: set_real,F: real > rat,G: real > rat] :
( ( finite_finite_real @ A2 )
=> ( ! [I3: real] :
( ( member_real @ I3 @ A2 )
=> ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
& ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A2 != bot_bot_set_real )
=> ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8225_prod__mono__strict,axiom,
! [A2: set_complex,F: complex > nat,G: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A2 != bot_bot_set_complex )
=> ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8226_prod__mono__strict,axiom,
! [A2: set_int,F: int > nat,G: int > nat] :
( ( finite_finite_int @ A2 )
=> ( ! [I3: int] :
( ( member_int @ I3 @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A2 != bot_bot_set_int )
=> ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ) ) ).
% prod_mono_strict
thf(fact_8227_even__prod__iff,axiom,
! [A2: set_nat,F: nat > code_integer] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3455450783089532116nteger @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8228_even__prod__iff,axiom,
! [A2: set_complex,F: complex > code_integer] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups8682486955453173170nteger @ F @ A2 ) )
= ( ? [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8229_even__prod__iff,axiom,
! [A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A2 ) )
= ( ? [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8230_even__prod__iff,axiom,
! [A2: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A2 ) )
= ( ? [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8231_even__prod__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8232_even__prod__iff,axiom,
! [A2: set_nat,F: nat > int] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8233_even__prod__iff,axiom,
! [A2: set_int,F: int > int] :
( ( finite_finite_int @ A2 )
=> ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A2 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X2 ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_8234_arccos__lbound,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y4 ) ) ) ) ).
% arccos_lbound
thf(fact_8235_arccos__less__arccos,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_real @ ( arccos @ Y4 ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_8236_arccos__less__mono,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
=> ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y4 ) )
= ( ord_less_real @ Y4 @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_8237_arccos__ubound,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y4 ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_8238_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > real,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8239_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > rat,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8240_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > nat,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8241_prod_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > int,P6: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_8242_arcsin__less__arcsin,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_8243_arcsin__less__mono,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
=> ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y4 ) )
= ( ord_less_real @ X @ Y4 ) ) ) ) ).
% arcsin_less_mono
thf(fact_8244_cos__arccos__abs,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ Y4 ) @ one_one_real )
=> ( ( cos_real @ ( arccos @ Y4 ) )
= Y4 ) ) ).
% cos_arccos_abs
thf(fact_8245_fold__atLeastAtMost__nat_Osimps,axiom,
( set_fo2584398358068434914at_nat
= ( ^ [F3: nat > nat > nat,A4: nat,B3: nat,Acc: nat] : ( if_nat @ ( ord_less_nat @ B3 @ A4 ) @ Acc @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B3 @ ( F3 @ A4 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_8246_fold__atLeastAtMost__nat_Oelims,axiom,
! [X: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y4: nat] :
( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb2 @ Xc )
= Y4 )
=> ( ( ( ord_less_nat @ Xb2 @ Xa2 )
=> ( Y4 = Xc ) )
& ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
=> ( Y4
= ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_8247_norm__prod__diff,axiom,
! [I5: set_int,Z2: int > real,W2: int > real] :
( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z2 @ I5 ) @ ( groups2316167850115554303t_real @ W2 @ I5 ) ) )
@ ( groups8778361861064173332t_real
@ ^ [I4: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8248_norm__prod__diff,axiom,
! [I5: set_real,Z2: real > real,W2: real > real] :
( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z2 @ I5 ) @ ( groups1681761925125756287l_real @ W2 @ I5 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I4: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8249_norm__prod__diff,axiom,
! [I5: set_complex,Z2: complex > real,W2: complex > real] :
( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups766887009212190081x_real @ Z2 @ I5 ) @ ( groups766887009212190081x_real @ W2 @ I5 ) ) )
@ ( groups5808333547571424918x_real
@ ^ [I4: complex] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8250_norm__prod__diff,axiom,
! [I5: set_Pr1261947904930325089at_nat,Z2: product_prod_nat_nat > real,W2: product_prod_nat_nat > real] :
( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups6036352826371341000t_real @ Z2 @ I5 ) @ ( groups6036352826371341000t_real @ W2 @ I5 ) ) )
@ ( groups4567486121110086003t_real
@ ^ [I4: product_prod_nat_nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8251_norm__prod__diff,axiom,
! [I5: set_int,Z2: int > complex,W2: int > complex] :
( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z2 @ I5 ) @ ( groups7440179247065528705omplex @ W2 @ I5 ) ) )
@ ( groups8778361861064173332t_real
@ ^ [I4: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8252_norm__prod__diff,axiom,
! [I5: set_real,Z2: real > complex,W2: real > complex] :
( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: real] :
( ( member_real @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z2 @ I5 ) @ ( groups713298508707869441omplex @ W2 @ I5 ) ) )
@ ( groups8097168146408367636l_real
@ ^ [I4: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8253_norm__prod__diff,axiom,
! [I5: set_complex,Z2: complex > complex,W2: complex > complex] :
( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: complex] :
( ( member_complex @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups3708469109370488835omplex @ Z2 @ I5 ) @ ( groups3708469109370488835omplex @ W2 @ I5 ) ) )
@ ( groups5808333547571424918x_real
@ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8254_norm__prod__diff,axiom,
! [I5: set_Pr1261947904930325089at_nat,Z2: product_prod_nat_nat > complex,W2: product_prod_nat_nat > complex] :
( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups8110221916422527690omplex @ Z2 @ I5 ) @ ( groups8110221916422527690omplex @ W2 @ I5 ) ) )
@ ( groups4567486121110086003t_real
@ ^ [I4: product_prod_nat_nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8255_norm__prod__diff,axiom,
! [I5: set_nat,Z2: nat > real,W2: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W2 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z2 @ I5 ) @ ( groups129246275422532515t_real @ W2 @ I5 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z2 @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8256_norm__prod__diff,axiom,
! [I5: set_nat,Z2: nat > complex,W2: nat > complex] :
( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z2 @ I3 ) ) @ one_one_real ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W2 @ I3 ) ) @ one_one_real ) )
=> ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z2 @ I5 ) @ ( groups6464643781859351333omplex @ W2 @ I5 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z2 @ I4 ) @ ( W2 @ I4 ) ) )
@ I5 ) ) ) ) ).
% norm_prod_diff
thf(fact_8257_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1408675320244567234ct_nat @ M )
= ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
@ ( groups708209901874060359at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_8258_prod__mono2,axiom,
! [B5: set_int,A2: set_int,F: int > real] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A2 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8259_prod__mono2,axiom,
! [B5: set_real,A2: set_real,F: real > real] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8260_prod__mono2,axiom,
! [B5: set_complex,A2: set_complex,F: complex > real] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8261_prod__mono2,axiom,
! [B5: set_int,A2: set_int,F: int > rat] :
( ( finite_finite_int @ B5 )
=> ( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ! [B2: int] :
( ( member_int @ B2 @ ( minus_minus_set_int @ B5 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A3: int] :
( ( member_int @ A3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8262_prod__mono2,axiom,
! [B5: set_real,A2: set_real,F: real > rat] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8263_prod__mono2,axiom,
! [B5: set_complex,A2: set_complex,F: complex > rat] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8264_prod__mono2,axiom,
! [B5: set_real,A2: set_real,F: real > int] :
( ( finite_finite_real @ B5 )
=> ( ( ord_less_eq_set_real @ A2 @ B5 )
=> ( ! [B2: real] :
( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A2 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
=> ( ! [A3: real] :
( ( member_real @ A3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
=> ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8265_prod__mono2,axiom,
! [B5: set_complex,A2: set_complex,F: complex > int] :
( ( finite3207457112153483333omplex @ B5 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B5 )
=> ( ! [B2: complex] :
( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
=> ( ord_less_eq_int @ one_one_int @ ( F @ B2 ) ) )
=> ( ! [A3: complex] :
( ( member_complex @ A3 @ A2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
=> ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8266_prod__mono2,axiom,
! [B5: set_nat,A2: set_nat,F: nat > real] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A2 ) )
=> ( ord_less_eq_real @ one_one_real @ ( F @ B2 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
=> ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8267_prod__mono2,axiom,
! [B5: set_nat,A2: set_nat,F: nat > rat] :
( ( finite_finite_nat @ B5 )
=> ( ( ord_less_eq_set_nat @ A2 @ B5 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A2 ) )
=> ( ord_less_eq_rat @ one_one_rat @ ( F @ B2 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
=> ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) ) ) ) ) ).
% prod_mono2
thf(fact_8268_arccos__lt__bounded,axiom,
! [Y4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_real @ Y4 @ one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y4 ) )
& ( ord_less_real @ ( arccos @ Y4 ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_8269_arccos__bounded,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y4 ) )
& ( ord_less_eq_real @ ( arccos @ Y4 ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_8270_sin__arccos__nonzero,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ( sin_real @ ( arccos @ X ) )
!= zero_zero_real ) ) ) ).
% sin_arccos_nonzero
thf(fact_8271_arccos__minus,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( arccos @ ( uminus_uminus_real @ X ) )
= ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% arccos_minus
thf(fact_8272_cos__arcsin__nonzero,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ( cos_real @ ( arcsin @ X ) )
!= zero_zero_real ) ) ) ).
% cos_arcsin_nonzero
thf(fact_8273_pochhammer__Suc__prod,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8274_pochhammer__Suc__prod,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8275_pochhammer__Suc__prod,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8276_pochhammer__Suc__prod,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod
thf(fact_8277_pochhammer__prod__rev,axiom,
( comm_s7457072308508201937r_real
= ( ^ [A4: real,N2: nat] :
( groups129246275422532515t_real
@ ^ [I4: nat] : ( plus_plus_real @ A4 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8278_pochhammer__prod__rev,axiom,
( comm_s4028243227959126397er_rat
= ( ^ [A4: rat,N2: nat] :
( groups73079841787564623at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ A4 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8279_pochhammer__prod__rev,axiom,
( comm_s4663373288045622133er_nat
= ( ^ [A4: nat,N2: nat] :
( groups708209901874060359at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A4 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8280_pochhammer__prod__rev,axiom,
( comm_s4660882817536571857er_int
= ( ^ [A4: int,N2: nat] :
( groups705719431365010083at_int
@ ^ [I4: nat] : ( plus_plus_int @ A4 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_8281_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [X2: nat] : X2
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_8282_prod_Oin__pairs,axiom,
! [G: nat > real,M: nat,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups129246275422532515t_real
@ ^ [I4: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_8283_prod_Oin__pairs,axiom,
! [G: nat > rat,M: nat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_8284_prod_Oin__pairs,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_8285_prod_Oin__pairs,axiom,
! [G: nat > int,M: nat,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% prod.in_pairs
thf(fact_8286_sum__atLeastAtMost__code,axiom,
! [F: nat > complex,A: nat,B: nat] :
( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1517530859248394432omplex
@ ^ [A4: nat] : ( plus_plus_complex @ ( F @ A4 ) )
@ A
@ B
@ zero_zero_complex ) ) ).
% sum_atLeastAtMost_code
thf(fact_8287_sum__atLeastAtMost__code,axiom,
! [F: nat > rat,A: nat,B: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo1949268297981939178at_rat
@ ^ [A4: nat] : ( plus_plus_rat @ ( F @ A4 ) )
@ A
@ B
@ zero_zero_rat ) ) ).
% sum_atLeastAtMost_code
thf(fact_8288_sum__atLeastAtMost__code,axiom,
! [F: nat > int,A: nat,B: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2581907887559384638at_int
@ ^ [A4: nat] : ( plus_plus_int @ ( F @ A4 ) )
@ A
@ B
@ zero_zero_int ) ) ).
% sum_atLeastAtMost_code
thf(fact_8289_sum__atLeastAtMost__code,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo2584398358068434914at_nat
@ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
@ A
@ B
@ zero_zero_nat ) ) ).
% sum_atLeastAtMost_code
thf(fact_8290_sum__atLeastAtMost__code,axiom,
! [F: nat > real,A: nat,B: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
= ( set_fo3111899725591712190t_real
@ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
@ A
@ B
@ zero_zero_real ) ) ).
% sum_atLeastAtMost_code
thf(fact_8291_pochhammer__Suc__prod__rev,axiom,
! [A: real,N: nat] :
( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
= ( groups129246275422532515t_real
@ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8292_pochhammer__Suc__prod__rev,axiom,
! [A: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
= ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8293_pochhammer__Suc__prod__rev,axiom,
! [A: nat,N: nat] :
( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8294_pochhammer__Suc__prod__rev,axiom,
! [A: int,N: nat] :
( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I4 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_8295_arccos,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y4 ) )
& ( ord_less_eq_real @ ( arccos @ Y4 ) @ pi )
& ( ( cos_real @ ( arccos @ Y4 ) )
= Y4 ) ) ) ) ).
% arccos
thf(fact_8296_arccos__minus__abs,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( ( arccos @ ( uminus_uminus_real @ X ) )
= ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% arccos_minus_abs
thf(fact_8297_arccos__le__pi2,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_eq_real @ ( arccos @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_8298_arcsin__lt__bounded,axiom,
! [Y4: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_real @ Y4 @ one_one_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
& ( ord_less_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_8299_arcsin__lbound,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) ) ) ) ).
% arcsin_lbound
thf(fact_8300_arcsin__ubound,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ord_less_eq_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_8301_arcsin__bounded,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
& ( ord_less_eq_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_8302_arcsin__sin,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
=> ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( arcsin @ ( sin_real @ X ) )
= X ) ) ) ).
% arcsin_sin
thf(fact_8303_fact__code,axiom,
( semiri1406184849735516958ct_int
= ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_8304_fact__code,axiom,
( semiri773545260158071498ct_rat
= ( ^ [N2: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_8305_fact__code,axiom,
( semiri2265585572941072030t_real
= ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_8306_fact__code,axiom,
( semiri1408675320244567234ct_nat
= ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% fact_code
thf(fact_8307_arcsin,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
& ( ord_less_eq_real @ ( arcsin @ Y4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ ( arcsin @ Y4 ) )
= Y4 ) ) ) ) ).
% arcsin
thf(fact_8308_arcsin__pi,axiom,
! [Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ one_one_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y4 ) )
& ( ord_less_eq_real @ ( arcsin @ Y4 ) @ pi )
& ( ( sin_real @ ( arcsin @ Y4 ) )
= Y4 ) ) ) ) ).
% arcsin_pi
thf(fact_8309_arcsin__le__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y4 )
= ( ord_less_eq_real @ X @ ( sin_real @ Y4 ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_8310_le__arcsin__iff,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_real @ Y4 @ ( arcsin @ X ) )
= ( ord_less_eq_real @ ( sin_real @ Y4 ) @ X ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_8311_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K2: int] :
( ( arccos @ ( cos_real @ Theta ) )
!= ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_8312_sin__arccos,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ( sin_real @ ( arccos @ X ) )
= ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_8313_gchoose__row__sum__weighted,axiom,
! [R2: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_8314_gchoose__row__sum__weighted,axiom,
! [R2: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_8315_gchoose__row__sum__weighted,axiom,
! [R2: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
= ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_8316_central__binomial__lower__bound,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_8317_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq_real
@ ( abs_abs_real
@ ( minus_minus_real @ ( sin_real @ X )
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) )
@ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_8318_divmod__BitM__2__eq,axiom,
! [M: num] :
( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
= ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% divmod_BitM_2_eq
thf(fact_8319_cot__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% cot_less_zero
thf(fact_8320_inverse__mult__distrib,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).
% inverse_mult_distrib
thf(fact_8321_inverse__mult__distrib,axiom,
! [A: complex,B: complex] :
( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
= ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) ) ) ).
% inverse_mult_distrib
thf(fact_8322_inverse__mult__distrib,axiom,
! [A: rat,B: rat] :
( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) ) ) ).
% inverse_mult_distrib
thf(fact_8323_inverse__1,axiom,
( ( inverse_inverse_real @ one_one_real )
= one_one_real ) ).
% inverse_1
thf(fact_8324_inverse__1,axiom,
( ( invers8013647133539491842omplex @ one_one_complex )
= one_one_complex ) ).
% inverse_1
thf(fact_8325_inverse__1,axiom,
( ( inverse_inverse_rat @ one_one_rat )
= one_one_rat ) ).
% inverse_1
thf(fact_8326_inverse__eq__1__iff,axiom,
! [X: real] :
( ( ( inverse_inverse_real @ X )
= one_one_real )
= ( X = one_one_real ) ) ).
% inverse_eq_1_iff
thf(fact_8327_inverse__eq__1__iff,axiom,
! [X: complex] :
( ( ( invers8013647133539491842omplex @ X )
= one_one_complex )
= ( X = one_one_complex ) ) ).
% inverse_eq_1_iff
thf(fact_8328_inverse__eq__1__iff,axiom,
! [X: rat] :
( ( ( inverse_inverse_rat @ X )
= one_one_rat )
= ( X = one_one_rat ) ) ).
% inverse_eq_1_iff
thf(fact_8329_inverse__divide,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ B @ A ) ) ).
% inverse_divide
thf(fact_8330_inverse__divide,axiom,
! [A: complex,B: complex] :
( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
= ( divide1717551699836669952omplex @ B @ A ) ) ).
% inverse_divide
thf(fact_8331_inverse__divide,axiom,
! [A: rat,B: rat] :
( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B ) )
= ( divide_divide_rat @ B @ A ) ) ).
% inverse_divide
thf(fact_8332_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_8333_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
thf(fact_8334_inverse__nonnegative__iff__nonnegative,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_8335_inverse__nonnegative__iff__nonnegative,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
= ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_8336_inverse__nonpositive__iff__nonpositive,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_8337_inverse__nonpositive__iff__nonpositive,axiom,
! [A: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
= ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_8338_inverse__less__iff__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_8339_inverse__less__iff__less,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_8340_inverse__less__iff__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_8341_inverse__less__iff__less__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_rat @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_8342_inverse__negative__iff__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_8343_inverse__negative__iff__negative,axiom,
! [A: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
= ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% inverse_negative_iff_negative
thf(fact_8344_inverse__positive__iff__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_8345_inverse__positive__iff__positive,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
= ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_8346_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
= zero_zero_complex ) ).
% gbinomial_0(2)
thf(fact_8347_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
= zero_zero_real ) ).
% gbinomial_0(2)
thf(fact_8348_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
= zero_zero_rat ) ).
% gbinomial_0(2)
thf(fact_8349_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_8350_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
= zero_zero_int ) ).
% gbinomial_0(2)
thf(fact_8351_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_8352_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_8353_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= zero_zero_nat )
= ( ord_less_nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_8354_gbinomial__0_I1_J,axiom,
! [A: complex] :
( ( gbinomial_complex @ A @ zero_zero_nat )
= one_one_complex ) ).
% gbinomial_0(1)
thf(fact_8355_gbinomial__0_I1_J,axiom,
! [A: real] :
( ( gbinomial_real @ A @ zero_zero_nat )
= one_one_real ) ).
% gbinomial_0(1)
thf(fact_8356_gbinomial__0_I1_J,axiom,
! [A: rat] :
( ( gbinomial_rat @ A @ zero_zero_nat )
= one_one_rat ) ).
% gbinomial_0(1)
thf(fact_8357_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_8358_gbinomial__0_I1_J,axiom,
! [A: int] :
( ( gbinomial_int @ A @ zero_zero_nat )
= one_one_int ) ).
% gbinomial_0(1)
thf(fact_8359_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_8360_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_8361_prod__eq__1__iff,axiom,
! [A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( ( groups861055069439313189ex_nat @ F @ A2 )
= one_one_nat )
= ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ( F @ X2 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_8362_prod__eq__1__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups708209901874060359at_nat @ F @ A2 )
= one_one_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_8363_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
= ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_8364_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
= ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_8365_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
= ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_8366_dbl__dec__simps_I5_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% dbl_dec_simps(5)
thf(fact_8367_inverse__le__iff__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_8368_inverse__le__iff__le__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_8369_inverse__le__iff__le,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_8370_inverse__le__iff__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ zero_zero_rat @ B )
=> ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_8371_right__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
= one_one_real ) ) ).
% right_inverse
thf(fact_8372_right__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
= one_one_complex ) ) ).
% right_inverse
thf(fact_8373_right__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
= one_one_rat ) ) ).
% right_inverse
thf(fact_8374_left__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% left_inverse
thf(fact_8375_left__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
= one_one_complex ) ) ).
% left_inverse
thf(fact_8376_left__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
= one_one_rat ) ) ).
% left_inverse
thf(fact_8377_inverse__eq__divide__numeral,axiom,
! [W2: num] :
( ( inverse_inverse_real @ ( numeral_numeral_real @ W2 ) )
= ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W2 ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_8378_inverse__eq__divide__numeral,axiom,
! [W2: num] :
( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W2 ) )
= ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W2 ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_8379_inverse__eq__divide__numeral,axiom,
! [W2: num] :
( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W2 ) )
= ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W2 ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_8380_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
= ( ord_less_eq_nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_8381_prod__pos__nat__iff,axiom,
! [A2: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) )
= ( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_8382_prod__pos__nat__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_8383_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_8384_inverse__eq__divide__neg__numeral,axiom,
! [W2: num] :
( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
= ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_8385_inverse__eq__divide__neg__numeral,axiom,
! [W2: num] :
( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
= ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_8386_inverse__eq__divide__neg__numeral,axiom,
! [W2: num] :
( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
= ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_8387_cot__npi,axiom,
! [N: nat] :
( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
= zero_zero_real ) ).
% cot_npi
thf(fact_8388_cot__periodic,axiom,
! [X: real] :
( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
= ( cot_real @ X ) ) ).
% cot_periodic
thf(fact_8389_mult__commute__imp__mult__inverse__commute,axiom,
! [Y4: real,X: real] :
( ( ( times_times_real @ Y4 @ X )
= ( times_times_real @ X @ Y4 ) )
=> ( ( times_times_real @ ( inverse_inverse_real @ Y4 ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ Y4 ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_8390_mult__commute__imp__mult__inverse__commute,axiom,
! [Y4: complex,X: complex] :
( ( ( times_times_complex @ Y4 @ X )
= ( times_times_complex @ X @ Y4 ) )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y4 ) @ X )
= ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y4 ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_8391_mult__commute__imp__mult__inverse__commute,axiom,
! [Y4: rat,X: rat] :
( ( ( times_times_rat @ Y4 @ X )
= ( times_times_rat @ X @ Y4 ) )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ Y4 ) @ X )
= ( times_times_rat @ X @ ( inverse_inverse_rat @ Y4 ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_8392_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ one_one_nat )
= N ) ).
% choose_one
thf(fact_8393_power__inverse,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
= ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).
% power_inverse
thf(fact_8394_power__inverse,axiom,
! [A: complex,N: nat] :
( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N )
= ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N ) ) ) ).
% power_inverse
thf(fact_8395_power__inverse,axiom,
! [A: rat,N: nat] :
( ( power_power_rat @ ( inverse_inverse_rat @ A ) @ N )
= ( inverse_inverse_rat @ ( power_power_rat @ A @ N ) ) ) ).
% power_inverse
thf(fact_8396_semiring__norm_I26_J,axiom,
( ( bitM @ one )
= one ) ).
% semiring_norm(26)
thf(fact_8397_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_8398_inverse__less__imp__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less
thf(fact_8399_inverse__less__imp__less,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ B @ A ) ) ) ).
% inverse_less_imp_less
thf(fact_8400_less__imp__inverse__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less
thf(fact_8401_less__imp__inverse__less,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% less_imp_inverse_less
thf(fact_8402_inverse__less__imp__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_8403_inverse__less__imp__less__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ B @ A ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_8404_less__imp__inverse__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_8405_less__imp__inverse__less__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ A @ B )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_8406_inverse__negative__imp__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
=> ( ( A != zero_zero_real )
=> ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% inverse_negative_imp_negative
thf(fact_8407_inverse__negative__imp__negative,axiom,
! [A: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
=> ( ( A != zero_zero_rat )
=> ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% inverse_negative_imp_negative
thf(fact_8408_inverse__positive__imp__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
=> ( ( A != zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ A ) ) ) ).
% inverse_positive_imp_positive
thf(fact_8409_inverse__positive__imp__positive,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
=> ( ( A != zero_zero_rat )
=> ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).
% inverse_positive_imp_positive
thf(fact_8410_negative__imp__inverse__negative,axiom,
! [A: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).
% negative_imp_inverse_negative
thf(fact_8411_negative__imp__inverse__negative,axiom,
! [A: rat] :
( ( ord_less_rat @ A @ zero_zero_rat )
=> ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).
% negative_imp_inverse_negative
thf(fact_8412_positive__imp__inverse__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).
% positive_imp_inverse_positive
thf(fact_8413_positive__imp__inverse__positive,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).
% positive_imp_inverse_positive
thf(fact_8414_nonzero__inverse__mult__distrib,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
= ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_8415_nonzero__inverse__mult__distrib,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
= ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_8416_nonzero__inverse__mult__distrib,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
= ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_8417_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_8418_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_8419_inverse__numeral__1,axiom,
( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
= ( numeral_numeral_real @ one ) ) ).
% inverse_numeral_1
thf(fact_8420_inverse__numeral__1,axiom,
( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
= ( numera6690914467698888265omplex @ one ) ) ).
% inverse_numeral_1
thf(fact_8421_inverse__numeral__1,axiom,
( ( inverse_inverse_rat @ ( numeral_numeral_rat @ one ) )
= ( numeral_numeral_rat @ one ) ) ).
% inverse_numeral_1
thf(fact_8422_inverse__unique,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= one_one_real )
=> ( ( inverse_inverse_real @ A )
= B ) ) ).
% inverse_unique
thf(fact_8423_inverse__unique,axiom,
! [A: complex,B: complex] :
( ( ( times_times_complex @ A @ B )
= one_one_complex )
=> ( ( invers8013647133539491842omplex @ A )
= B ) ) ).
% inverse_unique
thf(fact_8424_inverse__unique,axiom,
! [A: rat,B: rat] :
( ( ( times_times_rat @ A @ B )
= one_one_rat )
=> ( ( inverse_inverse_rat @ A )
= B ) ) ).
% inverse_unique
thf(fact_8425_field__class_Ofield__divide__inverse,axiom,
( divide_divide_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ A4 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_8426_field__class_Ofield__divide__inverse,axiom,
( divide1717551699836669952omplex
= ( ^ [A4: complex,B3: complex] : ( times_times_complex @ A4 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_8427_field__class_Ofield__divide__inverse,axiom,
( divide_divide_rat
= ( ^ [A4: rat,B3: rat] : ( times_times_rat @ A4 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_8428_divide__inverse,axiom,
( divide_divide_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ A4 @ ( inverse_inverse_real @ B3 ) ) ) ) ).
% divide_inverse
thf(fact_8429_divide__inverse,axiom,
( divide1717551699836669952omplex
= ( ^ [A4: complex,B3: complex] : ( times_times_complex @ A4 @ ( invers8013647133539491842omplex @ B3 ) ) ) ) ).
% divide_inverse
thf(fact_8430_divide__inverse,axiom,
( divide_divide_rat
= ( ^ [A4: rat,B3: rat] : ( times_times_rat @ A4 @ ( inverse_inverse_rat @ B3 ) ) ) ) ).
% divide_inverse
thf(fact_8431_divide__inverse__commute,axiom,
( divide_divide_real
= ( ^ [A4: real,B3: real] : ( times_times_real @ ( inverse_inverse_real @ B3 ) @ A4 ) ) ) ).
% divide_inverse_commute
thf(fact_8432_divide__inverse__commute,axiom,
( divide1717551699836669952omplex
= ( ^ [A4: complex,B3: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B3 ) @ A4 ) ) ) ).
% divide_inverse_commute
thf(fact_8433_divide__inverse__commute,axiom,
( divide_divide_rat
= ( ^ [A4: rat,B3: rat] : ( times_times_rat @ ( inverse_inverse_rat @ B3 ) @ A4 ) ) ) ).
% divide_inverse_commute
thf(fact_8434_inverse__eq__divide,axiom,
( inverse_inverse_real
= ( divide_divide_real @ one_one_real ) ) ).
% inverse_eq_divide
thf(fact_8435_inverse__eq__divide,axiom,
( invers8013647133539491842omplex
= ( divide1717551699836669952omplex @ one_one_complex ) ) ).
% inverse_eq_divide
thf(fact_8436_inverse__eq__divide,axiom,
( inverse_inverse_rat
= ( divide_divide_rat @ one_one_rat ) ) ).
% inverse_eq_divide
thf(fact_8437_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_8438_power__mult__power__inverse__commute,axiom,
! [X: real,M: nat,N: nat] :
( ( times_times_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) )
= ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X ) @ N ) @ ( power_power_real @ X @ M ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_8439_power__mult__power__inverse__commute,axiom,
! [X: complex,M: nat,N: nat] :
( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) )
= ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ N ) @ ( power_power_complex @ X @ M ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_8440_power__mult__power__inverse__commute,axiom,
! [X: rat,M: nat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) )
= ( times_times_rat @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ N ) @ ( power_power_rat @ X @ M ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_8441_power__mult__inverse__distrib,axiom,
! [X: real,M: nat] :
( ( times_times_real @ ( power_power_real @ X @ M ) @ ( inverse_inverse_real @ X ) )
= ( times_times_real @ ( inverse_inverse_real @ X ) @ ( power_power_real @ X @ M ) ) ) ).
% power_mult_inverse_distrib
thf(fact_8442_power__mult__inverse__distrib,axiom,
! [X: complex,M: nat] :
( ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( invers8013647133539491842omplex @ X ) )
= ( times_times_complex @ ( invers8013647133539491842omplex @ X ) @ ( power_power_complex @ X @ M ) ) ) ).
% power_mult_inverse_distrib
thf(fact_8443_power__mult__inverse__distrib,axiom,
! [X: rat,M: nat] :
( ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( inverse_inverse_rat @ X ) )
= ( times_times_rat @ ( inverse_inverse_rat @ X ) @ ( power_power_rat @ X @ M ) ) ) ).
% power_mult_inverse_distrib
thf(fact_8444_choose__mult__lemma,axiom,
! [M: nat,R2: nat,K: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_8445_mult__inverse__of__nat__commute,axiom,
! [Xa2: nat,X: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa2 ) ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa2 ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_8446_mult__inverse__of__nat__commute,axiom,
! [Xa2: nat,X: complex] :
( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa2 ) ) @ X )
= ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa2 ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_8447_mult__inverse__of__nat__commute,axiom,
! [Xa2: nat,X: rat] :
( ( times_times_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa2 ) ) @ X )
= ( times_times_rat @ X @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ Xa2 ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_8448_binomial__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq_nat @ R2 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_8449_mult__inverse__of__int__commute,axiom,
! [Xa2: int,X: real] :
( ( times_times_real @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa2 ) ) @ X )
= ( times_times_real @ X @ ( inverse_inverse_real @ ( ring_1_of_int_real @ Xa2 ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_8450_mult__inverse__of__int__commute,axiom,
! [Xa2: int,X: complex] :
( ( times_times_complex @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa2 ) ) @ X )
= ( times_times_complex @ X @ ( invers8013647133539491842omplex @ ( ring_17405671764205052669omplex @ Xa2 ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_8451_mult__inverse__of__int__commute,axiom,
! [Xa2: int,X: rat] :
( ( times_times_rat @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa2 ) ) @ X )
= ( times_times_rat @ X @ ( inverse_inverse_rat @ ( ring_1_of_int_rat @ Xa2 ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_8452_divide__real__def,axiom,
( divide_divide_real
= ( ^ [X2: real,Y: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y ) ) ) ) ).
% divide_real_def
thf(fact_8453_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_8454_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_8455_le__imp__inverse__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_8456_le__imp__inverse__le__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_8457_inverse__le__imp__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_8458_inverse__le__imp__le__neg,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
=> ( ( ord_less_rat @ B @ zero_zero_rat )
=> ( ord_less_eq_rat @ B @ A ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_8459_le__imp__inverse__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% le_imp_inverse_le
thf(fact_8460_le__imp__inverse__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ A @ B )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).
% le_imp_inverse_le
thf(fact_8461_inverse__le__imp__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ B @ A ) ) ) ).
% inverse_le_imp_le
thf(fact_8462_inverse__le__imp__le,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
=> ( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ord_less_eq_rat @ B @ A ) ) ) ).
% inverse_le_imp_le
thf(fact_8463_inverse__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% inverse_le_1_iff
thf(fact_8464_inverse__le__1__iff,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
= ( ( ord_less_eq_rat @ X @ zero_zero_rat )
| ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% inverse_le_1_iff
thf(fact_8465_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_8466_one__less__inverse,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ A @ one_one_real )
=> ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% one_less_inverse
thf(fact_8467_one__less__inverse,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_rat @ A @ one_one_rat )
=> ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% one_less_inverse
thf(fact_8468_one__less__inverse__iff,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_real @ X @ one_one_real ) ) ) ).
% one_less_inverse_iff
thf(fact_8469_one__less__inverse__iff,axiom,
! [X: rat] :
( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
= ( ( ord_less_rat @ zero_zero_rat @ X )
& ( ord_less_rat @ X @ one_one_rat ) ) ) ).
% one_less_inverse_iff
thf(fact_8470_field__class_Ofield__inverse,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
= one_one_real ) ) ).
% field_class.field_inverse
thf(fact_8471_field__class_Ofield__inverse,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
= one_one_complex ) ) ).
% field_class.field_inverse
thf(fact_8472_field__class_Ofield__inverse,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
= one_one_rat ) ) ).
% field_class.field_inverse
thf(fact_8473_inverse__add,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% inverse_add
thf(fact_8474_inverse__add,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
= ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% inverse_add
thf(fact_8475_inverse__add,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% inverse_add
thf(fact_8476_division__ring__inverse__add,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_8477_division__ring__inverse__add,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
= ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_8478_division__ring__inverse__add,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_8479_division__ring__inverse__diff,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_8480_division__ring__inverse__diff,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
= ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_8481_division__ring__inverse__diff,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_8482_nonzero__inverse__eq__divide,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
= ( divide_divide_real @ one_one_real @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_8483_nonzero__inverse__eq__divide,axiom,
! [A: complex] :
( ( A != zero_zero_complex )
=> ( ( invers8013647133539491842omplex @ A )
= ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_8484_nonzero__inverse__eq__divide,axiom,
! [A: rat] :
( ( A != zero_zero_rat )
=> ( ( inverse_inverse_rat @ A )
= ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_8485_Suc__times__binomial__add,axiom,
! [A: nat,B: nat] :
( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
= ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% Suc_times_binomial_add
thf(fact_8486_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_8487_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_8488_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_8489_gbinomial__Suc__Suc,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
= ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_8490_gbinomial__Suc__Suc,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
= ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_8491_gbinomial__Suc__Suc,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
= ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_8492_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
= ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_8493_gbinomial__of__nat__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ K )
= ( gbinomial_rat @ ( semiri681578069525770553at_rat @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_8494_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_8495_inverse__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_real @ B @ A ) )
& ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_real @ A @ B ) ) ) ) ).
% inverse_less_iff
thf(fact_8496_inverse__less__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_rat @ B @ A ) )
& ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
=> ( ord_less_rat @ A @ B ) ) ) ) ).
% inverse_less_iff
thf(fact_8497_inverse__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) )
& ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
=> ( ord_less_eq_real @ A @ B ) ) ) ) ).
% inverse_le_iff
thf(fact_8498_inverse__le__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
=> ( ord_less_eq_rat @ B @ A ) )
& ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
=> ( ord_less_eq_rat @ A @ B ) ) ) ) ).
% inverse_le_iff
thf(fact_8499_one__le__inverse__iff,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X ) )
= ( ( ord_less_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% one_le_inverse_iff
thf(fact_8500_one__le__inverse__iff,axiom,
! [X: rat] :
( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X ) )
= ( ( ord_less_rat @ zero_zero_rat @ X )
& ( ord_less_eq_rat @ X @ one_one_rat ) ) ) ).
% one_le_inverse_iff
thf(fact_8501_inverse__less__1__iff,axiom,
! [X: real] :
( ( ord_less_real @ ( inverse_inverse_real @ X ) @ one_one_real )
= ( ( ord_less_eq_real @ X @ zero_zero_real )
| ( ord_less_real @ one_one_real @ X ) ) ) ).
% inverse_less_1_iff
thf(fact_8502_inverse__less__1__iff,axiom,
! [X: rat] :
( ( ord_less_rat @ ( inverse_inverse_rat @ X ) @ one_one_rat )
= ( ( ord_less_eq_rat @ X @ zero_zero_rat )
| ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% inverse_less_1_iff
thf(fact_8503_one__le__inverse,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).
% one_le_inverse
thf(fact_8504_one__le__inverse,axiom,
! [A: rat] :
( ( ord_less_rat @ zero_zero_rat @ A )
=> ( ( ord_less_eq_rat @ A @ one_one_rat )
=> ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).
% one_le_inverse
thf(fact_8505_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_8506_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_8507_inverse__diff__inverse,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_8508_inverse__diff__inverse,axiom,
! [A: complex,B: complex] :
( ( A != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
= ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_8509_inverse__diff__inverse,axiom,
! [A: rat,B: rat] :
( ( A != zero_zero_rat )
=> ( ( B != zero_zero_rat )
=> ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
= ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_8510_reals__Archimedean,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N3: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ X ) ) ).
% reals_Archimedean
thf(fact_8511_reals__Archimedean,axiom,
! [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N3: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ X ) ) ).
% reals_Archimedean
thf(fact_8512_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorption
thf(fact_8513_gbinomial__addition__formula,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ A @ ( suc @ K ) )
= ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% gbinomial_addition_formula
thf(fact_8514_gbinomial__addition__formula,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ A @ ( suc @ K ) )
= ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% gbinomial_addition_formula
thf(fact_8515_gbinomial__addition__formula,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ A @ ( suc @ K ) )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% gbinomial_addition_formula
thf(fact_8516_gbinomial__absorb__comp,axiom,
! [A: complex,K: nat] :
( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
= ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_8517_gbinomial__absorb__comp,axiom,
! [A: real,K: nat] :
( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
= ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_8518_gbinomial__absorb__comp,axiom,
! [A: rat,K: nat] :
( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
= ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% gbinomial_absorb_comp
thf(fact_8519_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
=> ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_8520_gbinomial__ge__n__over__k__pow__k,axiom,
! [K: nat,A: rat] :
( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
=> ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_8521_gbinomial__mult__1,axiom,
! [A: real,K: nat] :
( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
= ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_8522_gbinomial__mult__1,axiom,
! [A: rat,K: nat] :
( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_8523_gbinomial__mult__1_H,axiom,
! [A: real,K: nat] :
( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
= ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_8524_gbinomial__mult__1_H,axiom,
! [A: rat,K: nat] :
( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
= ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_8525_forall__pos__mono__1,axiom,
! [P: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less_real @ D3 @ E2 )
=> ( ( P @ D3 )
=> ( P @ E2 ) ) )
=> ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E )
=> ( P @ E ) ) ) ) ).
% forall_pos_mono_1
thf(fact_8526_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_8527_prod__int__plus__eq,axiom,
! [I2: nat,J: nat] :
( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J ) ) )
= ( groups1705073143266064639nt_int
@ ^ [X2: int] : X2
@ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_8528_summable__exp,axiom,
! [X: complex] :
( summable_complex
@ ^ [N2: nat] : ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri5044797733671781792omplex @ N2 ) ) @ ( power_power_complex @ X @ N2 ) ) ) ).
% summable_exp
thf(fact_8529_summable__exp,axiom,
! [X: real] :
( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ).
% summable_exp
thf(fact_8530_binomial__ge__n__over__k__pow__k,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_8531_binomial__ge__n__over__k__pow__k,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_8532_binomial__mono,axiom,
! [K: nat,K6: nat,N: nat] :
( ( ord_less_eq_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% binomial_mono
thf(fact_8533_binomial__maximum_H,axiom,
! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_8534_binomial__maximum,axiom,
! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% binomial_maximum
thf(fact_8535_binomial__antimono,axiom,
! [K: nat,K6: nat,N: nat] :
( ( ord_less_eq_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
=> ( ( ord_less_eq_nat @ K6 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_8536_binomial__le__pow2,axiom,
! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_8537_numeral__BitM,axiom,
! [N: num] :
( ( numera6690914467698888265omplex @ ( bitM @ N ) )
= ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% numeral_BitM
thf(fact_8538_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bitM @ N ) )
= ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% numeral_BitM
thf(fact_8539_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_rat @ ( bitM @ N ) )
= ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% numeral_BitM
thf(fact_8540_numeral__BitM,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bitM @ N ) )
= ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% numeral_BitM
thf(fact_8541_odd__numeral__BitM,axiom,
! [W2: num] :
~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W2 ) ) ) ).
% odd_numeral_BitM
thf(fact_8542_odd__numeral__BitM,axiom,
! [W2: num] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W2 ) ) ) ).
% odd_numeral_BitM
thf(fact_8543_odd__numeral__BitM,axiom,
! [W2: num] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W2 ) ) ) ).
% odd_numeral_BitM
thf(fact_8544_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_8545_ex__inverse__of__nat__less,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_8546_ex__inverse__of__nat__less,axiom,
! [X: rat] :
( ( ord_less_rat @ zero_zero_rat @ X )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N3 ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_8547_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_8548_power__diff__conv__inverse,axiom,
! [X: real,M: nat,N: nat] :
( ( X != zero_zero_real )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( power_power_real @ X @ ( minus_minus_nat @ N @ M ) )
= ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ ( inverse_inverse_real @ X ) @ M ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_8549_power__diff__conv__inverse,axiom,
! [X: complex,M: nat,N: nat] :
( ( X != zero_zero_complex )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( power_power_complex @ X @ ( minus_minus_nat @ N @ M ) )
= ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X ) @ M ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_8550_power__diff__conv__inverse,axiom,
! [X: rat,M: nat,N: nat] :
( ( X != zero_zero_rat )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( power_power_rat @ X @ ( minus_minus_nat @ N @ M ) )
= ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ ( inverse_inverse_rat @ X ) @ M ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_8551_cot__def,axiom,
( cot_complex
= ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X2 ) @ ( sin_complex @ X2 ) ) ) ) ).
% cot_def
thf(fact_8552_cot__def,axiom,
( cot_real
= ( ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) ) ) ) ).
% cot_def
thf(fact_8553_Suc__times__gbinomial,axiom,
! [K: nat,A: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
= ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_8554_Suc__times__gbinomial,axiom,
! [K: nat,A: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
= ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_8555_Suc__times__gbinomial,axiom,
! [K: nat,A: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
= ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% Suc_times_gbinomial
thf(fact_8556_gbinomial__absorption,axiom,
! [K: nat,A: complex] :
( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
= ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_8557_gbinomial__absorption,axiom,
! [K: nat,A: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
= ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_8558_gbinomial__absorption,axiom,
! [K: nat,A: rat] :
( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
= ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% gbinomial_absorption
thf(fact_8559_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_8560_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A: real] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
= ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_8561_gbinomial__trinomial__revision,axiom,
! [K: nat,M: nat,A: rat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
= ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_8562_binomial__less__binomial__Suc,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_8563_binomial__strict__mono,axiom,
! [K: nat,K6: nat,N: nat] :
( ( ord_less_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
=> ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% binomial_strict_mono
thf(fact_8564_binomial__strict__antimono,axiom,
! [K: nat,K6: nat,N: nat] :
( ( ord_less_nat @ K @ K6 )
=> ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
=> ( ( ord_less_eq_nat @ K6 @ N )
=> ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_8565_central__binomial__odd,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_8566_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_8567_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_8568_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_8569_fact__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% fact_binomial
thf(fact_8570_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
= ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_8571_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
= ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_8572_binomial__fact,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
= ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% binomial_fact
thf(fact_8573_gbinomial__rec,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
= ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_8574_gbinomial__rec,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
= ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_8575_gbinomial__rec,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
= ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% gbinomial_rec
thf(fact_8576_gbinomial__factors,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% gbinomial_factors
thf(fact_8577_gbinomial__factors,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
= ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% gbinomial_factors
thf(fact_8578_gbinomial__factors,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
= ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% gbinomial_factors
thf(fact_8579_gbinomial__negated__upper,axiom,
( gbinomial_complex
= ( ^ [A4: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A4 ) @ one_one_complex ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_8580_gbinomial__negated__upper,axiom,
( gbinomial_real
= ( ^ [A4: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A4 ) @ one_one_real ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_8581_gbinomial__negated__upper,axiom,
( gbinomial_rat
= ( ^ [A4: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A4 ) @ one_one_rat ) @ K3 ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_8582_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_8583_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_8584_gbinomial__index__swap,axiom,
! [K: nat,N: nat] :
( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).
% gbinomial_index_swap
thf(fact_8585_exp__plus__inverse__exp,axiom,
! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_8586_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% choose_two
thf(fact_8587_gbinomial__minus,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_8588_gbinomial__minus,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_8589_gbinomial__minus,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% gbinomial_minus
thf(fact_8590_plus__inverse__ge__2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_8591_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( inverse_inverse_real @ X ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_8592_gbinomial__reduce__nat,axiom,
! [K: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_complex @ A @ K )
= ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_8593_gbinomial__reduce__nat,axiom,
! [K: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_real @ A @ K )
= ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_8594_gbinomial__reduce__nat,axiom,
! [K: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_rat @ A @ K )
= ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_8595_gbinomial__pochhammer,axiom,
( gbinomial_complex
= ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A4 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_8596_gbinomial__pochhammer,axiom,
( gbinomial_rat
= ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A4 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_8597_gbinomial__pochhammer,axiom,
( gbinomial_real
= ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A4 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_8598_gbinomial__pochhammer_H,axiom,
( gbinomial_complex
= ( ^ [A4: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_8599_gbinomial__pochhammer_H,axiom,
( gbinomial_rat
= ( ^ [A4: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_8600_gbinomial__pochhammer_H,axiom,
( gbinomial_real
= ( ^ [A4: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_8601_tan__cot,axiom,
! [X: real] :
( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
= ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% tan_cot
thf(fact_8602_real__le__x__sinh,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_8603_real__le__abs__sinh,axiom,
! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_8604_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] :
( ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8605_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] :
( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8606_gbinomial__sum__up__index,axiom,
! [K: nat,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
= ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% gbinomial_sum_up_index
thf(fact_8607_gbinomial__Suc,axiom,
! [A: complex,K: nat] :
( ( gbinomial_complex @ A @ ( suc @ K ) )
= ( divide1717551699836669952omplex
@ ( groups6464643781859351333omplex
@ ^ [I4: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8608_gbinomial__Suc,axiom,
! [A: rat,K: nat] :
( ( gbinomial_rat @ A @ ( suc @ K ) )
= ( divide_divide_rat
@ ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8609_gbinomial__Suc,axiom,
! [A: real,K: nat] :
( ( gbinomial_real @ A @ ( suc @ K ) )
= ( divide_divide_real
@ ( groups129246275422532515t_real
@ ^ [I4: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8610_gbinomial__Suc,axiom,
! [A: nat,K: nat] :
( ( gbinomial_nat @ A @ ( suc @ K ) )
= ( divide_divide_nat
@ ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8611_gbinomial__Suc,axiom,
! [A: int,K: nat] :
( ( gbinomial_int @ A @ ( suc @ K ) )
= ( divide_divide_int
@ ( groups705719431365010083at_int
@ ^ [I4: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
@ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).
% gbinomial_Suc
thf(fact_8612_tan__sec,axiom,
! [X: real] :
( ( ( cos_real @ X )
!= zero_zero_real )
=> ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% tan_sec
thf(fact_8613_tan__sec,axiom,
! [X: complex] :
( ( ( cos_complex @ X )
!= zero_zero_complex )
=> ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% tan_sec
thf(fact_8614_gbinomial__absorption_H,axiom,
! [K: nat,A: complex] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_complex @ A @ K )
= ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_8615_gbinomial__absorption_H,axiom,
! [K: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_real @ A @ K )
= ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_8616_gbinomial__absorption_H,axiom,
! [K: nat,A: rat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( gbinomial_rat @ A @ K )
= ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_8617_cot__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
=> ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% cot_gt_zero
thf(fact_8618_tan__cot_H,axiom,
! [X: real] :
( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
= ( cot_real @ X ) ) ).
% tan_cot'
thf(fact_8619_binomial__code,axiom,
( binomial
= ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_8620_gbinomial__code,axiom,
( gbinomial_complex
= ( ^ [A4: complex,K3: nat] :
( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
@ ( divide1717551699836669952omplex
@ ( set_fo1517530859248394432omplex
@ ^ [L: nat] : ( times_times_complex @ ( minus_minus_complex @ A4 @ ( semiri8010041392384452111omplex @ L ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_complex )
@ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_8621_gbinomial__code,axiom,
( gbinomial_rat
= ( ^ [A4: rat,K3: nat] :
( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
@ ( divide_divide_rat
@ ( set_fo1949268297981939178at_rat
@ ^ [L: nat] : ( times_times_rat @ ( minus_minus_rat @ A4 @ ( semiri681578069525770553at_rat @ L ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_rat )
@ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_8622_gbinomial__code,axiom,
( gbinomial_real
= ( ^ [A4: real,K3: nat] :
( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
@ ( divide_divide_real
@ ( set_fo3111899725591712190t_real
@ ^ [L: nat] : ( times_times_real @ ( minus_minus_real @ A4 @ ( semiri5074537144036343181t_real @ L ) ) )
@ zero_zero_nat
@ ( minus_minus_nat @ K3 @ one_one_nat )
@ one_one_real )
@ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).
% gbinomial_code
thf(fact_8623_gbinomial__partial__row__sum,axiom,
! [A: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8624_gbinomial__partial__row__sum,axiom,
! [A: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8625_gbinomial__partial__row__sum,axiom,
! [A: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_8626_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) @ zero_zero_complex )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8627_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) @ zero_zero_int )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8628_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) @ zero_zero_rat )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8629_choose__even__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) @ zero_zero_real )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_even_sum
thf(fact_8630_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] :
( if_complex
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
@ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8631_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] :
( if_int
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
@ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) )
@ zero_zero_int )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8632_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] :
( if_rat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
@ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) )
@ zero_zero_rat )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8633_choose__odd__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] :
( if_real
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
@ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ N ) ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% choose_odd_sum
thf(fact_8634_gbinomial__r__part__sum,axiom,
! [M: nat] :
( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% gbinomial_r_part_sum
thf(fact_8635_gbinomial__r__part__sum,axiom,
! [M: nat] :
( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% gbinomial_r_part_sum
thf(fact_8636_gbinomial__r__part__sum,axiom,
! [M: nat] :
( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% gbinomial_r_part_sum
thf(fact_8637_exp__first__two__terms,axiom,
( exp_real
= ( ^ [X2: real] :
( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 )
@ ( suminf_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_8638_exp__first__two__terms,axiom,
( exp_complex
= ( ^ [X2: complex] :
( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ X2 )
@ ( suminf_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_complex @ X2 @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_8639_mult__scaleR__right,axiom,
! [X: real,A: real,Y4: real] :
( ( times_times_real @ X @ ( real_V1485227260804924795R_real @ A @ Y4 ) )
= ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y4 ) ) ) ).
% mult_scaleR_right
thf(fact_8640_mult__scaleR__right,axiom,
! [X: complex,A: real,Y4: complex] :
( ( times_times_complex @ X @ ( real_V2046097035970521341omplex @ A @ Y4 ) )
= ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y4 ) ) ) ).
% mult_scaleR_right
thf(fact_8641_mult__scaleR__left,axiom,
! [A: real,X: real,Y4: real] :
( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X ) @ Y4 )
= ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X @ Y4 ) ) ) ).
% mult_scaleR_left
thf(fact_8642_mult__scaleR__left,axiom,
! [A: real,X: complex,Y4: complex] :
( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ Y4 )
= ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X @ Y4 ) ) ) ).
% mult_scaleR_left
thf(fact_8643_atMost__iff,axiom,
! [I2: real,K: real] :
( ( member_real @ I2 @ ( set_ord_atMost_real @ K ) )
= ( ord_less_eq_real @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8644_atMost__iff,axiom,
! [I2: set_nat,K: set_nat] :
( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8645_atMost__iff,axiom,
! [I2: rat,K: rat] :
( ( member_rat @ I2 @ ( set_ord_atMost_rat @ K ) )
= ( ord_less_eq_rat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8646_atMost__iff,axiom,
! [I2: num,K: num] :
( ( member_num @ I2 @ ( set_ord_atMost_num @ K ) )
= ( ord_less_eq_num @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8647_atMost__iff,axiom,
! [I2: int,K: int] :
( ( member_int @ I2 @ ( set_ord_atMost_int @ K ) )
= ( ord_less_eq_int @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8648_atMost__iff,axiom,
! [I2: nat,K: nat] :
( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I2 @ K ) ) ).
% atMost_iff
thf(fact_8649_scaleR__one,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ one_one_real @ X )
= X ) ).
% scaleR_one
thf(fact_8650_scaleR__one,axiom,
! [X: complex] :
( ( real_V2046097035970521341omplex @ one_one_real @ X )
= X ) ).
% scaleR_one
thf(fact_8651_scaleR__scaleR,axiom,
! [A: real,B: real,X: real] :
( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B @ X ) )
= ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B ) @ X ) ) ).
% scaleR_scaleR
thf(fact_8652_scaleR__scaleR,axiom,
! [A: real,B: real,X: complex] :
( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B @ X ) )
= ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B ) @ X ) ) ).
% scaleR_scaleR
thf(fact_8653_atMost__subset__iff,axiom,
! [X: set_nat,Y4: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y4 ) )
= ( ord_less_eq_set_nat @ X @ Y4 ) ) ).
% atMost_subset_iff
thf(fact_8654_atMost__subset__iff,axiom,
! [X: rat,Y4: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y4 ) )
= ( ord_less_eq_rat @ X @ Y4 ) ) ).
% atMost_subset_iff
thf(fact_8655_atMost__subset__iff,axiom,
! [X: num,Y4: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y4 ) )
= ( ord_less_eq_num @ X @ Y4 ) ) ).
% atMost_subset_iff
thf(fact_8656_atMost__subset__iff,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y4 ) )
= ( ord_less_eq_int @ X @ Y4 ) ) ).
% atMost_subset_iff
thf(fact_8657_atMost__subset__iff,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y4 ) )
= ( ord_less_eq_nat @ X @ Y4 ) ) ).
% atMost_subset_iff
thf(fact_8658_scaleR__eq__iff,axiom,
! [B: real,U2: real,A: real] :
( ( ( plus_plus_real @ B @ ( real_V1485227260804924795R_real @ U2 @ A ) )
= ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U2 @ B ) ) )
= ( ( A = B )
| ( U2 = one_one_real ) ) ) ).
% scaleR_eq_iff
thf(fact_8659_scaleR__eq__iff,axiom,
! [B: complex,U2: real,A: complex] :
( ( ( plus_plus_complex @ B @ ( real_V2046097035970521341omplex @ U2 @ A ) )
= ( plus_plus_complex @ A @ ( real_V2046097035970521341omplex @ U2 @ B ) ) )
= ( ( A = B )
| ( U2 = one_one_real ) ) ) ).
% scaleR_eq_iff
thf(fact_8660_scaleR__power,axiom,
! [X: real,Y4: real,N: nat] :
( ( power_power_real @ ( real_V1485227260804924795R_real @ X @ Y4 ) @ N )
= ( real_V1485227260804924795R_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y4 @ N ) ) ) ).
% scaleR_power
thf(fact_8661_scaleR__power,axiom,
! [X: real,Y4: complex,N: nat] :
( ( power_power_complex @ ( real_V2046097035970521341omplex @ X @ Y4 ) @ N )
= ( real_V2046097035970521341omplex @ ( power_power_real @ X @ N ) @ ( power_power_complex @ Y4 @ N ) ) ) ).
% scaleR_power
thf(fact_8662_Icc__subset__Iic__iff,axiom,
! [L2: set_nat,H2: set_nat,H3: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L2 @ H2 ) @ ( set_or4236626031148496127et_nat @ H3 ) )
= ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
| ( ord_less_eq_set_nat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8663_Icc__subset__Iic__iff,axiom,
! [L2: rat,H2: rat,H3: rat] :
( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L2 @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
= ( ~ ( ord_less_eq_rat @ L2 @ H2 )
| ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8664_Icc__subset__Iic__iff,axiom,
! [L2: num,H2: num,H3: num] :
( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L2 @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
= ( ~ ( ord_less_eq_num @ L2 @ H2 )
| ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8665_Icc__subset__Iic__iff,axiom,
! [L2: nat,H2: nat,H3: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L2 @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
= ( ~ ( ord_less_eq_nat @ L2 @ H2 )
| ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8666_Icc__subset__Iic__iff,axiom,
! [L2: int,H2: int,H3: int] :
( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L2 @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
= ( ~ ( ord_less_eq_int @ L2 @ H2 )
| ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8667_Icc__subset__Iic__iff,axiom,
! [L2: real,H2: real,H3: real] :
( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L2 @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
= ( ~ ( ord_less_eq_real @ L2 @ H2 )
| ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_8668_sum_OatMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8669_sum_OatMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8670_sum_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8671_sum_OatMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_8672_scaleR__minus1__left,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) @ X )
= ( uminus_uminus_real @ X ) ) ).
% scaleR_minus1_left
thf(fact_8673_scaleR__minus1__left,axiom,
! [X: complex] :
( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ one_one_real ) @ X )
= ( uminus1482373934393186551omplex @ X ) ) ).
% scaleR_minus1_left
thf(fact_8674_scaleR__collapse,axiom,
! [U2: real,A: real] :
( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U2 ) @ A ) @ ( real_V1485227260804924795R_real @ U2 @ A ) )
= A ) ).
% scaleR_collapse
thf(fact_8675_scaleR__collapse,axiom,
! [U2: real,A: complex] :
( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ ( minus_minus_real @ one_one_real @ U2 ) @ A ) @ ( real_V2046097035970521341omplex @ U2 @ A ) )
= A ) ).
% scaleR_collapse
thf(fact_8676_prod_OatMost__Suc,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8677_prod_OatMost__Suc,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8678_prod_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8679_prod_OatMost__Suc,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% prod.atMost_Suc
thf(fact_8680_norm__scaleR,axiom,
! [A: real,X: real] :
( ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ A @ X ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% norm_scaleR
thf(fact_8681_norm__scaleR,axiom,
! [A: real,X: complex] :
( ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ A @ X ) )
= ( times_times_real @ ( abs_abs_real @ A ) @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% norm_scaleR
thf(fact_8682_scaleR__times,axiom,
! [U2: num,W2: num,A: real] :
( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ U2 ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ A ) )
= ( real_V1485227260804924795R_real @ ( times_times_real @ ( numeral_numeral_real @ U2 ) @ ( numeral_numeral_real @ W2 ) ) @ A ) ) ).
% scaleR_times
thf(fact_8683_scaleR__times,axiom,
! [U2: num,W2: num,A: complex] :
( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ U2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ A ) )
= ( real_V2046097035970521341omplex @ ( times_times_real @ ( numeral_numeral_real @ U2 ) @ ( numeral_numeral_real @ W2 ) ) @ A ) ) ).
% scaleR_times
thf(fact_8684_inverse__scaleR__times,axiom,
! [V: num,W2: num,A: real] :
( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ A ) )
= ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ W2 ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% inverse_scaleR_times
thf(fact_8685_inverse__scaleR__times,axiom,
! [V: num,W2: num,A: complex] :
( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ A ) )
= ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ W2 ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% inverse_scaleR_times
thf(fact_8686_fraction__scaleR__times,axiom,
! [U2: num,V: num,W2: num,A: real] :
( ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ U2 ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ A ) )
= ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U2 ) @ ( numeral_numeral_real @ W2 ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% fraction_scaleR_times
thf(fact_8687_fraction__scaleR__times,axiom,
! [U2: num,V: num,W2: num,A: complex] :
( ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ U2 ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ A ) )
= ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U2 ) @ ( numeral_numeral_real @ W2 ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).
% fraction_scaleR_times
thf(fact_8688_scaleR__half__double,axiom,
! [A: real] :
( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
= A ) ).
% scaleR_half_double
thf(fact_8689_scaleR__half__double,axiom,
! [A: complex] :
( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ A @ A ) )
= A ) ).
% scaleR_half_double
thf(fact_8690_scaleR__right__distrib,axiom,
! [A: real,X: real,Y4: real] :
( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X @ Y4 ) )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y4 ) ) ) ).
% scaleR_right_distrib
thf(fact_8691_scaleR__right__distrib,axiom,
! [A: real,X: complex,Y4: complex] :
( ( real_V2046097035970521341omplex @ A @ ( plus_plus_complex @ X @ Y4 ) )
= ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ A @ Y4 ) ) ) ).
% scaleR_right_distrib
thf(fact_8692_real__scaleR__def,axiom,
real_V1485227260804924795R_real = times_times_real ).
% real_scaleR_def
thf(fact_8693_divide__complex__def,axiom,
( divide1717551699836669952omplex
= ( ^ [X2: complex,Y: complex] : ( times_times_complex @ X2 @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% divide_complex_def
thf(fact_8694_not__empty__eq__Iic__eq__empty,axiom,
! [H2: int] :
( bot_bot_set_int
!= ( set_ord_atMost_int @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_8695_not__empty__eq__Iic__eq__empty,axiom,
! [H2: real] :
( bot_bot_set_real
!= ( set_ord_atMost_real @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_8696_not__empty__eq__Iic__eq__empty,axiom,
! [H2: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H2 ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_8697_atMost__def,axiom,
( set_ord_atMost_real
= ( ^ [U3: real] :
( collect_real
@ ^ [X2: real] : ( ord_less_eq_real @ X2 @ U3 ) ) ) ) ).
% atMost_def
thf(fact_8698_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U3: set_nat] :
( collect_set_nat
@ ^ [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ U3 ) ) ) ) ).
% atMost_def
thf(fact_8699_atMost__def,axiom,
( set_ord_atMost_rat
= ( ^ [U3: rat] :
( collect_rat
@ ^ [X2: rat] : ( ord_less_eq_rat @ X2 @ U3 ) ) ) ) ).
% atMost_def
thf(fact_8700_atMost__def,axiom,
( set_ord_atMost_num
= ( ^ [U3: num] :
( collect_num
@ ^ [X2: num] : ( ord_less_eq_num @ X2 @ U3 ) ) ) ) ).
% atMost_def
thf(fact_8701_atMost__def,axiom,
( set_ord_atMost_int
= ( ^ [U3: int] :
( collect_int
@ ^ [X2: int] : ( ord_less_eq_int @ X2 @ U3 ) ) ) ) ).
% atMost_def
thf(fact_8702_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U3: nat] :
( collect_nat
@ ^ [X2: nat] : ( ord_less_eq_nat @ X2 @ U3 ) ) ) ) ).
% atMost_def
thf(fact_8703_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_8704_scaleR__left_Oadd,axiom,
! [X: real,Y4: real,Xa2: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X @ Y4 ) @ Xa2 )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ X @ Xa2 ) @ ( real_V1485227260804924795R_real @ Y4 @ Xa2 ) ) ) ).
% scaleR_left.add
thf(fact_8705_scaleR__left_Oadd,axiom,
! [X: real,Y4: real,Xa2: complex] :
( ( real_V2046097035970521341omplex @ ( plus_plus_real @ X @ Y4 ) @ Xa2 )
= ( plus_plus_complex @ ( real_V2046097035970521341omplex @ X @ Xa2 ) @ ( real_V2046097035970521341omplex @ Y4 @ Xa2 ) ) ) ).
% scaleR_left.add
thf(fact_8706_scaleR__left__distrib,axiom,
! [A: real,B: real,X: real] :
( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ).
% scaleR_left_distrib
thf(fact_8707_scaleR__left__distrib,axiom,
! [A: real,B: real,X: complex] :
( ( real_V2046097035970521341omplex @ ( plus_plus_real @ A @ B ) @ X )
= ( plus_plus_complex @ ( real_V2046097035970521341omplex @ A @ X ) @ ( real_V2046097035970521341omplex @ B @ X ) ) ) ).
% scaleR_left_distrib
thf(fact_8708_complex__scaleR,axiom,
! [R2: real,A: real,B: real] :
( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B ) )
= ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B ) ) ) ).
% complex_scaleR
thf(fact_8709_scaleR__right__mono,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ X ) ) ) ) ).
% scaleR_right_mono
thf(fact_8710_scaleR__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_8711_scaleR__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B @ A ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_8712_scaleR__le__cancel__left__neg,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_8713_scaleR__le__cancel__left__pos,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_8714_scaleR__left__mono,axiom,
! [X: real,Y4: real,A: real] :
( ( ord_less_eq_real @ X @ Y4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ A @ Y4 ) ) ) ) ).
% scaleR_left_mono
thf(fact_8715_scaleR__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( real_V1485227260804924795R_real @ C @ B ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_8716_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_8717_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: real,E: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E ) @ D ) )
= ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_8718_Iic__subset__Iio__iff,axiom,
! [A: rat,B: rat] :
( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
= ( ord_less_rat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8719_Iic__subset__Iio__iff,axiom,
! [A: num,B: num] :
( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
= ( ord_less_num @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8720_Iic__subset__Iio__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8721_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8722_Iic__subset__Iio__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_8723_exp__series__add__commuting,axiom,
! [X: real,Y4: real,N: nat] :
( ( ( times_times_real @ X @ Y4 )
= ( times_times_real @ Y4 @ X ) )
=> ( ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y4 ) @ N ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ I4 ) ) @ ( power_power_real @ X @ I4 ) ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ I4 ) ) ) @ ( power_power_real @ Y4 @ ( minus_minus_nat @ N @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% exp_series_add_commuting
thf(fact_8724_exp__series__add__commuting,axiom,
! [X: complex,Y4: complex,N: nat] :
( ( ( times_times_complex @ X @ Y4 )
= ( times_times_complex @ Y4 @ X ) )
=> ( ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y4 ) @ N ) )
= ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ I4 ) ) @ ( power_power_complex @ X @ I4 ) ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ I4 ) ) ) @ ( power_power_complex @ Y4 @ ( minus_minus_nat @ N @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% exp_series_add_commuting
thf(fact_8725_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ K3 @ M )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_8726_scaleR__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( A = zero_zero_real ) ) ) ).
% scaleR_le_0_iff
thf(fact_8727_zero__le__scaleR__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( A = zero_zero_real ) ) ) ).
% zero_le_scaleR_iff
thf(fact_8728_scaleR__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_8729_scaleR__nonpos__nonneg,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_8730_scaleR__nonneg__nonpos,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_8731_scaleR__nonneg__nonneg,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_8732_split__scaleR__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).
% split_scaleR_pos_le
thf(fact_8733_split__scaleR__neg__le,axiom,
! [A: real,X: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ X @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ X ) ) )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ zero_zero_real ) ) ).
% split_scaleR_neg_le
thf(fact_8734_scaleR__mono_H,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_8735_scaleR__mono,axiom,
! [A: real,B: real,X: real,Y4: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ X @ Y4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ ( real_V1485227260804924795R_real @ B @ Y4 ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_8736_scaleR__left__le__one__le,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ A @ one_one_real )
=> ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X ) @ X ) ) ) ).
% scaleR_left_le_one_le
thf(fact_8737_scaleR__2,axiom,
! [X: real] :
( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
= ( plus_plus_real @ X @ X ) ) ).
% scaleR_2
thf(fact_8738_scaleR__2,axiom,
! [X: complex] :
( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X )
= ( plus_plus_complex @ X @ X ) ) ).
% scaleR_2
thf(fact_8739_real__vector__affinity__eq,axiom,
! [M: real,X: real,C: real,Y4: real] :
( ( M != zero_zero_real )
=> ( ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C )
= Y4 )
= ( X
= ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_8740_real__vector__affinity__eq,axiom,
! [M: real,X: complex,C: complex,Y4: complex] :
( ( M != zero_zero_real )
=> ( ( ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M @ X ) @ C )
= Y4 )
= ( X
= ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_8741_real__vector__eq__affinity,axiom,
! [M: real,Y4: real,X: real,C: real] :
( ( M != zero_zero_real )
=> ( ( Y4
= ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X ) @ C ) )
= ( ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) )
= X ) ) ) ).
% real_vector_eq_affinity
thf(fact_8742_real__vector__eq__affinity,axiom,
! [M: real,Y4: complex,X: complex,C: complex] :
( ( M != zero_zero_real )
=> ( ( Y4
= ( plus_plus_complex @ ( real_V2046097035970521341omplex @ M @ X ) @ C ) )
= ( ( minus_minus_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ Y4 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ M ) @ C ) )
= X ) ) ) ).
% real_vector_eq_affinity
thf(fact_8743_pos__divideR__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
= ( ord_less_eq_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_8744_pos__le__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
= ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% pos_le_divideR_eq
thf(fact_8745_neg__divideR__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
= ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% neg_divideR_le_eq
thf(fact_8746_neg__le__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
= ( ord_less_eq_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_8747_neg__less__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
= ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_8748_neg__divideR__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% neg_divideR_less_eq
thf(fact_8749_pos__less__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).
% pos_less_divideR_eq
thf(fact_8750_pos__divideR__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
= ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_8751_sum_OatMost__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8752_sum_OatMost__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8753_sum_OatMost__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8754_sum_OatMost__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_8755_sum__telescope,axiom,
! [F: nat > rat,I2: nat] :
( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_8756_sum__telescope,axiom,
! [F: nat > int,I2: nat] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_8757_sum__telescope,axiom,
! [F: nat > real,I2: nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
@ ( set_ord_atMost_nat @ I2 ) )
= ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).
% sum_telescope
thf(fact_8758_polyfun__eq__coeffs,axiom,
! [C: nat > complex,N: nat,D: nat > complex] :
( ( ! [X2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( D @ I4 ) @ ( power_power_complex @ X2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( C @ I4 )
= ( D @ I4 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_8759_polyfun__eq__coeffs,axiom,
! [C: nat > real,N: nat,D: nat > real] :
( ( ! [X2: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( D @ I4 ) @ ( power_power_real @ X2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( C @ I4 )
= ( D @ I4 ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_8760_bounded__imp__summable,axiom,
! [A: nat > int,B5: int] :
( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
=> ( summable_int @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8761_bounded__imp__summable,axiom,
! [A: nat > nat,B5: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
=> ( summable_nat @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8762_bounded__imp__summable,axiom,
! [A: nat > real,B5: real] :
( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
=> ( summable_real @ A ) ) ) ).
% bounded_imp_summable
thf(fact_8763_prod_OatMost__Suc__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8764_prod_OatMost__Suc__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8765_prod_OatMost__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8766_prod_OatMost__Suc__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_8767_sum_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( A @ I4 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_8768_sum_Onested__swap_H,axiom,
! [A: nat > nat > real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( groups6591440286371151544t_real @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( A @ I4 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_8769_prod_Onested__swap_H,axiom,
! [A: nat > nat > nat,N: nat] :
( ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( groups708209901874060359at_nat @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] :
( groups708209901874060359at_nat
@ ^ [I4: nat] : ( A @ I4 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_8770_prod_Onested__swap_H,axiom,
! [A: nat > nat > int,N: nat] :
( ( groups705719431365010083at_int
@ ^ [I4: nat] : ( groups705719431365010083at_int @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] :
( groups705719431365010083at_int
@ ^ [I4: nat] : ( A @ I4 @ J3 )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.nested_swap'
thf(fact_8771_sum__choose__lower,axiom,
! [R2: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_8772_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% choose_rising_sum(1)
thf(fact_8773_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_8774_summable__exp__generic,axiom,
! [X: real] :
( summable_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ).
% summable_exp_generic
thf(fact_8775_summable__exp__generic,axiom,
! [X: complex] :
( summable_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X @ N2 ) ) ) ).
% summable_exp_generic
thf(fact_8776_sin__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N2 ) @ ( power_power_real @ X @ N2 ) )
@ ( sin_real @ X ) ) ).
% sin_converges
thf(fact_8777_sin__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N2 ) @ ( power_power_complex @ X @ N2 ) )
@ ( sin_complex @ X ) ) ).
% sin_converges
thf(fact_8778_sin__def,axiom,
( sin_real
= ( ^ [X2: real] :
( suminf_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).
% sin_def
thf(fact_8779_sin__def,axiom,
( sin_complex
= ( ^ [X2: complex] :
( suminf_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).
% sin_def
thf(fact_8780_cos__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N2 ) @ ( power_power_real @ X @ N2 ) )
@ ( cos_real @ X ) ) ).
% cos_converges
thf(fact_8781_cos__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N2 ) @ ( power_power_complex @ X @ N2 ) )
@ ( cos_complex @ X ) ) ).
% cos_converges
thf(fact_8782_cos__def,axiom,
( cos_real
= ( ^ [X2: real] :
( suminf_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).
% cos_def
thf(fact_8783_cos__def,axiom,
( cos_complex
= ( ^ [X2: complex] :
( suminf_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).
% cos_def
thf(fact_8784_summable__norm__sin,axiom,
! [X: real] :
( summable_real
@ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% summable_norm_sin
thf(fact_8785_summable__norm__sin,axiom,
! [X: complex] :
( summable_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% summable_norm_sin
thf(fact_8786_summable__norm__cos,axiom,
! [X: real] :
( summable_real
@ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( cos_coeff @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% summable_norm_cos
thf(fact_8787_summable__norm__cos,axiom,
! [X: complex] :
( summable_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( cos_coeff @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% summable_norm_cos
thf(fact_8788_neg__minus__divideR__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
= ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_8789_neg__le__minus__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_8790_pos__minus__divideR__le__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_8791_pos__le__minus__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
= ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_8792_pos__less__minus__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_8793_pos__minus__divideR__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_8794_neg__less__minus__divideR__eq,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_8795_neg__minus__divideR__less__eq,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
= ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_8796_polyfun__eq__0,axiom,
! [C: nat > complex,N: nat] :
( ( ! [X2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) )
= ( ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( C @ I4 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_0
thf(fact_8797_polyfun__eq__0,axiom,
! [C: nat > real,N: nat] :
( ( ! [X2: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) )
= ( ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
=> ( ( C @ I4 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_0
thf(fact_8798_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > complex,N: nat,K: nat] :
( ! [W: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ W @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( C @ K )
= zero_zero_complex ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_8799_zero__polynom__imp__zero__coeffs,axiom,
! [C: nat > real,N: nat,K: nat] :
( ! [W: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ W @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( C @ K )
= zero_zero_real ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_8800_sum_OatMost__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_rat @ ( G @ zero_zero_nat )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8801_sum_OatMost__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int @ ( G @ zero_zero_nat )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8802_sum_OatMost__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8803_sum_OatMost__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_real @ ( G @ zero_zero_nat )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_8804_sum__up__index__split,axiom,
! [F: nat > rat,M: nat,N: nat] :
( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8805_sum__up__index__split,axiom,
! [F: nat > int,M: nat,N: nat] :
( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8806_sum__up__index__split,axiom,
! [F: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8807_sum__up__index__split,axiom,
! [F: nat > real,M: nat,N: nat] :
( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_8808_prod_OatMost__shift,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( G @ zero_zero_nat )
@ ( groups129246275422532515t_real
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8809_prod_OatMost__shift,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_rat @ ( G @ zero_zero_nat )
@ ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8810_prod_OatMost__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_nat @ ( G @ zero_zero_nat )
@ ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8811_prod_OatMost__shift,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( G @ zero_zero_nat )
@ ( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% prod.atMost_shift
thf(fact_8812_gbinomial__parallel__sum,axiom,
! [A: complex,N: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8813_gbinomial__parallel__sum,axiom,
! [A: rat,N: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8814_gbinomial__parallel__sum,axiom,
! [A: real,N: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K3 ) ) @ K3 )
@ ( set_ord_atMost_nat @ N ) )
= ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ N ) ) ).
% gbinomial_parallel_sum
thf(fact_8815_sum_Otriangle__reindex__eq,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_8816_sum_Otriangle__reindex__eq,axiom,
! [G: nat > nat > real,N: nat] :
( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_8817_prod_Otriangle__reindex__eq,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] :
( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_8818_prod_Otriangle__reindex__eq,axiom,
! [G: nat > nat > int,N: nat] :
( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] :
( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_8819_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_8820_vandermonde,axiom,
! [M: nat,N: nat,R2: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R2 ) )
= ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).
% vandermonde
thf(fact_8821_exp__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) )
@ ( exp_real @ X ) ) ).
% exp_converges
thf(fact_8822_exp__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X @ N2 ) )
@ ( exp_complex @ X ) ) ).
% exp_converges
thf(fact_8823_exp__def,axiom,
( exp_real
= ( ^ [X2: real] :
( suminf_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).
% exp_def
thf(fact_8824_exp__def,axiom,
( exp_complex
= ( ^ [X2: complex] :
( suminf_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).
% exp_def
thf(fact_8825_summable__norm__exp,axiom,
! [X: real] :
( summable_real
@ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% summable_norm_exp
thf(fact_8826_summable__norm__exp,axiom,
! [X: complex] :
( summable_real
@ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% summable_norm_exp
thf(fact_8827_sin__minus__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ N2 ) ) )
@ ( sin_real @ X ) ) ).
% sin_minus_converges
thf(fact_8828_sin__minus__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ N2 ) ) )
@ ( sin_complex @ X ) ) ).
% sin_minus_converges
thf(fact_8829_cos__minus__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ N2 ) )
@ ( cos_real @ X ) ) ).
% cos_minus_converges
thf(fact_8830_cos__minus__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ N2 ) )
@ ( cos_complex @ X ) ) ).
% cos_minus_converges
thf(fact_8831_sum__gp__basic,axiom,
! [X: complex,N: nat] :
( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8832_sum__gp__basic,axiom,
! [X: rat,N: nat] :
( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8833_sum__gp__basic,axiom,
! [X: int,N: nat] :
( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8834_sum__gp__basic,axiom,
! [X: real,N: nat] :
( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) ) )
= ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ).
% sum_gp_basic
thf(fact_8835_polyfun__finite__roots,axiom,
! [C: nat > complex,N: nat] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( C @ I4 )
!= zero_zero_complex ) ) ) ) ).
% polyfun_finite_roots
thf(fact_8836_polyfun__finite__roots,axiom,
! [C: nat > real,N: nat] :
( ( finite_finite_real
@ ( collect_real
@ ^ [X2: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq_nat @ I4 @ N )
& ( ( C @ I4 )
!= zero_zero_real ) ) ) ) ).
% polyfun_finite_roots
thf(fact_8837_polyfun__roots__finite,axiom,
! [C: nat > complex,K: nat,N: nat] :
( ( ( C @ K )
!= zero_zero_complex )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_8838_polyfun__roots__finite,axiom,
! [C: nat > real,K: nat,N: nat] :
( ( ( C @ K )
!= zero_zero_real )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( finite_finite_real
@ ( collect_real
@ ^ [Z4: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_8839_polyfun__linear__factor__root,axiom,
! [C: nat > complex,A: complex,N: nat] :
( ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex )
=> ~ ! [B2: nat > complex] :
~ ! [Z5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_complex @ ( minus_minus_complex @ Z5 @ A )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( B2 @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8840_polyfun__linear__factor__root,axiom,
! [C: nat > rat,A: rat,N: nat] :
( ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat )
=> ~ ! [B2: nat > rat] :
~ ! [Z5: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_rat @ ( minus_minus_rat @ Z5 @ A )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( B2 @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8841_polyfun__linear__factor__root,axiom,
! [C: nat > int,A: int,N: nat] :
( ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int )
=> ~ ! [B2: nat > int] :
~ ! [Z5: int] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_int @ ( minus_minus_int @ Z5 @ A )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( B2 @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8842_polyfun__linear__factor__root,axiom,
! [C: nat > real,A: real,N: nat] :
( ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real )
=> ~ ! [B2: nat > real] :
~ ! [Z5: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_real @ ( minus_minus_real @ Z5 @ A )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( B2 @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_8843_polyfun__linear__factor,axiom,
! [C: nat > complex,N: nat,A: complex] :
? [B2: nat > complex] :
! [Z5: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_complex
@ ( times_times_complex @ ( minus_minus_complex @ Z5 @ A )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( B2 @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8844_polyfun__linear__factor,axiom,
! [C: nat > rat,N: nat,A: rat] :
? [B2: nat > rat] :
! [Z5: rat] :
( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_rat
@ ( times_times_rat @ ( minus_minus_rat @ Z5 @ A )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( B2 @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8845_polyfun__linear__factor,axiom,
! [C: nat > int,N: nat,A: int] :
? [B2: nat > int] :
! [Z5: int] :
( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_int
@ ( times_times_int @ ( minus_minus_int @ Z5 @ A )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( B2 @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8846_polyfun__linear__factor,axiom,
! [C: nat > real,N: nat,A: real] :
? [B2: nat > real] :
! [Z5: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_real
@ ( times_times_real @ ( minus_minus_real @ Z5 @ A )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( B2 @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
@ ( set_ord_lessThan_nat @ N ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ A @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% polyfun_linear_factor
thf(fact_8847_sum__power__shift,axiom,
! [M: nat,N: nat,X: complex] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8848_sum__power__shift,axiom,
! [M: nat,N: nat,X: rat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8849_sum__power__shift,axiom,
! [M: nat,N: nat,X: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8850_sum__power__shift,axiom,
! [M: nat,N: nat,X: real] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_8851_sum_Otriangle__reindex,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.triangle_reindex
thf(fact_8852_sum_Otriangle__reindex,axiom,
! [G: nat > nat > real,N: nat] :
( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.triangle_reindex
thf(fact_8853_prod_Otriangle__reindex,axiom,
! [G: nat > nat > nat,N: nat] :
( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [K3: nat] :
( groups708209901874060359at_nat
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.triangle_reindex
thf(fact_8854_prod_Otriangle__reindex,axiom,
! [G: nat > nat > int,N: nat] :
( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [K3: nat] :
( groups705719431365010083at_int
@ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% prod.triangle_reindex
thf(fact_8855_summable__Cauchy__product,axiom,
! [A: nat > complex,B: nat > complex] :
( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
=> ( summable_complex
@ ^ [K3: nat] :
( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B @ ( minus_minus_nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_8856_summable__Cauchy__product,axiom,
! [A: nat > real,B: nat > real] :
( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
=> ( summable_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B @ ( minus_minus_nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_8857_choose__row__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% choose_row_sum
thf(fact_8858_Cauchy__product,axiom,
! [A: nat > complex,B: nat > complex] :
( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
=> ( ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B ) )
= ( suminf_complex
@ ^ [K3: nat] :
( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B @ ( minus_minus_nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_8859_Cauchy__product,axiom,
! [A: nat > real,B: nat > real] :
( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
=> ( ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B ) )
= ( suminf_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B @ ( minus_minus_nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost_nat @ K3 ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_8860_binomial,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial
thf(fact_8861_sum_Oin__pairs__0,axiom,
! [G: nat > rat,N: nat] :
( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8862_sum_Oin__pairs__0,axiom,
! [G: nat > int,N: nat] :
( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8863_sum_Oin__pairs__0,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8864_sum_Oin__pairs__0,axiom,
! [G: nat > real,N: nat] :
( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% sum.in_pairs_0
thf(fact_8865_polynomial__product,axiom,
! [M: nat,A: nat > complex,N: nat,B: nat > complex,X: complex] :
( ! [I3: nat] :
( ( ord_less_nat @ M @ I3 )
=> ( ( A @ I3 )
= zero_zero_complex ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_complex ) )
=> ( ( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups2073611262835488442omplex
@ ^ [R5: nat] :
( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_complex @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8866_polynomial__product,axiom,
! [M: nat,A: nat > rat,N: nat,B: nat > rat,X: rat] :
( ! [I3: nat] :
( ( ord_less_nat @ M @ I3 )
=> ( ( A @ I3 )
= zero_zero_rat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_rat ) )
=> ( ( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups2906978787729119204at_rat
@ ^ [R5: nat] :
( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_rat @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8867_polynomial__product,axiom,
! [M: nat,A: nat > int,N: nat,B: nat > int,X: int] :
( ! [I3: nat] :
( ( ord_less_nat @ M @ I3 )
=> ( ( A @ I3 )
= zero_zero_int ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_int ) )
=> ( ( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups3539618377306564664at_int
@ ^ [R5: nat] :
( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_int @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8868_polynomial__product,axiom,
! [M: nat,A: nat > real,N: nat,B: nat > real,X: real] :
( ! [I3: nat] :
( ( ord_less_nat @ M @ I3 )
=> ( ( A @ I3 )
= zero_zero_real ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_real ) )
=> ( ( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups6591440286371151544t_real
@ ^ [R5: nat] :
( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_real @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product
thf(fact_8869_prod_Oin__pairs__0,axiom,
! [G: nat > real,N: nat] :
( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups129246275422532515t_real
@ ^ [I4: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8870_prod_Oin__pairs__0,axiom,
! [G: nat > rat,N: nat] :
( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups73079841787564623at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8871_prod_Oin__pairs__0,axiom,
! [G: nat > nat,N: nat] :
( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8872_prod_Oin__pairs__0,axiom,
! [G: nat > int,N: nat] :
( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
= ( groups705719431365010083at_int
@ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% prod.in_pairs_0
thf(fact_8873_polyfun__eq__const,axiom,
! [C: nat > complex,N: nat,K: complex] :
( ( ! [X2: complex] :
( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= K ) )
= ( ( ( C @ zero_zero_nat )
= K )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
=> ( ( C @ X2 )
= zero_zero_complex ) ) ) ) ).
% polyfun_eq_const
thf(fact_8874_polyfun__eq__const,axiom,
! [C: nat > real,N: nat,K: real] :
( ( ! [X2: real] :
( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= K ) )
= ( ( ( C @ zero_zero_nat )
= K )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
=> ( ( C @ X2 )
= zero_zero_real ) ) ) ) ).
% polyfun_eq_const
thf(fact_8875_gbinomial__sum__lower__neg,axiom,
! [A: complex,M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ M ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_8876_gbinomial__sum__lower__neg,axiom,
! [A: rat,M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_8877_gbinomial__sum__lower__neg,axiom,
! [A: real,M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ M ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_8878_binomial__ring,axiom,
! [A: complex,B: complex,N: nat] :
( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N )
= ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K3 ) ) @ ( power_power_complex @ A @ K3 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8879_binomial__ring,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N )
= ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( power_power_int @ A @ K3 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8880_binomial__ring,axiom,
! [A: rat,B: rat,N: nat] :
( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( power_power_rat @ A @ K3 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8881_binomial__ring,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
= ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8882_binomial__ring,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( power_power_real @ A @ K3 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% binomial_ring
thf(fact_8883_pochhammer__binomial__sum,axiom,
! [A: int,B: int,N: nat] :
( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N )
= ( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( comm_s4660882817536571857er_int @ A @ K3 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8884_pochhammer__binomial__sum,axiom,
! [A: rat,B: rat,N: nat] :
( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8885_pochhammer__binomial__sum,axiom,
! [A: real,B: real,N: nat] :
( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( comm_s7457072308508201937r_real @ A @ K3 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% pochhammer_binomial_sum
thf(fact_8886_polynomial__product__nat,axiom,
! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ M @ I3 )
=> ( ( A @ I3 )
= zero_zero_nat ) )
=> ( ! [J2: nat] :
( ( ord_less_nat @ N @ J2 )
=> ( ( B @ J2 )
= zero_zero_nat ) )
=> ( ( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X @ I4 ) )
@ ( set_ord_atMost_nat @ M ) )
@ ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [R5: nat] :
( times_times_nat
@ ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost_nat @ R5 ) )
@ ( power_power_nat @ X @ R5 ) )
@ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_8887_choose__square__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
@ ( set_ord_atMost_nat @ N ) )
= ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% choose_square_sum
thf(fact_8888_Cauchy__product__sums,axiom,
! [A: nat > complex,B: nat > complex] :
( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V1022390504157884413omplex @ ( B @ K3 ) ) )
=> ( sums_complex
@ ^ [K3: nat] :
( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B @ ( minus_minus_nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_8889_Cauchy__product__sums,axiom,
! [A: nat > real,B: nat > real] :
( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( A @ K3 ) ) )
=> ( ( summable_real
@ ^ [K3: nat] : ( real_V7735802525324610683m_real @ ( B @ K3 ) ) )
=> ( sums_real
@ ^ [K3: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B @ ( minus_minus_nat @ K3 @ I4 ) ) )
@ ( set_ord_atMost_nat @ K3 ) )
@ ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_8890_complex__inverse,axiom,
! [A: real,B: real] :
( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
= ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% complex_inverse
thf(fact_8891_sum_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > complex,H2: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups2073611262835488442omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8892_sum_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > rat,H2: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups2906978787729119204at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8893_sum_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > int,H2: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups3539618377306564664at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8894_sum_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > nat,H2: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8895_sum_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > real,H2: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups6591440286371151544t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_8896_prod_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > complex,H2: nat > complex] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups6464643781859351333omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups6464643781859351333omplex
@ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8897_prod_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > real,H2: nat > real] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups129246275422532515t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups129246275422532515t_real
@ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8898_prod_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > rat,H2: nat > rat] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ one_one_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups73079841787564623at_rat
@ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8899_prod_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > nat,H2: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups708209901874060359at_nat
@ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8900_prod_Ozero__middle,axiom,
! [P6: nat,K: nat,G: nat > int,H2: nat > int] :
( ( ord_less_eq_nat @ one_one_nat @ P6 )
=> ( ( ord_less_eq_nat @ K @ P6 )
=> ( ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P6 ) )
= ( groups705719431365010083at_int
@ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P6 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_8901_gbinomial__partial__sum__poly,axiom,
! [M: nat,A: complex,X: complex,Y4: complex] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ Y4 @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y4 ) @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_8902_gbinomial__partial__sum__poly,axiom,
! [M: nat,A: rat,X: rat,Y4: rat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ Y4 @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X ) @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_8903_gbinomial__partial__sum__poly,axiom,
! [M: nat,A: real,X: real,Y4: real] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ Y4 @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ X ) @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y4 ) @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_8904_exp__first__term,axiom,
( exp_real
= ( ^ [X2: real] :
( plus_plus_real @ one_one_real
@ ( suminf_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_8905_exp__first__term,axiom,
( exp_complex
= ( ^ [X2: complex] :
( plus_plus_complex @ one_one_complex
@ ( suminf_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( suc @ N2 ) ) ) @ ( power_power_complex @ X2 @ ( suc @ N2 ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_8906_root__polyfun,axiom,
! [N: nat,Z2: int,A: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_int @ Z2 @ N )
= A )
= ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( if_int @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I4 = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ) ).
% root_polyfun
thf(fact_8907_root__polyfun,axiom,
! [N: nat,Z2: complex,A: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_complex @ Z2 @ N )
= A )
= ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( if_complex @ ( I4 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I4 = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ) ).
% root_polyfun
thf(fact_8908_root__polyfun,axiom,
! [N: nat,Z2: rat,A: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_rat @ Z2 @ N )
= A )
= ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( if_rat @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I4 = N ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ) ).
% root_polyfun
thf(fact_8909_root__polyfun,axiom,
! [N: nat,Z2: code_integer,A: code_integer] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_8256067586552552935nteger @ Z2 @ N )
= A )
= ( ( groups7501900531339628137nteger
@ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I4 = N ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_z3403309356797280102nteger ) ) ) ).
% root_polyfun
thf(fact_8910_root__polyfun,axiom,
! [N: nat,Z2: real,A: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( ( power_power_real @ Z2 @ N )
= A )
= ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( if_real @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I4 = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z2 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ) ).
% root_polyfun
thf(fact_8911_sum__gp0,axiom,
! [X: complex,N: nat] :
( ( ( X = one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
& ( ( X != one_one_complex )
=> ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_8912_sum__gp0,axiom,
! [X: rat,N: nat] :
( ( ( X = one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
& ( ( X != one_one_rat )
=> ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_8913_sum__gp0,axiom,
! [X: real,N: nat] :
( ( ( X = one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
& ( ( X != one_one_real )
=> ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) )
= ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% sum_gp0
thf(fact_8914_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ I4 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ).
% choose_alternating_linear_sum
thf(fact_8915_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups7501900531339628137nteger
@ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ I4 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_z3403309356797280102nteger ) ) ).
% choose_alternating_linear_sum
thf(fact_8916_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ I4 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ).
% choose_alternating_linear_sum
thf(fact_8917_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ I4 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ).
% choose_alternating_linear_sum
thf(fact_8918_choose__alternating__linear__sum,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ I4 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ).
% choose_alternating_linear_sum
thf(fact_8919_gbinomial__sum__nat__pow2,axiom,
! [M: nat] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_8920_gbinomial__sum__nat__pow2,axiom,
! [M: nat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_8921_gbinomial__sum__nat__pow2,axiom,
! [M: nat] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K3 ) )
@ ( set_ord_atMost_nat @ M ) )
= ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_8922_gbinomial__partial__sum__poly__xpos,axiom,
! [M: nat,A: complex,X: complex,Y4: complex] :
( ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ Y4 @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A ) @ one_one_complex ) @ K3 ) @ ( power_power_complex @ X @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y4 ) @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_8923_gbinomial__partial__sum__poly__xpos,axiom,
! [M: nat,A: rat,X: rat,Y4: rat] :
( ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ Y4 @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A ) @ one_one_rat ) @ K3 ) @ ( power_power_rat @ X @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y4 ) @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_8924_gbinomial__partial__sum__poly__xpos,axiom,
! [M: nat,A: real,X: real,Y4: real] :
( ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ Y4 @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) )
= ( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A ) @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y4 ) @ ( minus_minus_nat @ M @ K3 ) ) )
@ ( set_ord_atMost_nat @ M ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_8925_polyfun__diff__alt,axiom,
! [N: nat,A: nat > complex,X: complex,Y4: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_complex
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y4 )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] :
( groups2073611262835488442omplex
@ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y4 @ K3 ) ) @ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8926_polyfun__diff__alt,axiom,
! [N: nat,A: nat > rat,X: rat,Y4: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y4 )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] :
( groups2906978787729119204at_rat
@ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_rat @ Y4 @ K3 ) ) @ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8927_polyfun__diff__alt,axiom,
! [N: nat,A: nat > int,X: int,Y4: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y4 )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( groups3539618377306564664at_int
@ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y4 @ K3 ) ) @ ( power_power_int @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8928_polyfun__diff__alt,axiom,
! [N: nat,A: nat > real,X: real,Y4: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y4 )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( groups6591440286371151544t_real
@ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y4 @ K3 ) ) @ ( power_power_real @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_8929_binomial__r__part__sum,axiom,
! [M: nat] :
( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% binomial_r_part_sum
thf(fact_8930_choose__linear__sum,axiom,
! [N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% choose_linear_sum
thf(fact_8931_exp__first__terms,axiom,
! [K: nat] :
( exp_real
= ( ^ [X2: real] :
( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( suminf_real
@ ^ [N2: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N2 @ K ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ N2 @ K ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_8932_exp__first__terms,axiom,
! [K: nat] :
( exp_complex
= ( ^ [X2: complex] :
( plus_plus_complex
@ ( groups2073611262835488442omplex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X2 @ N2 ) )
@ ( set_ord_lessThan_nat @ K ) )
@ ( suminf_complex
@ ^ [N2: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N2 @ K ) ) ) @ ( power_power_complex @ X2 @ ( plus_plus_nat @ N2 @ K ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_8933_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_complex ) ) ).
% choose_alternating_sum
thf(fact_8934_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups7501900531339628137nteger
@ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_z3403309356797280102nteger ) ) ).
% choose_alternating_sum
thf(fact_8935_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_int ) ) ).
% choose_alternating_sum
thf(fact_8936_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_rat ) ) ).
% choose_alternating_sum
thf(fact_8937_choose__alternating__sum,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) )
= zero_zero_real ) ) ).
% choose_alternating_sum
thf(fact_8938_polyfun__extremal__lemma,axiom,
! [E: real,C: nat > complex,N: nat] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M8: real] :
! [Z5: complex] :
( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z5 ) )
=> ( ord_less_eq_real
@ ( real_V1022390504157884413omplex
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
@ ( times_times_real @ E @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_8939_polyfun__extremal__lemma,axiom,
! [E: real,C: nat > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ E )
=> ? [M8: real] :
! [Z5: real] :
( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z5 ) )
=> ( ord_less_eq_real
@ ( real_V7735802525324610683m_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
@ ( times_times_real @ E @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_8940_polyfun__diff,axiom,
! [N: nat,A: nat > complex,X: complex,Y4: complex] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_complex
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_complex @ ( minus_minus_complex @ X @ Y4 )
@ ( groups2073611262835488442omplex
@ ^ [J3: nat] :
( times_times_complex
@ ( groups2073611262835488442omplex
@ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y4 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( power_power_complex @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff
thf(fact_8941_polyfun__diff,axiom,
! [N: nat,A: nat > rat,X: rat,Y4: rat] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_rat
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_rat @ ( minus_minus_rat @ X @ Y4 )
@ ( groups2906978787729119204at_rat
@ ^ [J3: nat] :
( times_times_rat
@ ( groups2906978787729119204at_rat
@ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y4 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( power_power_rat @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff
thf(fact_8942_polyfun__diff,axiom,
! [N: nat,A: nat > int,X: int,Y4: int] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_int
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_int @ ( minus_minus_int @ X @ Y4 )
@ ( groups3539618377306564664at_int
@ ^ [J3: nat] :
( times_times_int
@ ( groups3539618377306564664at_int
@ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y4 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( power_power_int @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff
thf(fact_8943_polyfun__diff,axiom,
! [N: nat,A: nat > real,X: real,Y4: real] :
( ( ord_less_eq_nat @ one_one_nat @ N )
=> ( ( minus_minus_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X @ I4 ) )
@ ( set_ord_atMost_nat @ N ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y4 @ I4 ) )
@ ( set_ord_atMost_nat @ N ) ) )
= ( times_times_real @ ( minus_minus_real @ X @ Y4 )
@ ( groups6591440286371151544t_real
@ ^ [J3: nat] :
( times_times_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y4 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
@ ( power_power_real @ X @ J3 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% polyfun_diff
thf(fact_8944_cos__x__cos__y,axiom,
! [X: real,Y4: real] :
( sums_real
@ ^ [P4: nat] :
( groups6591440286371151544t_real
@ ^ [N2: nat] :
( if_real
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
@ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y4 @ ( minus_minus_nat @ P4 @ N2 ) ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y4 ) ) ) ).
% cos_x_cos_y
thf(fact_8945_cos__x__cos__y,axiom,
! [X: complex,Y4: complex] :
( sums_complex
@ ^ [P4: nat] :
( groups2073611262835488442omplex
@ ^ [N2: nat] :
( if_complex
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
& ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
@ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y4 @ ( minus_minus_nat @ P4 @ N2 ) ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y4 ) ) ) ).
% cos_x_cos_y
thf(fact_8946_sums__cos__x__plus__y,axiom,
! [X: real,Y4: real] :
( sums_real
@ ^ [P4: nat] :
( groups6591440286371151544t_real
@ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y4 @ ( minus_minus_nat @ P4 @ N2 ) ) ) @ zero_zero_real )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( cos_real @ ( plus_plus_real @ X @ Y4 ) ) ) ).
% sums_cos_x_plus_y
thf(fact_8947_sums__cos__x__plus__y,axiom,
! [X: complex,Y4: complex] :
( sums_complex
@ ^ [P4: nat] :
( groups2073611262835488442omplex
@ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y4 @ ( minus_minus_nat @ P4 @ N2 ) ) ) @ zero_zero_complex )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( cos_complex @ ( plus_plus_complex @ X @ Y4 ) ) ) ).
% sums_cos_x_plus_y
thf(fact_8948_sin__x__sin__y,axiom,
! [X: real,Y4: real] :
( sums_real
@ ^ [P4: nat] :
( groups6591440286371151544t_real
@ ^ [N2: nat] :
( if_real
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
@ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) ) @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ Y4 @ ( minus_minus_nat @ P4 @ N2 ) ) )
@ zero_zero_real )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y4 ) ) ) ).
% sin_x_sin_y
thf(fact_8949_sin__x__sin__y,axiom,
! [X: complex,Y4: complex] :
( sums_complex
@ ^ [P4: nat] :
( groups2073611262835488442omplex
@ ^ [N2: nat] :
( if_complex
@ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P4 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
@ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P4 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P4 ) ) ) @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_complex @ Y4 @ ( minus_minus_nat @ P4 @ N2 ) ) )
@ zero_zero_complex )
@ ( set_ord_atMost_nat @ P4 ) )
@ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ Y4 ) ) ) ).
% sin_x_sin_y
thf(fact_8950_sinh__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) )
@ ( sinh_real @ X ) ) ).
% sinh_converges
thf(fact_8951_sinh__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X @ N2 ) ) )
@ ( sinh_complex @ X ) ) ).
% sinh_converges
thf(fact_8952_cosh__converges,axiom,
! [X: real] :
( sums_real
@ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) @ zero_zero_real )
@ ( cosh_real @ X ) ) ).
% cosh_converges
thf(fact_8953_cosh__converges,axiom,
! [X: complex] :
( sums_complex
@ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ X @ N2 ) ) @ zero_zero_complex )
@ ( cosh_complex @ X ) ) ).
% cosh_converges
thf(fact_8954_i__even__power,axiom,
! [N: nat] :
( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% i_even_power
thf(fact_8955_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_8956_arctan__def,axiom,
( arctan
= ( ^ [Y: real] :
( the_real
@ ^ [X2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
& ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( tan_real @ X2 )
= Y ) ) ) ) ) ).
% arctan_def
thf(fact_8957_cosh__0,axiom,
( ( cosh_complex @ zero_zero_complex )
= one_one_complex ) ).
% cosh_0
thf(fact_8958_cosh__0,axiom,
( ( cosh_real @ zero_zero_real )
= one_one_real ) ).
% cosh_0
thf(fact_8959_log__one,axiom,
! [A: real] :
( ( log @ A @ one_one_real )
= zero_zero_real ) ).
% log_one
thf(fact_8960_norm__ii,axiom,
( ( real_V1022390504157884413omplex @ imaginary_unit )
= one_one_real ) ).
% norm_ii
thf(fact_8961_complex__i__mult__minus,axiom,
! [X: complex] :
( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
= ( uminus1482373934393186551omplex @ X ) ) ).
% complex_i_mult_minus
thf(fact_8962_zero__less__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_8963_log__less__zero__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_8964_one__less__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
= ( ord_less_real @ A @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_8965_log__less__one__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
= ( ord_less_real @ X @ A ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_8966_log__less__cancel__iff,axiom,
! [A: real,X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) )
= ( ord_less_real @ X @ Y4 ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_8967_log__eq__one,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ A )
= one_one_real ) ) ) ).
% log_eq_one
thf(fact_8968_divide__numeral__i,axiom,
! [Z2: complex,N: num] :
( ( divide1717551699836669952omplex @ Z2 @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
= ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% divide_numeral_i
thf(fact_8969_divide__i,axiom,
! [X: complex] :
( ( divide1717551699836669952omplex @ X @ imaginary_unit )
= ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% divide_i
thf(fact_8970_i__squared,axiom,
( ( times_times_complex @ imaginary_unit @ imaginary_unit )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% i_squared
thf(fact_8971_log__le__cancel__iff,axiom,
! [A: real,X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) )
= ( ord_less_eq_real @ X @ Y4 ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_8972_log__le__one__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ A ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_8973_one__le__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
= ( ord_less_eq_real @ A @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_8974_log__le__zero__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_8975_zero__le__log__cancel__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_8976_log__pow__cancel,axiom,
! [A: real,B: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( power_power_real @ A @ B ) )
= ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% log_pow_cancel
thf(fact_8977_power2__i,axiom,
( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% power2_i
thf(fact_8978_tanh__def,axiom,
( tanh_complex
= ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X2 ) @ ( cosh_complex @ X2 ) ) ) ) ).
% tanh_def
thf(fact_8979_tanh__def,axiom,
( tanh_real
= ( ^ [X2: real] : ( divide_divide_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ) ) ).
% tanh_def
thf(fact_8980_sinh__plus__cosh,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) )
= ( exp_complex @ X ) ) ).
% sinh_plus_cosh
thf(fact_8981_sinh__plus__cosh,axiom,
! [X: real] :
( ( plus_plus_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) )
= ( exp_real @ X ) ) ).
% sinh_plus_cosh
thf(fact_8982_cosh__plus__sinh,axiom,
! [X: complex] :
( ( plus_plus_complex @ ( cosh_complex @ X ) @ ( sinh_complex @ X ) )
= ( exp_complex @ X ) ) ).
% cosh_plus_sinh
thf(fact_8983_cosh__plus__sinh,axiom,
! [X: real] :
( ( plus_plus_real @ ( cosh_real @ X ) @ ( sinh_real @ X ) )
= ( exp_real @ X ) ) ).
% cosh_plus_sinh
thf(fact_8984_cosh__add,axiom,
! [X: real,Y4: real] :
( ( cosh_real @ ( plus_plus_real @ X @ Y4 ) )
= ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% cosh_add
thf(fact_8985_sinh__add,axiom,
! [X: real,Y4: real] :
( ( sinh_real @ ( plus_plus_real @ X @ Y4 ) )
= ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% sinh_add
thf(fact_8986_cosh__diff,axiom,
! [X: real,Y4: real] :
( ( cosh_real @ ( minus_minus_real @ X @ Y4 ) )
= ( minus_minus_real @ ( times_times_real @ ( cosh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( sinh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% cosh_diff
thf(fact_8987_sinh__diff,axiom,
! [X: real,Y4: real] :
( ( sinh_real @ ( minus_minus_real @ X @ Y4 ) )
= ( minus_minus_real @ ( times_times_real @ ( sinh_real @ X ) @ ( cosh_real @ Y4 ) ) @ ( times_times_real @ ( cosh_real @ X ) @ ( sinh_real @ Y4 ) ) ) ) ).
% sinh_diff
thf(fact_8988_complex__i__not__one,axiom,
imaginary_unit != one_one_complex ).
% complex_i_not_one
thf(fact_8989_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% cosh_real_ge_1
thf(fact_8990_sinh__double,axiom,
! [X: complex] :
( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
= ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X ) ) @ ( cosh_complex @ X ) ) ) ).
% sinh_double
thf(fact_8991_sinh__double,axiom,
! [X: real] :
( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X ) ) @ ( cosh_real @ X ) ) ) ).
% sinh_double
thf(fact_8992_i__times__eq__iff,axiom,
! [W2: complex,Z2: complex] :
( ( ( times_times_complex @ imaginary_unit @ W2 )
= Z2 )
= ( W2
= ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) ) ) ).
% i_times_eq_iff
thf(fact_8993_log__ln,axiom,
( ln_ln_real
= ( log @ ( exp_real @ one_one_real ) ) ) ).
% log_ln
thf(fact_8994_cosh__square__eq,axiom,
! [X: real] :
( ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% cosh_square_eq
thf(fact_8995_cosh__square__eq,axiom,
! [X: complex] :
( ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% cosh_square_eq
thf(fact_8996_sinh__square__eq,axiom,
! [X: complex] :
( ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).
% sinh_square_eq
thf(fact_8997_sinh__square__eq,axiom,
! [X: real] :
( ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).
% sinh_square_eq
thf(fact_8998_hyperbolic__pythagoras,axiom,
! [X: complex] :
( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_complex ) ).
% hyperbolic_pythagoras
thf(fact_8999_hyperbolic__pythagoras,axiom,
! [X: real] :
( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ).
% hyperbolic_pythagoras
thf(fact_9000_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% imaginary_unit.code
thf(fact_9001_Complex__eq__i,axiom,
! [X: real,Y4: real] :
( ( ( complex2 @ X @ Y4 )
= imaginary_unit )
= ( ( X = zero_zero_real )
& ( Y4 = one_one_real ) ) ) ).
% Complex_eq_i
thf(fact_9002_cosh__double,axiom,
! [X: complex] :
( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
= ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cosh_double
thf(fact_9003_cosh__double,axiom,
! [X: real] :
( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
= ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cosh_double
thf(fact_9004_i__mult__Complex,axiom,
! [A: real,B: real] :
( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
= ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% i_mult_Complex
thf(fact_9005_Complex__mult__i,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
= ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% Complex_mult_i
thf(fact_9006_log__base__change,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ B @ X )
= ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% log_base_change
thf(fact_9007_less__log__of__power,axiom,
! [B: real,N: nat,M: real] :
( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% less_log_of_power
thf(fact_9008_log__of__power__eq,axiom,
! [M: nat,B: real,N: nat] :
( ( ( semiri5074537144036343181t_real @ M )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_9009_log__mult,axiom,
! [A: real,X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( log @ A @ ( times_times_real @ X @ Y4 ) )
= ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) ) ) ) ) ) ) ).
% log_mult
thf(fact_9010_log__divide,axiom,
! [A: real,X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( log @ A @ ( divide_divide_real @ X @ Y4 ) )
= ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y4 ) ) ) ) ) ) ) ).
% log_divide
thf(fact_9011_le__log__of__power,axiom,
! [B: real,N: nat,M: real] :
( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% le_log_of_power
thf(fact_9012_log__nat__power,axiom,
! [X: real,B: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ B @ ( power_power_real @ X @ N ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% log_nat_power
thf(fact_9013_log__inverse,axiom,
! [A: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A @ ( inverse_inverse_real @ X ) )
= ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_9014_log2__of__power__eq,axiom,
! [M: nat,N: nat] :
( ( M
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( semiri5074537144036343181t_real @ N )
= ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% log2_of_power_eq
thf(fact_9015_log__of__power__less,axiom,
! [M: nat,B: real,N: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_9016_log__eq__div__ln__mult__log,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ A @ X )
= ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_9017_log__of__power__le,axiom,
! [M: nat,B: real,N: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_9018_tanh__add,axiom,
! [X: complex,Y4: complex] :
( ( ( cosh_complex @ X )
!= zero_zero_complex )
=> ( ( ( cosh_complex @ Y4 )
!= zero_zero_complex )
=> ( ( tanh_complex @ ( plus_plus_complex @ X @ Y4 ) )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y4 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X ) @ ( tanh_complex @ Y4 ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_9019_tanh__add,axiom,
! [X: real,Y4: real] :
( ( ( cosh_real @ X )
!= zero_zero_real )
=> ( ( ( cosh_real @ Y4 )
!= zero_zero_real )
=> ( ( tanh_real @ ( plus_plus_real @ X @ Y4 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X ) @ ( tanh_real @ Y4 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X ) @ ( tanh_real @ Y4 ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_9020_less__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% less_log2_of_power
thf(fact_9021_le__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% le_log2_of_power
thf(fact_9022_cosh__field__def,axiom,
( cosh_real
= ( ^ [Z4: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z4 ) @ ( exp_real @ ( uminus_uminus_real @ Z4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% cosh_field_def
thf(fact_9023_cosh__field__def,axiom,
( cosh_complex
= ( ^ [Z4: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z4 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% cosh_field_def
thf(fact_9024_sinh__field__def,axiom,
( sinh_real
= ( ^ [Z4: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z4 ) @ ( exp_real @ ( uminus_uminus_real @ Z4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% sinh_field_def
thf(fact_9025_sinh__field__def,axiom,
( sinh_complex
= ( ^ [Z4: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z4 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).
% sinh_field_def
thf(fact_9026_log2__of__power__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_9027_cosh__zero__iff,axiom,
! [X: real] :
( ( ( cosh_real @ X )
= zero_zero_real )
= ( ( power_power_real @ ( exp_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( uminus_uminus_real @ one_one_real ) ) ) ).
% cosh_zero_iff
thf(fact_9028_cosh__zero__iff,axiom,
! [X: complex] :
( ( ( cosh_complex @ X )
= zero_zero_complex )
= ( ( power_power_complex @ ( exp_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% cosh_zero_iff
thf(fact_9029_pi__half,axiom,
( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
= ( the_real
@ ^ [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ X2 )
= zero_zero_real ) ) ) ) ).
% pi_half
thf(fact_9030_pi__def,axiom,
( pi
= ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
@ ( the_real
@ ^ [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
& ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
& ( ( cos_real @ X2 )
= zero_zero_real ) ) ) ) ) ).
% pi_def
thf(fact_9031_cosh__def,axiom,
( cosh_real
= ( ^ [X2: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ) ).
% cosh_def
thf(fact_9032_cosh__def,axiom,
( cosh_complex
= ( ^ [X2: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) ) ) ) ).
% cosh_def
thf(fact_9033_cosh__ln__real,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( cosh_real @ ( ln_ln_real @ X ) )
= ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% cosh_ln_real
thf(fact_9034_log2__of__power__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_9035_sinh__def,axiom,
( sinh_real
= ( ^ [X2: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ) ).
% sinh_def
thf(fact_9036_sinh__def,axiom,
( sinh_complex
= ( ^ [X2: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) ) ) ) ).
% sinh_def
thf(fact_9037_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
= ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_9038_sinh__ln__real,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( sinh_real @ ( ln_ln_real @ X ) )
= ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% sinh_ln_real
thf(fact_9039_arcsin__def,axiom,
( arcsin
= ( ^ [Y: real] :
( the_real
@ ^ [X2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
& ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
& ( ( sin_real @ X2 )
= Y ) ) ) ) ) ).
% arcsin_def
thf(fact_9040_Arg__minus__ii,axiom,
( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
= ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% Arg_minus_ii
thf(fact_9041_ceiling__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
= ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_9042_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% Arg_ii
thf(fact_9043_ceiling__log__nat__eq__if,axiom,
! [B: nat,N: nat,K: nat] :
( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
=> ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_9044_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% ceiling_log2_div2
thf(fact_9045_ceiling__numeral,axiom,
! [V: num] :
( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
= ( numeral_numeral_int @ V ) ) ).
% ceiling_numeral
thf(fact_9046_ceiling__numeral,axiom,
! [V: num] :
( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
= ( numeral_numeral_int @ V ) ) ).
% ceiling_numeral
thf(fact_9047_ceiling__one,axiom,
( ( archim2889992004027027881ng_rat @ one_one_rat )
= one_one_int ) ).
% ceiling_one
thf(fact_9048_ceiling__one,axiom,
( ( archim7802044766580827645g_real @ one_one_real )
= one_one_int ) ).
% ceiling_one
thf(fact_9049_ceiling__add__of__int,axiom,
! [X: rat,Z2: int] :
( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) )
= ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z2 ) ) ).
% ceiling_add_of_int
thf(fact_9050_ceiling__add__of__int,axiom,
! [X: real,Z2: int] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z2 ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z2 ) ) ).
% ceiling_add_of_int
thf(fact_9051_ceiling__le__zero,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_le_zero
thf(fact_9052_ceiling__le__zero,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
= ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% ceiling_le_zero
thf(fact_9053_zero__less__ceiling,axiom,
! [X: rat] :
( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% zero_less_ceiling
thf(fact_9054_zero__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% zero_less_ceiling
thf(fact_9055_ceiling__le__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).
% ceiling_le_numeral
thf(fact_9056_ceiling__le__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).
% ceiling_le_numeral
thf(fact_9057_ceiling__less__one,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% ceiling_less_one
thf(fact_9058_ceiling__less__one,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
= ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).
% ceiling_less_one
thf(fact_9059_one__le__ceiling,axiom,
! [X: rat] :
( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ zero_zero_rat @ X ) ) ).
% one_le_ceiling
thf(fact_9060_one__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ zero_zero_real @ X ) ) ).
% one_le_ceiling
thf(fact_9061_numeral__less__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).
% numeral_less_ceiling
thf(fact_9062_numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).
% numeral_less_ceiling
thf(fact_9063_ceiling__le__one,axiom,
! [X: real] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
= ( ord_less_eq_real @ X @ one_one_real ) ) ).
% ceiling_le_one
thf(fact_9064_ceiling__le__one,axiom,
! [X: rat] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
= ( ord_less_eq_rat @ X @ one_one_rat ) ) ).
% ceiling_le_one
thf(fact_9065_one__less__ceiling,axiom,
! [X: rat] :
( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ one_one_rat @ X ) ) ).
% one_less_ceiling
thf(fact_9066_one__less__ceiling,axiom,
! [X: real] :
( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ).
% one_less_ceiling
thf(fact_9067_ceiling__add__numeral,axiom,
! [X: real,V: num] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% ceiling_add_numeral
thf(fact_9068_ceiling__add__numeral,axiom,
! [X: rat,V: num] :
( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
= ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% ceiling_add_numeral
thf(fact_9069_ceiling__neg__numeral,axiom,
! [V: num] :
( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% ceiling_neg_numeral
thf(fact_9070_ceiling__neg__numeral,axiom,
! [V: num] :
( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).
% ceiling_neg_numeral
thf(fact_9071_ceiling__add__one,axiom,
! [X: rat] :
( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
= ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% ceiling_add_one
thf(fact_9072_ceiling__add__one,axiom,
! [X: real] :
( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
= ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% ceiling_add_one
thf(fact_9073_ceiling__diff__numeral,axiom,
! [X: real,V: num] :
( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
= ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% ceiling_diff_numeral
thf(fact_9074_ceiling__diff__numeral,axiom,
! [X: rat,V: num] :
( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
= ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).
% ceiling_diff_numeral
thf(fact_9075_ceiling__diff__one,axiom,
! [X: rat] :
( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
= ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).
% ceiling_diff_one
thf(fact_9076_ceiling__diff__one,axiom,
! [X: real] :
( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
= ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).
% ceiling_diff_one
thf(fact_9077_ceiling__numeral__power,axiom,
! [X: num,N: nat] :
( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% ceiling_numeral_power
thf(fact_9078_ceiling__numeral__power,axiom,
! [X: num,N: nat] :
( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ).
% ceiling_numeral_power
thf(fact_9079_ceiling__less__zero,axiom,
! [X: real] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
= ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% ceiling_less_zero
thf(fact_9080_ceiling__less__zero,axiom,
! [X: rat] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
= ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% ceiling_less_zero
thf(fact_9081_zero__le__ceiling,axiom,
! [X: real] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).
% zero_le_ceiling
thf(fact_9082_zero__le__ceiling,axiom,
! [X: rat] :
( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).
% zero_le_ceiling
thf(fact_9083_ceiling__less__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% ceiling_less_numeral
thf(fact_9084_ceiling__less__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% ceiling_less_numeral
thf(fact_9085_numeral__le__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).
% numeral_le_ceiling
thf(fact_9086_numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).
% numeral_le_ceiling
thf(fact_9087_ceiling__le__neg__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_9088_ceiling__le__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_9089_neg__numeral__less__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).
% neg_numeral_less_ceiling
thf(fact_9090_neg__numeral__less__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).
% neg_numeral_less_ceiling
thf(fact_9091_ceiling__less__neg__numeral,axiom,
! [X: real,V: num] :
( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_9092_ceiling__less__neg__numeral,axiom,
! [X: rat,V: num] :
( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
= ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_9093_neg__numeral__le__ceiling,axiom,
! [V: num,X: real] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
= ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).
% neg_numeral_le_ceiling
thf(fact_9094_neg__numeral__le__ceiling,axiom,
! [V: num,X: rat] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
= ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).
% neg_numeral_le_ceiling
thf(fact_9095_ceiling__mono,axiom,
! [Y4: real,X: real] :
( ( ord_less_eq_real @ Y4 @ X )
=> ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y4 ) @ ( archim7802044766580827645g_real @ X ) ) ) ).
% ceiling_mono
thf(fact_9096_ceiling__mono,axiom,
! [Y4: rat,X: rat] :
( ( ord_less_eq_rat @ Y4 @ X )
=> ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y4 ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% ceiling_mono
thf(fact_9097_le__of__int__ceiling,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).
% le_of_int_ceiling
thf(fact_9098_le__of__int__ceiling,axiom,
! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).
% le_of_int_ceiling
thf(fact_9099_cis__minus__pi__half,axiom,
( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
= ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% cis_minus_pi_half
thf(fact_9100_ceiling__log__eq__powr__iff,axiom,
! [X: real,B: real,K: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
= ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
& ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_9101_floor__log__nat__eq__powr__iff,axiom,
! [B: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( semiri1314217659103216013at_int @ N ) )
= ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_9102_floor__log__nat__eq__if,axiom,
! [B: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
=> ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
=> ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
= ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_9103_even__set__encode__iff,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
= ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% even_set_encode_iff
thf(fact_9104_powr__less__cancel__iff,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
= ( ord_less_real @ A @ B ) ) ) ).
% powr_less_cancel_iff
thf(fact_9105_norm__cis,axiom,
! [A: real] :
( ( real_V1022390504157884413omplex @ ( cis @ A ) )
= one_one_real ) ).
% norm_cis
thf(fact_9106_cis__zero,axiom,
( ( cis @ zero_zero_real )
= one_one_complex ) ).
% cis_zero
thf(fact_9107_set__encode__empty,axiom,
( ( nat_set_encode @ bot_bot_set_nat )
= zero_zero_nat ) ).
% set_encode_empty
thf(fact_9108_powr__eq__one__iff,axiom,
! [A: real,X: real] :
( ( ord_less_real @ one_one_real @ A )
=> ( ( ( powr_real @ A @ X )
= one_one_real )
= ( X = zero_zero_real ) ) ) ).
% powr_eq_one_iff
thf(fact_9109_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr_real @ X @ one_one_real )
= X )
= ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_9110_powr__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ one_one_real )
= X ) ) ).
% powr_one
thf(fact_9111_powr__le__cancel__iff,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ) ).
% powr_le_cancel_iff
thf(fact_9112_numeral__powr__numeral__real,axiom,
! [M: num,N: num] :
( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_powr_numeral_real
thf(fact_9113_cis__pi,axiom,
( ( cis @ pi )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% cis_pi
thf(fact_9114_log__powr__cancel,axiom,
! [A: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( log @ A @ ( powr_real @ A @ Y4 ) )
= Y4 ) ) ) ).
% log_powr_cancel
thf(fact_9115_powr__log__cancel,axiom,
! [A: real,X: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ A @ ( log @ A @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_9116_powr__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
= ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_9117_cis__pi__half,axiom,
( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_9118_floor__one__divide__eq__div__numeral,axiom,
! [B: num] :
( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
= ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% floor_one_divide_eq_div_numeral
thf(fact_9119_cis__2pi,axiom,
( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
= one_one_complex ) ).
% cis_2pi
thf(fact_9120_square__powr__half,axiom,
! [X: real] :
( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( abs_abs_real @ X ) ) ).
% square_powr_half
thf(fact_9121_floor__minus__one__divide__eq__div__numeral,axiom,
! [B: num] :
( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
= ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_9122_powr__powr,axiom,
! [X: real,A: real,B: real] :
( ( powr_real @ ( powr_real @ X @ A ) @ B )
= ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% powr_powr
thf(fact_9123_powr__less__mono,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% powr_less_mono
thf(fact_9124_powr__less__cancel,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ A @ B ) ) ) ).
% powr_less_cancel
thf(fact_9125_powr__mono,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% powr_mono
thf(fact_9126_cis__mult,axiom,
! [A: real,B: real] :
( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
= ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% cis_mult
thf(fact_9127_gr__one__powr,axiom,
! [X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y4 ) ) ) ) ).
% gr_one_powr
thf(fact_9128_powr__inj,axiom,
! [A: real,X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( A != one_one_real )
=> ( ( ( powr_real @ A @ X )
= ( powr_real @ A @ Y4 ) )
= ( X = Y4 ) ) ) ) ).
% powr_inj
thf(fact_9129_ge__one__powr__ge__zero,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_9130_powr__mono__both,axiom,
! [A: real,B: real,X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ B ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_9131_powr__le1,axiom,
! [A: real,X: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% powr_le1
thf(fact_9132_powr__mult,axiom,
! [X: real,Y4: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( powr_real @ ( times_times_real @ X @ Y4 ) @ A )
= ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y4 @ A ) ) ) ) ) ).
% powr_mult
thf(fact_9133_divide__powr__uminus,axiom,
! [A: real,B: real,C: real] :
( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
= ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% divide_powr_uminus
thf(fact_9134_ln__powr,axiom,
! [X: real,Y4: real] :
( ( X != zero_zero_real )
=> ( ( ln_ln_real @ ( powr_real @ X @ Y4 ) )
= ( times_times_real @ Y4 @ ( ln_ln_real @ X ) ) ) ) ).
% ln_powr
thf(fact_9135_log__powr,axiom,
! [X: real,B: real,Y4: real] :
( ( X != zero_zero_real )
=> ( ( log @ B @ ( powr_real @ X @ Y4 ) )
= ( times_times_real @ Y4 @ ( log @ B @ X ) ) ) ) ).
% log_powr
thf(fact_9136_floor__log__eq__powr__iff,axiom,
! [X: real,B: real,K: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ one_one_real @ B )
=> ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
= K )
= ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
& ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_9137_powr__less__iff,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( powr_real @ B @ Y4 ) @ X )
= ( ord_less_real @ Y4 @ ( log @ B @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_9138_less__powr__iff,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ ( powr_real @ B @ Y4 ) )
= ( ord_less_real @ ( log @ B @ X ) @ Y4 ) ) ) ) ).
% less_powr_iff
thf(fact_9139_log__less__iff,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( log @ B @ X ) @ Y4 )
= ( ord_less_real @ X @ ( powr_real @ B @ Y4 ) ) ) ) ) ).
% log_less_iff
thf(fact_9140_less__log__iff,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y4 @ ( log @ B @ X ) )
= ( ord_less_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_9141_real__of__int__floor__add__one__gt,axiom,
! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_9142_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_9143_real__of__int__floor__add__one__ge,axiom,
! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_9144_real__of__int__floor__gt__diff__one,axiom,
! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_9145_real__of__int__floor__ge__diff__one,axiom,
! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_9146_DeMoivre,axiom,
! [A: real,N: nat] :
( ( power_power_complex @ ( cis @ A ) @ N )
= ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% DeMoivre
thf(fact_9147_powr__neg__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
= ( divide_divide_real @ one_one_real @ X ) ) ) ).
% powr_neg_one
thf(fact_9148_powr__mult__base,axiom,
! [X: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( times_times_real @ X @ ( powr_real @ X @ Y4 ) )
= ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y4 ) ) ) ) ).
% powr_mult_base
thf(fact_9149_powr__le__iff,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( powr_real @ B @ Y4 ) @ X )
= ( ord_less_eq_real @ Y4 @ ( log @ B @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_9150_le__powr__iff,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y4 ) )
= ( ord_less_eq_real @ ( log @ B @ X ) @ Y4 ) ) ) ) ).
% le_powr_iff
thf(fact_9151_log__le__iff,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y4 )
= ( ord_less_eq_real @ X @ ( powr_real @ B @ Y4 ) ) ) ) ) ).
% log_le_iff
thf(fact_9152_le__log__iff,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y4 @ ( log @ B @ X ) )
= ( ord_less_eq_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_9153_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
=> ( ( archim6058952711729229775r_real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_9154_ln__powr__bound,axiom,
! [X: real,A: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% ln_powr_bound
thf(fact_9155_ln__powr__bound2,axiom,
! [X: real,A: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_9156_log__add__eq__powr,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( plus_plus_real @ ( log @ B @ X ) @ Y4 )
= ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y4 ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_9157_add__log__eq__powr,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( plus_plus_real @ Y4 @ ( log @ B @ X ) )
= ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_9158_minus__log__eq__powr,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ Y4 @ ( log @ B @ X ) )
= ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y4 ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_9159_log__minus__eq__powr,axiom,
! [B: real,X: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ B )
=> ( ( B != one_one_real )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( minus_minus_real @ ( log @ B @ X ) @ Y4 )
= ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y4 ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_9160_powr__half__sqrt,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( sqrt @ X ) ) ) ).
% powr_half_sqrt
thf(fact_9161_powr__neg__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_9162_set__encode__def,axiom,
( nat_set_encode
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% set_encode_def
thf(fact_9163_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
= ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% floor_log2_div2
thf(fact_9164_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_betw_nat_complex
@ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
@ ( set_ord_lessThan_nat @ N )
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= one_one_complex ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_9165_exp__pi__i_H,axiom,
( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% exp_pi_i'
thf(fact_9166_exp__pi__i,axiom,
( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
= ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% exp_pi_i
thf(fact_9167_exp__two__pi__i_H,axiom,
( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
= one_one_complex ) ).
% exp_two_pi_i'
thf(fact_9168_exp__two__pi__i,axiom,
( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
= one_one_complex ) ).
% exp_two_pi_i
thf(fact_9169_complex__exp__exists,axiom,
! [Z2: complex] :
? [A3: complex,R3: real] :
( Z2
= ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A3 ) ) ) ).
% complex_exp_exists
thf(fact_9170_complex__of__real__mult__Complex,axiom,
! [R2: real,X: real,Y4: real] :
( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y4 ) )
= ( complex2 @ ( times_times_real @ R2 @ X ) @ ( times_times_real @ R2 @ Y4 ) ) ) ).
% complex_of_real_mult_Complex
thf(fact_9171_Complex__mult__complex__of__real,axiom,
! [X: real,Y4: real,R2: real] :
( ( times_times_complex @ ( complex2 @ X @ Y4 ) @ ( real_V4546457046886955230omplex @ R2 ) )
= ( complex2 @ ( times_times_real @ X @ R2 ) @ ( times_times_real @ Y4 @ R2 ) ) ) ).
% Complex_mult_complex_of_real
thf(fact_9172_Complex__add__complex__of__real,axiom,
! [X: real,Y4: real,R2: real] :
( ( plus_plus_complex @ ( complex2 @ X @ Y4 ) @ ( real_V4546457046886955230omplex @ R2 ) )
= ( complex2 @ ( plus_plus_real @ X @ R2 ) @ Y4 ) ) ).
% Complex_add_complex_of_real
thf(fact_9173_complex__of__real__add__Complex,axiom,
! [R2: real,X: real,Y4: real] :
( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y4 ) )
= ( complex2 @ ( plus_plus_real @ R2 @ X ) @ Y4 ) ) ).
% complex_of_real_add_Complex
thf(fact_9174_cis__conv__exp,axiom,
( cis
= ( ^ [B3: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).
% cis_conv_exp
thf(fact_9175_i__complex__of__real,axiom,
! [R2: real] :
( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
= ( complex2 @ zero_zero_real @ R2 ) ) ).
% i_complex_of_real
thf(fact_9176_complex__of__real__i,axiom,
! [R2: real] :
( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
= ( complex2 @ zero_zero_real @ R2 ) ) ).
% complex_of_real_i
thf(fact_9177_Complex__eq,axiom,
( complex2
= ( ^ [A4: real,B3: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A4 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).
% Complex_eq
thf(fact_9178_complex__split__polar,axiom,
! [Z2: complex] :
? [R3: real,A3: real] :
( Z2
= ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A3 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A3 ) ) ) ) ) ) ).
% complex_split_polar
thf(fact_9179_cmod__unit__one,axiom,
! [A: real] :
( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
= one_one_real ) ).
% cmod_unit_one
thf(fact_9180_cmod__complex__polar,axiom,
! [R2: real,A: real] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
= ( abs_abs_real @ R2 ) ) ).
% cmod_complex_polar
thf(fact_9181_csqrt__ii,axiom,
( ( csqrt @ imaginary_unit )
= ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% csqrt_ii
thf(fact_9182_modulo__int__unfold,axiom,
! [L2: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L2 )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn_int @ L2 )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L2 ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
& ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L2 ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( times_times_int @ ( sgn_sgn_int @ L2 )
@ ( minus_minus_int
@ ( semiri1314217659103216013at_int
@ ( times_times_nat @ N
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_9183_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(3)
thf(fact_9184_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_9185_csqrt__eq__1,axiom,
! [Z2: complex] :
( ( ( csqrt @ Z2 )
= one_one_complex )
= ( Z2 = one_one_complex ) ) ).
% csqrt_eq_1
thf(fact_9186_csqrt__1,axiom,
( ( csqrt @ one_one_complex )
= one_one_complex ) ).
% csqrt_1
thf(fact_9187_dvd__mult__sgn__iff,axiom,
! [L2: int,K: int,R2: int] :
( ( dvd_dvd_int @ L2 @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
= ( ( dvd_dvd_int @ L2 @ K )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_9188_dvd__sgn__mult__iff,axiom,
! [L2: int,R2: int,K: int] :
( ( dvd_dvd_int @ L2 @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
= ( ( dvd_dvd_int @ L2 @ K )
| ( R2 = zero_zero_int ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_9189_mult__sgn__dvd__iff,axiom,
! [L2: int,R2: int,K: int] :
( ( dvd_dvd_int @ ( times_times_int @ L2 @ ( sgn_sgn_int @ R2 ) ) @ K )
= ( ( dvd_dvd_int @ L2 @ K )
& ( ( R2 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_9190_sgn__mult__dvd__iff,axiom,
! [R2: int,L2: int,K: int] :
( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L2 ) @ K )
= ( ( dvd_dvd_int @ L2 @ K )
& ( ( R2 = zero_zero_int )
=> ( K = zero_zero_int ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_9191_power2__csqrt,axiom,
! [Z2: complex] :
( ( power_power_complex @ ( csqrt @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= Z2 ) ).
% power2_csqrt
thf(fact_9192_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% less_eq_mask
thf(fact_9193_int__sgnE,axiom,
! [K: int] :
~ ! [N3: nat,L4: int] :
( K
!= ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_sgnE
thf(fact_9194_sgn__mod,axiom,
! [L2: int,K: int] :
( ( L2 != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ L2 @ K )
=> ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
= ( sgn_sgn_int @ L2 ) ) ) ) ).
% sgn_mod
thf(fact_9195_less__mask,axiom,
! [N: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% less_mask
thf(fact_9196_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_9197_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L2: int] :
( ( V != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L2 ) ) )
= ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_9198_div__dvd__sgn__abs,axiom,
! [L2: int,K: int] :
( ( dvd_dvd_int @ L2 @ K )
=> ( ( divide_divide_int @ K @ L2 )
= ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_9199_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_9200_Suc__mask__eq__exp,axiom,
! [N: nat] :
( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_mask_eq_exp
thf(fact_9201_mask__nat__less__exp,axiom,
! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% mask_nat_less_exp
thf(fact_9202_eucl__rel__int__remainderI,axiom,
! [R2: int,L2: int,K: int,Q2: int] :
( ( ( sgn_sgn_int @ R2 )
= ( sgn_sgn_int @ L2 ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L2 ) )
=> ( ( K
= ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R2 ) )
=> ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_9203_mask__nat__def,axiom,
( bit_se2002935070580805687sk_nat
= ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% mask_nat_def
thf(fact_9204_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% mask_half_int
thf(fact_9205_mask__int__def,axiom,
( bit_se2000444600071755411sk_int
= ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% mask_int_def
thf(fact_9206_eucl__rel__int_Ocases,axiom,
! [A12: int,A23: int,A32: product_prod_int_int] :
( ( eucl_rel_int @ A12 @ A23 @ A32 )
=> ( ( ( A23 = zero_zero_int )
=> ( A32
!= ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
=> ( ! [Q3: int] :
( ( A32
= ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
=> ( ( A23 != zero_zero_int )
=> ( A12
!= ( times_times_int @ Q3 @ A23 ) ) ) )
=> ~ ! [R3: int,Q3: int] :
( ( A32
= ( product_Pair_int_int @ Q3 @ R3 ) )
=> ( ( ( sgn_sgn_int @ R3 )
= ( sgn_sgn_int @ A23 ) )
=> ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
=> ( A12
!= ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_9207_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
( ? [K3: int] :
( ( A1 = K3 )
& ( A22 = zero_zero_int )
& ( A33
= ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
| ? [L: int,K3: int,Q4: int] :
( ( A1 = K3 )
& ( A22 = L )
& ( A33
= ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
& ( L != zero_zero_int )
& ( K3
= ( times_times_int @ Q4 @ L ) ) )
| ? [R5: int,L: int,K3: int,Q4: int] :
( ( A1 = K3 )
& ( A22 = L )
& ( A33
= ( product_Pair_int_int @ Q4 @ R5 ) )
& ( ( sgn_sgn_int @ R5 )
= ( sgn_sgn_int @ L ) )
& ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
& ( K3
= ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_9208_num_Osize__gen_I2_J,axiom,
! [X22: num] :
( ( size_num @ ( bit0 @ X22 ) )
= ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_9209_divide__int__unfold,axiom,
! [L2: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn_int @ L2 )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_int ) )
& ( ~ ( ( ( sgn_sgn_int @ L2 )
= zero_zero_int )
| ( ( sgn_sgn_int @ K )
= zero_zero_int )
| ( N = zero_zero_nat ) )
=> ( ( ( ( sgn_sgn_int @ K )
= ( sgn_sgn_int @ L2 ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn_int @ K )
!= ( sgn_sgn_int @ L2 ) )
=> ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
= ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_9210_bij__betw__nth__root__unity,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= one_one_complex ) )
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= C ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_9211_modulo__int__def,axiom,
( modulo_modulo_int
= ( ^ [K3: int,L: int] :
( if_int @ ( L = zero_zero_int ) @ K3
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L ) )
@ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
@ ( times_times_int @ ( sgn_sgn_int @ L )
@ ( minus_minus_int
@ ( times_times_int @ ( abs_abs_int @ L )
@ ( zero_n2684676970156552555ol_int
@ ~ ( dvd_dvd_int @ L @ K3 ) ) )
@ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_9212_divide__int__def,axiom,
( divide_divide_int
= ( ^ [K3: int,L: int] :
( if_int @ ( L = zero_zero_int ) @ zero_zero_int
@ ( if_int
@ ( ( sgn_sgn_int @ K3 )
= ( sgn_sgn_int @ L ) )
@ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
@ ( uminus_uminus_int
@ ( semiri1314217659103216013at_int
@ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_int @ L @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_9213_powr__int,axiom,
! [X: real,I2: int] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ I2 ) )
= ( power_power_real @ X @ ( nat2 @ I2 ) ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( powr_real @ X @ ( ring_1_of_int_real @ I2 ) )
= ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I2 ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_9214_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% nat_numeral
thf(fact_9215_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_9216_real__root__eq__iff,axiom,
! [N: nat,X: real,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y4 ) )
= ( X = Y4 ) ) ) ).
% real_root_eq_iff
thf(fact_9217_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_9218_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_9219_real__root__less__iff,axiom,
! [N: nat,X: real,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) )
= ( ord_less_real @ X @ Y4 ) ) ) ).
% real_root_less_iff
thf(fact_9220_real__root__le__iff,axiom,
! [N: nat,X: real,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) )
= ( ord_less_eq_real @ X @ Y4 ) ) ) ).
% real_root_le_iff
thf(fact_9221_zless__nat__conj,axiom,
! [W2: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W2 @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_9222_real__root__one,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_9223_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( root @ N @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_9224_real__root__gt__0__iff,axiom,
! [N: nat,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y4 ) )
= ( ord_less_real @ zero_zero_real @ Y4 ) ) ) ).
% real_root_gt_0_iff
thf(fact_9225_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_9226_real__root__ge__0__iff,axiom,
! [N: nat,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y4 ) )
= ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) ).
% real_root_ge_0_iff
thf(fact_9227_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_9228_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_9229_real__root__gt__1__iff,axiom,
! [N: nat,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ one_one_real @ ( root @ N @ Y4 ) )
= ( ord_less_real @ one_one_real @ Y4 ) ) ) ).
% real_root_gt_1_iff
thf(fact_9230_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_9231_real__root__ge__1__iff,axiom,
! [N: nat,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y4 ) )
= ( ord_less_eq_real @ one_one_real @ Y4 ) ) ) ).
% real_root_ge_1_iff
thf(fact_9232_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_9233_diff__nat__numeral,axiom,
! [V: num,V3: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% diff_nat_numeral
thf(fact_9234_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y4: int] :
( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
= ( nat2 @ Y4 ) )
= ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
= Y4 ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_9235_nat__eq__numeral__power__cancel__iff,axiom,
! [Y4: int,X: num,N: nat] :
( ( ( nat2 @ Y4 )
= ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( Y4
= ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_9236_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_9237_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_9238_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_9239_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
= ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% nat_numeral_diff_1
thf(fact_9240_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
= ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_9241_nat__less__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_9242_nat__le__numeral__power__cancel__iff,axiom,
! [A: int,X: num,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
= ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_9243_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A: int] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_9244_sgn__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( sgn_sgn_real @ ( root @ N @ X ) )
= ( sgn_sgn_real @ X ) ) ) ).
% sgn_root
thf(fact_9245_real__root__mult,axiom,
! [N: nat,X: real,Y4: real] :
( ( root @ N @ ( times_times_real @ X @ Y4 ) )
= ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ).
% real_root_mult
thf(fact_9246_real__root__mult__exp,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ ( times_times_nat @ M @ N ) @ X )
= ( root @ M @ ( root @ N @ X ) ) ) ).
% real_root_mult_exp
thf(fact_9247_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_9248_nat__numeral__as__int,axiom,
( numeral_numeral_nat
= ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_9249_nat__mono,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ).
% nat_mono
thf(fact_9250_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_9251_root__sgn__power,axiom,
! [N: nat,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N ) ) )
= Y4 ) ) ).
% root_sgn_power
thf(fact_9252_sgn__power__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
= X ) ) ).
% sgn_power_root
thf(fact_9253_split__root,axiom,
! [P: real > $o,N: nat,X: real] :
( ( P @ ( root @ N @ X ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_real ) )
& ( ( ord_less_nat @ zero_zero_nat @ N )
=> ! [Y: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
= X )
=> ( P @ Y ) ) ) ) ) ).
% split_root
thf(fact_9254_real__root__less__mono,axiom,
! [N: nat,X: real,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ X @ Y4 )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ) ).
% real_root_less_mono
thf(fact_9255_real__root__le__mono,axiom,
! [N: nat,X: real,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ X @ Y4 )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y4 ) ) ) ) ).
% real_root_le_mono
thf(fact_9256_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( power_power_real @ X @ K ) )
= ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_9257_real__root__abs,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( root @ N @ ( abs_abs_real @ X ) )
= ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% real_root_abs
thf(fact_9258_nat__mono__iff,axiom,
! [Z2: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_9259_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_9260_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_9261_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_9262_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% int_minus
thf(fact_9263_nat__abs__mult__distrib,axiom,
! [W2: int,Z2: int] :
( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z2 ) ) )
= ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_9264_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_9265_nat__times__as__int,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_9266_nat__minus__as__int,axiom,
( minus_minus_nat
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_9267_nat__div__as__int,axiom,
( divide_divide_nat
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_9268_nat__mod__as__int,axiom,
( modulo_modulo_nat
= ( ^ [A4: nat,B3: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_9269_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_9270_real__root__strict__decreasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_9271_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% sqrt_def
thf(fact_9272_sgn__real__def,axiom,
( sgn_sgn_real
= ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% sgn_real_def
thf(fact_9273_root__abs__power,axiom,
! [N: nat,Y4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y4 @ N ) ) )
= ( abs_abs_real @ Y4 ) ) ) ).
% root_abs_power
thf(fact_9274_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W2 @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_9275_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_9276_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_9277_nat__add__distrib,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( nat2 @ ( plus_plus_int @ Z2 @ Z6 ) )
= ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_9278_nat__mult__distrib,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z6 ) )
= ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_9279_Suc__as__int,axiom,
( suc
= ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_9280_nat__abs__triangle__ineq,axiom,
! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_9281_nat__div__distrib_H,axiom,
! [Y4: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y4 ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ).
% nat_div_distrib'
thf(fact_9282_nat__div__distrib,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y4 ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ).
% nat_div_distrib
thf(fact_9283_nat__power__eq,axiom,
! [Z2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
= ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_9284_nat__mod__distrib,axiom,
! [X: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( nat2 @ ( modulo_modulo_int @ X @ Y4 ) )
= ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y4 ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_9285_div__abs__eq__div__nat,axiom,
! [K: int,L2: int] :
( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
= ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_9286_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_9287_le__nat__floor,axiom,
! [X: nat,A: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
=> ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% le_nat_floor
thf(fact_9288_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_9289_real__root__strict__increasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_9290_real__root__decreasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_9291_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_9292_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( root @ N @ ( power_power_real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_9293_real__root__pos__unique,axiom,
! [N: nat,Y4: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ( power_power_real @ Y4 @ N )
= X )
=> ( ( root @ N @ X )
= Y4 ) ) ) ) ).
% real_root_pos_unique
thf(fact_9294_odd__real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( root @ N @ ( power_power_real @ X @ N ) )
= X ) ) ).
% odd_real_root_power_cancel
thf(fact_9295_odd__real__root__unique,axiom,
! [N: nat,Y4: real,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ( power_power_real @ Y4 @ N )
= X )
=> ( ( root @ N @ X )
= Y4 ) ) ) ).
% odd_real_root_unique
thf(fact_9296_odd__real__root__pow,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( power_power_real @ ( root @ N @ X ) @ N )
= X ) ) ).
% odd_real_root_pow
thf(fact_9297_nat__2,axiom,
( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% nat_2
thf(fact_9298_sgn__power__injE,axiom,
! [A: real,N: nat,X: real,B: real] :
( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
= X )
=> ( ( X
= ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ).
% sgn_power_injE
thf(fact_9299_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_9300_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_9301_nat__mult__distrib__neg,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z2 @ Z6 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_9302_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_9303_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
=> ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_9304_real__root__increasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_9305_ln__root,axiom,
! [N: nat,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ln_ln_real @ ( root @ N @ B ) )
= ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% ln_root
thf(fact_9306_log__root,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( log @ B @ ( root @ N @ A ) )
= ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% log_root
thf(fact_9307_log__base__root,axiom,
! [N: nat,B: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( log @ ( root @ N @ B ) @ X )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% log_base_root
thf(fact_9308_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( root @ N @ X )
= ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_9309_even__nat__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% even_nat_iff
thf(fact_9310_arctan__inverse,axiom,
! [X: real] :
( ( X != zero_zero_real )
=> ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
= ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% arctan_inverse
thf(fact_9311_and__int__unfold,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L: int] :
( if_int
@ ( ( K3 = zero_zero_int )
| ( L = zero_zero_int ) )
@ zero_zero_int
@ ( if_int
@ ( K3
= ( uminus_uminus_int @ one_one_int ) )
@ L
@ ( if_int
@ ( L
= ( uminus_uminus_int @ one_one_int ) )
@ K3
@ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_9312_take__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% take_bit_of_Suc_0
thf(fact_9313_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= one_one_int ) ).
% and_minus_numerals(6)
thf(fact_9314_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= one_one_int ) ).
% and_minus_numerals(2)
thf(fact_9315_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= zero_zero_int ) ).
% and_minus_numerals(5)
thf(fact_9316_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= zero_zero_int ) ).
% and_minus_numerals(1)
thf(fact_9317_take__bit__mult,axiom,
! [N: nat,K: int,L2: int] :
( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
= ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% take_bit_mult
thf(fact_9318_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_9319_take__bit__nat__less__eq__self,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_9320_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_9321_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one )
= X ) ).
% pow.simps(1)
thf(fact_9322_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
!= zero_zero_int )
=> ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
= ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% take_bit_decr_eq
thf(fact_9323_even__and__iff__int,axiom,
! [K: int,L2: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
= ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
| ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).
% even_and_iff_int
thf(fact_9324_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
= zero_zero_int ) ) ).
% take_bit_eq_mask_iff
thf(fact_9325_take__bit__nat__eq__self,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2925701944663578781it_nat @ N @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_9326_take__bit__nat__less__exp,axiom,
! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_9327_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M: nat] :
( ( ( bit_se2925701944663578781it_nat @ N @ M )
= M )
= ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_9328_take__bit__nat__def,axiom,
( bit_se2925701944663578781it_nat
= ( ^ [N2: nat,M2: nat] : ( modulo_modulo_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% take_bit_nat_def
thf(fact_9329_take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_9330_take__bit__int__def,axiom,
( bit_se2923211474154528505it_int
= ( ^ [N2: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% take_bit_int_def
thf(fact_9331_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
= ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_9332_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_9333_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
= ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_9334_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
= ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_9335_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_9336_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= K )
= ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_9337_take__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_9338_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
!= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
= ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_9339_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_9340_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ K @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_9341_signed__take__bit__eq__take__bit__shift,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_9342_and__int__rec,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_9343_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2923211474154528505it_int @ N @ K )
= ( bit_se2000444600071755411sk_int @ N ) )
= ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_9344_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
= ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_9345_take__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_9346_and__int_Oelims,axiom,
! [X: int,Xa2: int,Y4: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
= Y4 )
=> ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y4
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y4
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_9347_and__int_Osimps,axiom,
( bit_se725231765392027082nd_int
= ( ^ [K3: int,L: int] :
( if_int
@ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
@ ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
@ ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_9348_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_9349_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% pred_numeral_inc
thf(fact_9350_and__nat__numerals_I1_J,axiom,
! [Y4: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
= zero_zero_nat ) ).
% and_nat_numerals(1)
thf(fact_9351_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% and_nat_numerals(3)
thf(fact_9352_and__nat__numerals_I2_J,axiom,
! [Y4: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
= one_one_nat ) ).
% and_nat_numerals(2)
thf(fact_9353_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% and_nat_numerals(4)
thf(fact_9354_and__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
= ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% and_Suc_0_eq
thf(fact_9355_Suc__0__and__eq,axiom,
! [N: nat] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
= ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Suc_0_and_eq
thf(fact_9356_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_9357_add__inc,axiom,
! [X: num,Y4: num] :
( ( plus_plus_num @ X @ ( inc @ Y4 ) )
= ( inc @ ( plus_plus_num @ X @ Y4 ) ) ) ).
% add_inc
thf(fact_9358_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_9359_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_9360_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_9361_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_9362_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_9363_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_9364_mult__inc,axiom,
! [X: num,Y4: num] :
( ( times_times_num @ X @ ( inc @ Y4 ) )
= ( plus_plus_num @ ( times_times_num @ X @ Y4 ) @ X ) ) ).
% mult_inc
thf(fact_9365_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
=> ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
= ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_9366_simp__from__to,axiom,
( set_or1266510415728281911st_int
= ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_9367_and__nat__unfold,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M2: nat,N2: nat] :
( if_nat
@ ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_9368_and__nat__rec,axiom,
( bit_se727722235901077358nd_nat
= ( ^ [M2: nat,N2: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
& ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_9369_and__int_Opsimps,axiom,
! [K: int,L2: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
=> ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K @ L2 )
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
& ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( ( bit_se725231765392027082nd_int @ K @ L2 )
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_9370_and__int_Opelims,axiom,
! [X: int,Xa2: int,Y4: int] :
( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
= Y4 )
=> ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
=> ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y4
= ( uminus_uminus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( Y4
= ( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
& ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% and_int.pelims
thf(fact_9371_signed__take__bit__eq__take__bit__minus,axiom,
( bit_ri631733984087533419it_int
= ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_9372_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_9373_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9374_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9375_bit__minus__numeral__int_I1_J,axiom,
! [W2: num,N: num] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( numeral_numeral_nat @ N ) )
= ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_9376_bit__minus__numeral__int_I2_J,axiom,
! [W2: num,N: num] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( numeral_numeral_nat @ N ) )
= ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_9377_set__encode__insert,axiom,
! [A2: set_nat,N: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ N @ A2 )
=> ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
= ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% set_encode_insert
thf(fact_9378_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_9379_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_9380_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_9381_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_9382_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_9383_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
= ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_9384_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or1269000886237332187st_nat @ M @ N )
= ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_9385_lessThan__nat__numeral,axiom,
! [K: num] :
( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
= ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_9386_atMost__nat__numeral,axiom,
! [K: num] :
( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
= ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% atMost_nat_numeral
thf(fact_9387_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( ( ord_less_nat @ N @ M )
=> ( ( bit_se1146084159140164899it_int @ K @ N )
=> ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_9388_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L2: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N )
= ( ( ( ord_less_nat @ N @ M )
& ( bit_se1146084159140164899it_int @ K @ N ) )
| ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_9389_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ N3 @ M3 )
=> ( ( bit_se1146084159140164899it_int @ K @ M3 )
= ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
=> ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_9390_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_9391_bit__int__def,axiom,
( bit_se1146084159140164899it_int
= ( ^ [K3: int,N2: nat] :
~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% bit_int_def
thf(fact_9392_set__decode__plus__power__2,axiom,
! [N: nat,Z2: nat] :
( ~ ( member_nat @ N @ ( nat_set_decode @ Z2 ) )
=> ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z2 ) )
= ( insert_nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_9393_set__bit__eq,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N2: nat,K3: int] :
( plus_plus_int @ K3
@ ( times_times_int
@ ( zero_n2684676970156552555ol_int
@ ~ ( bit_se1146084159140164899it_int @ K3 @ N2 ) )
@ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% set_bit_eq
thf(fact_9394_unset__bit__eq,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N2: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_9395_and__int_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
=> ( ! [K2: int,L4: int] :
( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
=> ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
& ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
=> ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
=> ( P @ K2 @ L4 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% and_int.pinduct
thf(fact_9396_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] :
( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
= ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_9397_upto_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
=> ( ! [I3: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I3 @ J2 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
=> ( P @ I3 @ J2 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_9398_or__int__unfold,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K3: int,L: int] :
( if_int
@ ( ( K3
= ( uminus_uminus_int @ one_one_int ) )
| ( L
= ( uminus_uminus_int @ one_one_int ) ) )
@ ( uminus_uminus_int @ one_one_int )
@ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_9399_cis__multiple__2pi,axiom,
! [N: real] :
( ( member_real @ N @ ring_1_Ints_real )
=> ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
= one_one_complex ) ) ).
% cis_multiple_2pi
thf(fact_9400_xor__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
= ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_9401_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_9402_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_9403_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_9404_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_9405_xor__nat__numerals_I2_J,axiom,
! [Y4: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) ) ).
% xor_nat_numerals(2)
thf(fact_9406_xor__nat__numerals_I1_J,axiom,
! [Y4: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% xor_nat_numerals(1)
thf(fact_9407_bit__Suc__0__iff,axiom,
! [N: nat] :
( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
= ( N = zero_zero_nat ) ) ).
% bit_Suc_0_iff
thf(fact_9408_not__bit__Suc__0__Suc,axiom,
! [N: nat] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% not_bit_Suc_0_Suc
thf(fact_9409_plus__and__or,axiom,
! [X: int,Y4: int] :
( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y4 ) @ ( bit_se1409905431419307370or_int @ X @ Y4 ) )
= ( plus_plus_int @ X @ Y4 ) ) ).
% plus_and_or
thf(fact_9410_not__bit__Suc__0__numeral,axiom,
! [N: num] :
~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% not_bit_Suc_0_numeral
thf(fact_9411_sin__times__pi__eq__0,axiom,
! [X: real] :
( ( ( sin_real @ ( times_times_real @ X @ pi ) )
= zero_zero_real )
= ( member_real @ X @ ring_1_Ints_real ) ) ).
% sin_times_pi_eq_0
thf(fact_9412_bit__nat__def,axiom,
( bit_se1148574629649215175it_nat
= ( ^ [M2: nat,N2: nat] :
~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% bit_nat_def
thf(fact_9413_OR__upper,axiom,
! [X: int,N: nat,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_int @ Y4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y4 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_9414_sin__integer__2pi,axiom,
! [N: real] :
( ( member_real @ N @ ring_1_Ints_real )
=> ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
= zero_zero_real ) ) ).
% sin_integer_2pi
thf(fact_9415_cos__integer__2pi,axiom,
! [N: real] :
( ( member_real @ N @ ring_1_Ints_real )
=> ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
= one_one_real ) ) ).
% cos_integer_2pi
thf(fact_9416_xor__nat__unfold,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_9417_or__int__rec,axiom,
( bit_se1409905431419307370or_int
= ( ^ [K3: int,L: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
| ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_9418_xor__nat__rec,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [M2: nat,N2: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
!= ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_9419_Suc__0__xor__eq,axiom,
! [N: nat] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( zero_n2687167440665602831ol_nat
@ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_9420_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_9421_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_9422_horner__sum__of__bool__2__less,axiom,
! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).
% horner_sum_of_bool_2_less
thf(fact_9423_or__nat__numerals_I2_J,axiom,
! [Y4: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% or_nat_numerals(2)
thf(fact_9424_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_9425_or__nat__numerals_I1_J,axiom,
! [Y4: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y4 ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y4 ) ) ) ).
% or_nat_numerals(1)
thf(fact_9426_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_9427_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_9428_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_9429_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_9430_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_9431_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one @ one )
= one ) ).
% or_not_num_neg.simps(1)
thf(fact_9432_or__not__num__neg_Osimps_I4_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
= ( bit0 @ one ) ) ).
% or_not_num_neg.simps(4)
thf(fact_9433_or__not__num__neg_Osimps_I6_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
= ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_9434_or__not__num__neg_Osimps_I7_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
= one ) ).
% or_not_num_neg.simps(7)
thf(fact_9435_or__not__num__neg_Osimps_I3_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(3)
thf(fact_9436_or__not__num__neg_Osimps_I5_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_9437_or__not__num__neg_Osimps_I2_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(2)
thf(fact_9438_or__not__num__neg_Osimps_I8_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_9439_or__not__num__neg_Oelims,axiom,
! [X: num,Xa2: num,Y4: num] :
( ( ( bit_or_not_num_neg @ X @ Xa2 )
= Y4 )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y4 != one ) ) )
=> ( ( ( X = one )
=> ! [M4: num] :
( ( Xa2
= ( bit0 @ M4 ) )
=> ( Y4
!= ( bit1 @ M4 ) ) ) )
=> ( ( ( X = one )
=> ! [M4: num] :
( ( Xa2
= ( bit1 @ M4 ) )
=> ( Y4
!= ( bit1 @ M4 ) ) ) )
=> ( ( ? [N3: num] :
( X
= ( bit0 @ N3 ) )
=> ( ( Xa2 = one )
=> ( Y4
!= ( bit0 @ one ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M4: num] :
( ( Xa2
= ( bit0 @ M4 ) )
=> ( Y4
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M4: num] :
( ( Xa2
= ( bit1 @ M4 ) )
=> ( Y4
!= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
=> ( ( ? [N3: num] :
( X
= ( bit1 @ N3 ) )
=> ( ( Xa2 = one )
=> ( Y4 != one ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M4: num] :
( ( Xa2
= ( bit0 @ M4 ) )
=> ( Y4
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
=> ~ ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M4: num] :
( ( Xa2
= ( bit1 @ M4 ) )
=> ( Y4
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_9440_XOR__upper,axiom,
! [X: int,N: nat,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ( ord_less_int @ Y4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
=> ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y4 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_9441_floor__real__def,axiom,
( archim6058952711729229775r_real
= ( ^ [X2: real] :
( the_int
@ ^ [Z4: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X2 )
& ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_9442_or__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
= ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% or_Suc_0_eq
thf(fact_9443_Suc__0__or__eq,axiom,
! [N: nat] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
= ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% Suc_0_or_eq
thf(fact_9444_or__nat__rec,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M2: nat,N2: nat] :
( plus_plus_nat
@ ( zero_n2687167440665602831ol_nat
@ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
| ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_9445_xor__int__rec,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K3: int,L: int] :
( plus_plus_int
@ ( zero_n2684676970156552555ol_int
@ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
!= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
@ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_9446_or__nat__unfold,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_9447_xor__int__unfold,axiom,
( bit_se6526347334894502574or_int
= ( ^ [K3: int,L: int] :
( if_int
@ ( K3
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ L )
@ ( if_int
@ ( L
= ( uminus_uminus_int @ one_one_int ) )
@ ( bit_ri7919022796975470100ot_int @ K3 )
@ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_9448_floor__rat__def,axiom,
( archim3151403230148437115or_rat
= ( ^ [X2: rat] :
( the_int
@ ^ [Z4: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 )
& ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_9449_push__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% push_bit_of_Suc_0
thf(fact_9450_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_9451_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_9452_set__bit__nat__def,axiom,
( bit_se7882103937844011126it_nat
= ( ^ [M2: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M2 @ one_one_nat ) ) ) ) ).
% set_bit_nat_def
thf(fact_9453_flip__bit__int__def,axiom,
( bit_se2159334234014336723it_int
= ( ^ [N2: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% flip_bit_int_def
thf(fact_9454_sgn__rat__def,axiom,
( sgn_sgn_rat
= ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% sgn_rat_def
thf(fact_9455_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ~ ! [S: rat] :
( ( ord_less_rat @ zero_zero_rat @ S )
=> ! [T3: rat] :
( ( ord_less_rat @ zero_zero_rat @ T3 )
=> ( R2
!= ( plus_plus_rat @ S @ T3 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_9456_unset__bit__int__def,axiom,
( bit_se4203085406695923979it_int
= ( ^ [N2: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_9457_flip__bit__nat__def,axiom,
( bit_se2161824704523386999it_nat
= ( ^ [M2: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M2 @ one_one_nat ) ) ) ) ).
% flip_bit_nat_def
thf(fact_9458_not__int__def,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).
% not_int_def
thf(fact_9459_and__not__numerals_I1_J,axiom,
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= zero_zero_int ) ).
% and_not_numerals(1)
thf(fact_9460_or__not__numerals_I1_J,axiom,
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% or_not_numerals(1)
thf(fact_9461_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
= ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_9462_bit__push__bit__iff__nat,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
= ( ( ord_less_eq_nat @ M @ N )
& ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_9463_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N2: nat,K3: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% concat_bit_eq
thf(fact_9464_set__bit__int__def,axiom,
( bit_se7879613467334960850it_int
= ( ^ [N2: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% set_bit_int_def
thf(fact_9465_not__int__div__2,axiom,
! [K: int] :
( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% not_int_div_2
thf(fact_9466_even__not__iff__int,axiom,
! [K: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
= ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% even_not_iff_int
thf(fact_9467_and__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(4)
thf(fact_9468_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= one_one_int ) ).
% and_not_numerals(2)
thf(fact_9469_or__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% or_not_numerals(4)
thf(fact_9470_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_9471_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
= ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_9472_and__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_9473_and__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(7)
thf(fact_9474_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_9475_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= zero_zero_int ) ).
% and_not_numerals(3)
thf(fact_9476_or__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
= ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% or_not_numerals(7)
thf(fact_9477_push__bit__int__def,axiom,
( bit_se545348938243370406it_int
= ( ^ [N2: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% push_bit_int_def
thf(fact_9478_push__bit__nat__def,axiom,
( bit_se547839408752420682it_nat
= ( ^ [N2: nat,M2: nat] : ( times_times_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% push_bit_nat_def
thf(fact_9479_and__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_9480_and__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_9481_or__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_9482_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_9483_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_9484_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_9485_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_9486_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_9487_not__int__rec,axiom,
( bit_ri7919022796975470100ot_int
= ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_9488_rat__inverse__code,axiom,
! [P6: rat] :
( ( quotient_of @ ( inverse_inverse_rat @ P6 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A4 ) @ B3 ) @ ( abs_abs_int @ A4 ) ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_inverse_code
thf(fact_9489_normalize__negative,axiom,
! [Q2: int,P6: int] :
( ( ord_less_int @ Q2 @ zero_zero_int )
=> ( ( normalize @ ( product_Pair_int_int @ P6 @ Q2 ) )
= ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P6 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% normalize_negative
thf(fact_9490_quotient__of__number_I3_J,axiom,
! [K: num] :
( ( quotient_of @ ( numeral_numeral_rat @ K ) )
= ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% quotient_of_number(3)
thf(fact_9491_normalize__denom__zero,axiom,
! [P6: int] :
( ( normalize @ ( product_Pair_int_int @ P6 @ zero_zero_int ) )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% normalize_denom_zero
thf(fact_9492_rat__one__code,axiom,
( ( quotient_of @ one_one_rat )
= ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% rat_one_code
thf(fact_9493_rat__zero__code,axiom,
( ( quotient_of @ zero_zero_rat )
= ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% rat_zero_code
thf(fact_9494_quotient__of__number_I5_J,axiom,
! [K: num] :
( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
= ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% quotient_of_number(5)
thf(fact_9495_quotient__of__number_I4_J,axiom,
( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
= ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% quotient_of_number(4)
thf(fact_9496_divide__rat__def,axiom,
( divide_divide_rat
= ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% divide_rat_def
thf(fact_9497_diff__rat__def,axiom,
( minus_minus_rat
= ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% diff_rat_def
thf(fact_9498_rat__times__code,axiom,
! [P6: rat,Q2: rat] :
( ( quotient_of @ ( times_times_rat @ P6 @ Q2 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int,C2: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ B3 ) @ ( times_times_int @ C2 @ D2 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_times_code
thf(fact_9499_rat__divide__code,axiom,
! [P6: rat,Q2: rat] :
( ( quotient_of @ ( divide_divide_rat @ P6 @ Q2 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int,C2: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_divide_code
thf(fact_9500_quotient__of__div,axiom,
! [R2: rat,N: int,D: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair_int_int @ N @ D ) )
=> ( R2
= ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% quotient_of_div
thf(fact_9501_rat__plus__code,axiom,
! [P6: rat,Q2: rat] :
( ( quotient_of @ ( plus_plus_rat @ P6 @ Q2 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int,C2: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B3 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_plus_code
thf(fact_9502_rat__minus__code,axiom,
! [P6: rat,Q2: rat] :
( ( quotient_of @ ( minus_minus_rat @ P6 @ Q2 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int,C2: int] :
( produc4245557441103728435nt_int
@ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ B3 @ C2 ) ) @ ( times_times_int @ C2 @ D2 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_minus_code
thf(fact_9503_quotient__of__denom__pos,axiom,
! [R2: rat,P6: int,Q2: int] :
( ( ( quotient_of @ R2 )
= ( product_Pair_int_int @ P6 @ Q2 ) )
=> ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% quotient_of_denom_pos
thf(fact_9504_rat__less__code,axiom,
( ord_less_rat
= ( ^ [P4: rat,Q4: rat] :
( produc4947309494688390418_int_o
@ ^ [A4: int,C2: int] :
( produc4947309494688390418_int_o
@ ^ [B3: int,D2: int] : ( ord_less_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ) ).
% rat_less_code
thf(fact_9505_rat__less__eq__code,axiom,
( ord_less_eq_rat
= ( ^ [P4: rat,Q4: rat] :
( produc4947309494688390418_int_o
@ ^ [A4: int,C2: int] :
( produc4947309494688390418_int_o
@ ^ [B3: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D2 ) @ ( times_times_int @ C2 @ B3 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P4 ) ) ) ) ).
% rat_less_eq_code
thf(fact_9506_rat__uminus__code,axiom,
! [P6: rat] :
( ( quotient_of @ ( uminus_uminus_rat @ P6 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A4 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_uminus_code
thf(fact_9507_rat__abs__code,axiom,
! [P6: rat] :
( ( quotient_of @ ( abs_abs_rat @ P6 ) )
= ( produc4245557441103728435nt_int
@ ^ [A4: int] : ( product_Pair_int_int @ ( abs_abs_int @ A4 ) )
@ ( quotient_of @ P6 ) ) ) ).
% rat_abs_code
thf(fact_9508_normalize__denom__pos,axiom,
! [R2: product_prod_int_int,P6: int,Q2: int] :
( ( ( normalize @ R2 )
= ( product_Pair_int_int @ P6 @ Q2 ) )
=> ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% normalize_denom_pos
thf(fact_9509_normalize__crossproduct,axiom,
! [Q2: int,S2: int,P6: int,R2: int] :
( ( Q2 != zero_zero_int )
=> ( ( S2 != zero_zero_int )
=> ( ( ( normalize @ ( product_Pair_int_int @ P6 @ Q2 ) )
= ( normalize @ ( product_Pair_int_int @ R2 @ S2 ) ) )
=> ( ( times_times_int @ P6 @ S2 )
= ( times_times_int @ R2 @ Q2 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_9510_quotient__of__int,axiom,
! [A: int] :
( ( quotient_of @ ( of_int @ A ) )
= ( product_Pair_int_int @ A @ one_one_int ) ) ).
% quotient_of_int
thf(fact_9511_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% Sum_Ico_nat
thf(fact_9512_Cauchy__iff2,axiom,
( topolo4055970368930404560y_real
= ( ^ [X6: nat > real] :
! [J3: nat] :
? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq_nat @ M9 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ M9 @ N2 )
=> ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_9513_VEBT_Osize_I3_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% VEBT.size(3)
thf(fact_9514_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
= ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_9515_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_9516_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_9517_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L2: nat,U2: nat] :
( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U2 ) )
= ( set_or1269000886237332187st_nat @ L2 @ U2 ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_9518_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_9519_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_9520_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_9521_prod__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_fact
thf(fact_9522_prod__Suc__Suc__fact,axiom,
! [N: nat] :
( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( semiri1408675320244567234ct_nat @ N ) ) ).
% prod_Suc_Suc_fact
thf(fact_9523_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
= ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_9524_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_9525_sum__power2,axiom,
! [K: nat] :
( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
= ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% sum_power2
thf(fact_9526_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A: nat > nat,B: nat > nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ N )
=> ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
=> ( ord_less_eq_nat
@ ( times_times_nat @ N
@ ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
@ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_9527_VEBT_Osize__gen_I1_J,axiom,
! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% VEBT.size_gen(1)
thf(fact_9528_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L2: int,U2: int] :
( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U2 @ one_one_int ) )
= ( set_or1266510415728281911st_int @ L2 @ U2 ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9529_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X222: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= zero_zero_nat ) ).
% VEBT.size_gen(2)
thf(fact_9530_Frct__code__post_I5_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
= ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% Frct_code_post(5)
thf(fact_9531_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_9532_valid__eq2,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_VEBT_valid @ T @ D )
=> ( vEBT_invar_vebt @ T @ D ) ) ).
% valid_eq2
thf(fact_9533_valid__eq1,axiom,
! [T: vEBT_VEBT,D: nat] :
( ( vEBT_invar_vebt @ T @ D )
=> ( vEBT_VEBT_valid @ T @ D ) ) ).
% valid_eq1
thf(fact_9534_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
= ( D = one_one_nat ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_9535_Frct__code__post_I1_J,axiom,
! [A: int] :
( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
= zero_zero_rat ) ).
% Frct_code_post(1)
thf(fact_9536_Frct__code__post_I2_J,axiom,
! [A: int] :
( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
= zero_zero_rat ) ).
% Frct_code_post(2)
thf(fact_9537_Frct__code__post_I8_J,axiom,
! [A: int,B: int] :
( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
= ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% Frct_code_post(8)
thf(fact_9538_Frct__code__post_I7_J,axiom,
! [A: int,B: int] :
( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
= ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% Frct_code_post(7)
thf(fact_9539_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
= one_one_rat ) ).
% Frct_code_post(3)
thf(fact_9540_Frct__code__post_I4_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
= ( numeral_numeral_rat @ K ) ) ).
% Frct_code_post(4)
thf(fact_9541_Frct__code__post_I6_J,axiom,
! [K: num,L2: num] :
( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
= ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).
% Frct_code_post(6)
thf(fact_9542_divmod__step__integer__def,axiom,
( unique4921790084139445826nteger
= ( ^ [L: num] :
( produc6916734918728496179nteger
@ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_9543_csqrt_Osimps_I1_J,axiom,
! [Z2: complex] :
( ( re @ ( csqrt @ Z2 ) )
= ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_9544_times__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
= zero_z3403309356797280102nteger ) ).
% times_integer_code(1)
thf(fact_9545_times__integer__code_I2_J,axiom,
! [L2: code_integer] :
( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L2 )
= zero_z3403309356797280102nteger ) ).
% times_integer_code(2)
thf(fact_9546_divmod__integer_H__def,axiom,
( unique3479559517661332726nteger
= ( ^ [M2: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_9547_full__exhaustive__integer_H_Ocases,axiom,
! [X: produc1908205239877642774nteger] :
~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D3: code_integer,I3: code_integer] :
( X
!= ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I3 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_9548_exhaustive__integer_H_Ocases,axiom,
! [X: produc8763457246119570046nteger] :
~ ! [F2: code_integer > option6357759511663192854e_term,D3: code_integer,I3: code_integer] :
( X
!= ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I3 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_9549_plus__integer__code_I2_J,axiom,
! [L2: code_integer] :
( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L2 )
= L2 ) ).
% plus_integer_code(2)
thf(fact_9550_plus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
= K ) ).
% plus_integer_code(1)
thf(fact_9551_sgn__integer__code,axiom,
( sgn_sgn_Code_integer
= ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% sgn_integer_code
thf(fact_9552_one__complex_Osimps_I1_J,axiom,
( ( re @ one_one_complex )
= one_one_real ) ).
% one_complex.simps(1)
thf(fact_9553_plus__complex_Osimps_I1_J,axiom,
! [X: complex,Y4: complex] :
( ( re @ ( plus_plus_complex @ X @ Y4 ) )
= ( plus_plus_real @ ( re @ X ) @ ( re @ Y4 ) ) ) ).
% plus_complex.simps(1)
thf(fact_9554_scaleR__complex_Osimps_I1_J,axiom,
! [R2: real,X: complex] :
( ( re @ ( real_V2046097035970521341omplex @ R2 @ X ) )
= ( times_times_real @ R2 @ ( re @ X ) ) ) ).
% scaleR_complex.simps(1)
thf(fact_9555_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_9556_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_9557_cmod__plus__Re__le__0__iff,axiom,
! [Z2: complex] :
( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ zero_zero_real )
= ( ( re @ Z2 )
= ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_9558_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A: real] :
( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
= ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% cos_n_Re_cis_pow_n
thf(fact_9559_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z4: complex] :
( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
@ ( times_times_real
@ ( if_real
@ ( ( im @ Z4 )
= zero_zero_real )
@ one_one_real
@ ( sgn_sgn_real @ ( im @ Z4 ) ) )
@ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_9560_csqrt_Osimps_I2_J,axiom,
! [Z2: complex] :
( ( im @ ( csqrt @ Z2 ) )
= ( times_times_real
@ ( if_real
@ ( ( im @ Z2 )
= zero_zero_real )
@ one_one_real
@ ( sgn_sgn_real @ ( im @ Z2 ) ) )
@ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_9561_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K3: int] :
( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
@ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
@ ( if_Code_integer
@ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= zero_zero_int )
@ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
@ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_9562_csqrt__of__real__nonpos,axiom,
! [X: complex] :
( ( ( im @ X )
= zero_zero_real )
=> ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
=> ( ( csqrt @ X )
= ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% csqrt_of_real_nonpos
thf(fact_9563_Im__i__times,axiom,
! [Z2: complex] :
( ( im @ ( times_times_complex @ imaginary_unit @ Z2 ) )
= ( re @ Z2 ) ) ).
% Im_i_times
thf(fact_9564_Re__i__times,axiom,
! [Z2: complex] :
( ( re @ ( times_times_complex @ imaginary_unit @ Z2 ) )
= ( uminus_uminus_real @ ( im @ Z2 ) ) ) ).
% Re_i_times
thf(fact_9565_csqrt__minus,axiom,
! [X: complex] :
( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
| ( ( ( im @ X )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
=> ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
= ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% csqrt_minus
thf(fact_9566_imaginary__unit_Osimps_I2_J,axiom,
( ( im @ imaginary_unit )
= one_one_real ) ).
% imaginary_unit.simps(2)
thf(fact_9567_one__complex_Osimps_I2_J,axiom,
( ( im @ one_one_complex )
= zero_zero_real ) ).
% one_complex.simps(2)
thf(fact_9568_plus__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus_int @ Xa2 @ X ) ) ) ).
% plus_integer.abs_eq
thf(fact_9569_times__integer_Oabs__eq,axiom,
! [Xa2: int,X: int] :
( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( times_times_int @ Xa2 @ X ) ) ) ).
% times_integer.abs_eq
thf(fact_9570_one__integer__def,axiom,
( one_one_Code_integer
= ( code_integer_of_int @ one_one_int ) ) ).
% one_integer_def
thf(fact_9571_plus__complex_Osimps_I2_J,axiom,
! [X: complex,Y4: complex] :
( ( im @ ( plus_plus_complex @ X @ Y4 ) )
= ( plus_plus_real @ ( im @ X ) @ ( im @ Y4 ) ) ) ).
% plus_complex.simps(2)
thf(fact_9572_scaleR__complex_Osimps_I2_J,axiom,
! [R2: real,X: complex] :
( ( im @ ( real_V2046097035970521341omplex @ R2 @ X ) )
= ( times_times_real @ R2 @ ( im @ X ) ) ) ).
% scaleR_complex.simps(2)
thf(fact_9573_times__complex_Osimps_I2_J,axiom,
! [X: complex,Y4: complex] :
( ( im @ ( times_times_complex @ X @ Y4 ) )
= ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y4 ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y4 ) ) ) ) ).
% times_complex.simps(2)
thf(fact_9574_times__complex_Osimps_I1_J,axiom,
! [X: complex,Y4: complex] :
( ( re @ ( times_times_complex @ X @ Y4 ) )
= ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y4 ) ) ) ) ).
% times_complex.simps(1)
thf(fact_9575_plus__complex_Ocode,axiom,
( plus_plus_complex
= ( ^ [X2: complex,Y: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( plus_plus_real @ ( im @ X2 ) @ ( im @ Y ) ) ) ) ) ).
% plus_complex.code
thf(fact_9576_scaleR__complex_Ocode,axiom,
( real_V2046097035970521341omplex
= ( ^ [R5: real,X2: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X2 ) ) @ ( times_times_real @ R5 @ ( im @ X2 ) ) ) ) ) ).
% scaleR_complex.code
thf(fact_9577_cmod__le,axiom,
! [Z2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z2 ) ) @ ( abs_abs_real @ ( im @ Z2 ) ) ) ) ).
% cmod_le
thf(fact_9578_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A: real] :
( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
= ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% sin_n_Im_cis_pow_n
thf(fact_9579_Re__exp,axiom,
! [Z2: complex] :
( ( re @ ( exp_complex @ Z2 ) )
= ( times_times_real @ ( exp_real @ ( re @ Z2 ) ) @ ( cos_real @ ( im @ Z2 ) ) ) ) ).
% Re_exp
thf(fact_9580_Im__exp,axiom,
! [Z2: complex] :
( ( im @ ( exp_complex @ Z2 ) )
= ( times_times_real @ ( exp_real @ ( re @ Z2 ) ) @ ( sin_real @ ( im @ Z2 ) ) ) ) ).
% Im_exp
thf(fact_9581_complex__eq,axiom,
! [A: complex] :
( A
= ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% complex_eq
thf(fact_9582_times__complex_Ocode,axiom,
( times_times_complex
= ( ^ [X2: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y ) ) ) ) ) ) ).
% times_complex.code
thf(fact_9583_exp__eq__polar,axiom,
( exp_complex
= ( ^ [Z4: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z4 ) ) ) @ ( cis @ ( im @ Z4 ) ) ) ) ) ).
% exp_eq_polar
thf(fact_9584_cmod__power2,axiom,
! [Z2: complex] :
( ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% cmod_power2
thf(fact_9585_Im__power2,axiom,
! [X: complex] :
( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% Im_power2
thf(fact_9586_Re__power2,axiom,
! [X: complex] :
( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% Re_power2
thf(fact_9587_complex__eq__0,axiom,
! [Z2: complex] :
( ( Z2 = zero_zero_complex )
= ( ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= zero_zero_real ) ) ).
% complex_eq_0
thf(fact_9588_norm__complex__def,axiom,
( real_V1022390504157884413omplex
= ( ^ [Z4: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% norm_complex_def
thf(fact_9589_inverse__complex_Osimps_I1_J,axiom,
! [X: complex] :
( ( re @ ( invers8013647133539491842omplex @ X ) )
= ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% inverse_complex.simps(1)
thf(fact_9590_complex__neq__0,axiom,
! [Z2: complex] :
( ( Z2 != zero_zero_complex )
= ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_9591_Re__divide,axiom,
! [X: complex,Y4: complex] :
( ( re @ ( divide1717551699836669952omplex @ X @ Y4 ) )
= ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y4 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Re_divide
thf(fact_9592_csqrt__unique,axiom,
! [W2: complex,Z2: complex] :
( ( ( power_power_complex @ W2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= Z2 )
=> ( ( ( ord_less_real @ zero_zero_real @ ( re @ W2 ) )
| ( ( ( re @ W2 )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ W2 ) ) ) )
=> ( ( csqrt @ Z2 )
= W2 ) ) ) ).
% csqrt_unique
thf(fact_9593_csqrt__square,axiom,
! [B: complex] :
( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
| ( ( ( re @ B )
= zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
=> ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= B ) ) ).
% csqrt_square
thf(fact_9594_inverse__complex_Osimps_I2_J,axiom,
! [X: complex] :
( ( im @ ( invers8013647133539491842omplex @ X ) )
= ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% inverse_complex.simps(2)
thf(fact_9595_Im__divide,axiom,
! [X: complex,Y4: complex] :
( ( im @ ( divide1717551699836669952omplex @ X @ Y4 ) )
= ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y4 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Im_divide
thf(fact_9596_complex__abs__le__norm,axiom,
! [Z2: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z2 ) ) @ ( abs_abs_real @ ( im @ Z2 ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ).
% complex_abs_le_norm
thf(fact_9597_complex__unit__circle,axiom,
! [Z2: complex] :
( ( Z2 != zero_zero_complex )
=> ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= one_one_real ) ) ).
% complex_unit_circle
thf(fact_9598_inverse__complex_Ocode,axiom,
( invers8013647133539491842omplex
= ( ^ [X2: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X2 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X2 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_9599_Complex__divide,axiom,
( divide1717551699836669952omplex
= ( ^ [X2: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_9600_Im__Reals__divide,axiom,
! [R2: complex,Z2: complex] :
( ( member_complex @ R2 @ real_V2521375963428798218omplex )
=> ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z2 ) )
= ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_9601_Re__Reals__divide,axiom,
! [R2: complex,Z2: complex] :
( ( member_complex @ R2 @ real_V2521375963428798218omplex )
=> ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z2 ) )
= ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% Re_Reals_divide
thf(fact_9602_real__eq__imaginary__iff,axiom,
! [Y4: complex,X: complex] :
( ( member_complex @ Y4 @ real_V2521375963428798218omplex )
=> ( ( member_complex @ X @ real_V2521375963428798218omplex )
=> ( ( X
= ( times_times_complex @ imaginary_unit @ Y4 ) )
= ( ( X = zero_zero_complex )
& ( Y4 = zero_zero_complex ) ) ) ) ) ).
% real_eq_imaginary_iff
thf(fact_9603_imaginary__eq__real__iff,axiom,
! [Y4: complex,X: complex] :
( ( member_complex @ Y4 @ real_V2521375963428798218omplex )
=> ( ( member_complex @ X @ real_V2521375963428798218omplex )
=> ( ( ( times_times_complex @ imaginary_unit @ Y4 )
= X )
= ( ( X = zero_zero_complex )
& ( Y4 = zero_zero_complex ) ) ) ) ) ).
% imaginary_eq_real_iff
thf(fact_9604_complex__mult__cnj,axiom,
! [Z2: complex] :
( ( times_times_complex @ Z2 @ ( cnj @ Z2 ) )
= ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% complex_mult_cnj
thf(fact_9605_integer__of__num_I3_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit1 @ N ) )
= ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% integer_of_num(3)
thf(fact_9606_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
@ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
@ ( produc1553301316500091796er_int
@ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_9607_complex__cnj__mult,axiom,
! [X: complex,Y4: complex] :
( ( cnj @ ( times_times_complex @ X @ Y4 ) )
= ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y4 ) ) ) ).
% complex_cnj_mult
thf(fact_9608_complex__cnj__one__iff,axiom,
! [Z2: complex] :
( ( ( cnj @ Z2 )
= one_one_complex )
= ( Z2 = one_one_complex ) ) ).
% complex_cnj_one_iff
thf(fact_9609_complex__cnj__one,axiom,
( ( cnj @ one_one_complex )
= one_one_complex ) ).
% complex_cnj_one
thf(fact_9610_complex__cnj__add,axiom,
! [X: complex,Y4: complex] :
( ( cnj @ ( plus_plus_complex @ X @ Y4 ) )
= ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y4 ) ) ) ).
% complex_cnj_add
thf(fact_9611_plus__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa2 ) )
= ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% plus_integer.rep_eq
thf(fact_9612_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa2: code_integer] :
( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa2 ) )
= ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% times_integer.rep_eq
thf(fact_9613_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ one_one_Code_integer )
= one_one_int ) ).
% one_integer.rep_eq
thf(fact_9614_complex__In__mult__cnj__zero,axiom,
! [Z2: complex] :
( ( im @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
= zero_zero_real ) ).
% complex_In_mult_cnj_zero
thf(fact_9615_Re__complex__div__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
= zero_zero_real )
= ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
= zero_zero_real ) ) ).
% Re_complex_div_eq_0
thf(fact_9616_Im__complex__div__eq__0,axiom,
! [A: complex,B: complex] :
( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
= zero_zero_real )
= ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
= zero_zero_real ) ) ).
% Im_complex_div_eq_0
thf(fact_9617_complex__mod__sqrt__Re__mult__cnj,axiom,
( real_V1022390504157884413omplex
= ( ^ [Z4: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z4 @ ( cnj @ Z4 ) ) ) ) ) ) ).
% complex_mod_sqrt_Re_mult_cnj
thf(fact_9618_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one )
= one_one_Code_integer ) ).
% integer_of_num_triv(1)
thf(fact_9619_Re__complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Re_complex_div_gt_0
thf(fact_9620_Re__complex__div__lt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Re_complex_div_lt_0
thf(fact_9621_Re__complex__div__le__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Re_complex_div_le_0
thf(fact_9622_Re__complex__div__ge__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Re_complex_div_ge_0
thf(fact_9623_Im__complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Im_complex_div_gt_0
thf(fact_9624_Im__complex__div__lt__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Im_complex_div_lt_0
thf(fact_9625_Im__complex__div__le__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
= ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% Im_complex_div_le_0
thf(fact_9626_Im__complex__div__ge__0,axiom,
! [A: complex,B: complex] :
( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% Im_complex_div_ge_0
thf(fact_9627_integer__of__num_I2_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit0 @ N ) )
= ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% integer_of_num(2)
thf(fact_9628_complex__mod__mult__cnj,axiom,
! [Z2: complex] :
( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
= ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% complex_mod_mult_cnj
thf(fact_9629_complex__div__gt__0,axiom,
! [A: complex,B: complex] :
( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
& ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
= ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% complex_div_gt_0
thf(fact_9630_integer__of__num__triv_I2_J,axiom,
( ( code_integer_of_num @ ( bit0 @ one ) )
= ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% integer_of_num_triv(2)
thf(fact_9631_complex__norm__square,axiom,
! [Z2: complex] :
( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ).
% complex_norm_square
thf(fact_9632_complex__add__cnj,axiom,
! [Z2: complex] :
( ( plus_plus_complex @ Z2 @ ( cnj @ Z2 ) )
= ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z2 ) ) ) ) ).
% complex_add_cnj
thf(fact_9633_complex__diff__cnj,axiom,
! [Z2: complex] :
( ( minus_minus_complex @ Z2 @ ( cnj @ Z2 ) )
= ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z2 ) ) ) @ imaginary_unit ) ) ).
% complex_diff_cnj
thf(fact_9634_complex__div__cnj,axiom,
( divide1717551699836669952omplex
= ( ^ [A4: complex,B3: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A4 @ ( cnj @ B3 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% complex_div_cnj
thf(fact_9635_cnj__add__mult__eq__Re,axiom,
! [Z2: complex,W2: complex] :
( ( plus_plus_complex @ ( times_times_complex @ Z2 @ ( cnj @ W2 ) ) @ ( times_times_complex @ ( cnj @ Z2 ) @ W2 ) )
= ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ W2 ) ) ) ) ) ) ).
% cnj_add_mult_eq_Re
thf(fact_9636_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L ) @ ( modulo364778990260209775nteger @ K3 @ L ) ) ) ) ).
% divmod_integer_def
thf(fact_9637_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
@ ( produc7336495610019696514er_num
@ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_9638_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
@ ( produc1555791787009142072er_nat
@ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
@ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_9639_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
@ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_9640_nat__of__integer__code__post_I3_J,axiom,
! [K: num] :
( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% nat_of_integer_code_post(3)
thf(fact_9641_nat__of__integer__code__post_I2_J,axiom,
( ( code_nat_of_integer @ one_one_Code_integer )
= one_one_nat ) ).
% nat_of_integer_code_post(2)
thf(fact_9642_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
@ ( produc9125791028180074456eger_o
@ ^ [R5: code_integer,S5: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S5 ) ) @ ( S5 = one_one_Code_integer ) )
@ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_9643_vebt__maxt_Opelims,axiom,
! [X: vEBT_VEBT,Y4: option_nat] :
( ( ( vEBT_vebt_maxt @ X )
= Y4 )
=> ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( ( B2
=> ( Y4
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( ( A3
=> ( Y4
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( Y4 = none_nat ) ) ) ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( ( Y4 = none_nat )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y4
= ( some_nat @ Ma2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_maxt.pelims
thf(fact_9644_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_9645_card__atMost,axiom,
! [U2: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U2 ) )
= ( suc @ U2 ) ) ).
% card_atMost
thf(fact_9646_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_9647_card__atLeastAtMost,axiom,
! [L2: nat,U2: nat] :
( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U2 ) )
= ( minus_minus_nat @ ( suc @ U2 ) @ L2 ) ) ).
% card_atLeastAtMost
thf(fact_9648_card__atLeastAtMost__int,axiom,
! [L2: int,U2: int] :
( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U2 ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U2 @ L2 ) @ one_one_int ) ) ) ).
% card_atLeastAtMost_int
thf(fact_9649_card__less,axiom,
! [M7: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_9650_card__less__Suc,axiom,
! [M7: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M7 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_9651_card__less__Suc2,axiom,
! [M7: set_nat,I2: nat] :
( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M7 )
& ( ord_less_nat @ K3 @ I2 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_9652_subset__card__intvl__is__intvl,axiom,
! [A2: set_nat,K: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
=> ( A2
= ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_9653_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% divmod_abs_code(6)
thf(fact_9654_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
= ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_9655_subset__eq__atLeast0__lessThan__card,axiom,
! [N4: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_9656_card__sum__le__nat__sum,axiom,
! [S3: set_nat] :
( ord_less_eq_nat
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ S3 ) ) ).
% card_sum_le_nat_sum
thf(fact_9657_card__nth__roots,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= C ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_9658_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N )
= one_one_complex ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_9659_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ).
% divmod_abs_def
thf(fact_9660_vebt__mint_Opelims,axiom,
! [X: vEBT_VEBT,Y4: option_nat] :
( ( ( vEBT_vebt_mint @ X )
= Y4 )
=> ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
=> ( ! [A3: $o,B2: $o] :
( ( X
= ( vEBT_Leaf @ A3 @ B2 ) )
=> ( ( ( A3
=> ( Y4
= ( some_nat @ zero_zero_nat ) ) )
& ( ~ A3
=> ( ( B2
=> ( Y4
= ( some_nat @ one_one_nat ) ) )
& ( ~ B2
=> ( Y4 = none_nat ) ) ) ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
=> ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
=> ( ( Y4 = none_nat )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
=> ( ( Y4
= ( some_nat @ Mi2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% vebt_mint.pelims
thf(fact_9661_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L: code_integer] :
( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S5 ) ) )
@ ( code_divmod_abs @ K3 @ L ) ) )
@ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
@ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
@ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S5 ) ) )
@ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_9662_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Y4: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y4 )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y4
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y4
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu3: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ $true ) )
=> ( ~ Y4
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu3 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ( Y4
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ( ~ Y4
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_9663_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
=> ( ! [Uu3: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ $true ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu3 @ $true ) ) )
=> ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_9664_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_9665_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_9666_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_9667_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max_nat @ ( suc @ N ) @ M )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
@ M ) ) ).
% max_Suc1
thf(fact_9668_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_9669_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K3: nat] : K3
@ ( minus_minus_nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_9670_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y4: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y4 )
=> ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y4
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y4
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_9671_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K3 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_9672_drop__bit__numeral__minus__bit1,axiom,
! [L2: num,K: num] :
( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_9673_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% drop_bit_minus_one
thf(fact_9674_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_9675_drop__bit__numeral__minus__bit0,axiom,
! [L2: num,K: num] :
( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_9676_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
= ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_9677_drop__bit__int__def,axiom,
( bit_se8568078237143864401it_int
= ( ^ [N2: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% drop_bit_int_def
thf(fact_9678_Suc__0__mod__numeral,axiom,
! [K: num] :
( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
= ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% Suc_0_mod_numeral
thf(fact_9679_Suc__0__div__numeral,axiom,
! [K: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
= ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% Suc_0_div_numeral
thf(fact_9680_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L: code_integer] :
( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
@ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
@ ( if_Pro6119634080678213985nteger
@ ( ( sgn_sgn_Code_integer @ K3 )
= ( sgn_sgn_Code_integer @ L ) )
@ ( code_divmod_abs @ K3 @ L )
@ ( produc6916734918728496179nteger
@ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S5 ) ) )
@ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_9681_drop__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
= ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_9682_fst__divmod__nat,axiom,
! [M: nat,N: nat] :
( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
= ( divide_divide_nat @ M @ N ) ) ).
% fst_divmod_nat
thf(fact_9683_drop__bit__nat__def,axiom,
( bit_se8570568707652914677it_nat
= ( ^ [N2: nat,M2: nat] : ( divide_divide_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_9684_card_Ocomp__fun__commute__on,axiom,
( ( comp_nat_nat_nat @ suc @ suc )
= ( comp_nat_nat_nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_9685_bezw__non__0,axiom,
! [Y4: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ Y4 )
=> ( ( bezw @ X @ Y4 )
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X @ Y4 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y4 ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_9686_bezw_Osimps,axiom,
( bezw
= ( ^ [X2: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_9687_bezw_Oelims,axiom,
! [X: nat,Xa2: nat,Y4: product_prod_int_int] :
( ( ( bezw @ X @ Xa2 )
= Y4 )
=> ( ( ( Xa2 = zero_zero_nat )
=> ( Y4
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y4
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_9688_rat__sgn__code,axiom,
! [P6: rat] :
( ( quotient_of @ ( sgn_sgn_rat @ P6 ) )
= ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P6 ) ) ) @ one_one_int ) ) ).
% rat_sgn_code
thf(fact_9689_minus__one__mod__numeral,axiom,
! [N: num] :
( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
= ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% minus_one_mod_numeral
thf(fact_9690_one__mod__minus__numeral,axiom,
! [N: num] :
( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% one_mod_minus_numeral
thf(fact_9691_bezw_Opelims,axiom,
! [X: nat,Xa2: nat,Y4: product_prod_int_int] :
( ( ( bezw @ X @ Xa2 )
= Y4 )
=> ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y4
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y4
= ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% bezw.pelims
thf(fact_9692_normalize__def,axiom,
( normalize
= ( ^ [P4: product_prod_int_int] :
( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P4 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) )
@ ( if_Pro3027730157355071871nt_int
@ ( ( product_snd_int_int @ P4 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_9693_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd_int @ M @ one_one_int )
= one_one_int ) ).
% gcd_1_int
thf(fact_9694_bezout__int,axiom,
! [X: int,Y4: int] :
? [U4: int,V2: int] :
( ( plus_plus_int @ ( times_times_int @ U4 @ X ) @ ( times_times_int @ V2 @ Y4 ) )
= ( gcd_gcd_int @ X @ Y4 ) ) ).
% bezout_int
thf(fact_9695_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N: int] :
( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
= ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% gcd_mult_distrib_int
thf(fact_9696_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I: nat] :
( ( ord_less_nat @ K2 @ I )
=> ( P @ I ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_9697_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y4: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y4 )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y4
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y4
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_9698_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ one_one_nat )
= one_one_nat ) ).
% gcd_1_nat
thf(fact_9699_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( suc @ zero_zero_nat ) ) ).
% gcd_Suc_0
thf(fact_9700_gcd__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
= ( ( M != zero_zero_nat )
| ( N != zero_zero_nat ) ) ) ).
% gcd_pos_nat
thf(fact_9701_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
= ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% gcd_mult_distrib_nat
thf(fact_9702_gcd__le2__nat,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% gcd_le2_nat
thf(fact_9703_gcd__le1__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% gcd_le1_nat
thf(fact_9704_gcd__diff1__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
% gcd_diff1_nat
thf(fact_9705_gcd__diff2__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
= ( gcd_gcd_nat @ M @ N ) ) ) ).
% gcd_diff2_nat
thf(fact_9706_bezout__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [X3: nat,Y3: nat] :
( ( times_times_nat @ A @ X3 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_nat
thf(fact_9707_bezout__gcd__nat_H,axiom,
! [B: nat,A: nat] :
? [X3: nat,Y3: nat] :
( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X3 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
= ( gcd_gcd_nat @ A @ B ) ) )
| ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X3 ) )
& ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
= ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_9708_bezw__aux,axiom,
! [X: nat,Y4: nat] :
( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y4 ) )
= ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y4 ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y4 ) ) @ ( semiri1314217659103216013at_int @ Y4 ) ) ) ) ).
% bezw_aux
thf(fact_9709_gcd__nat_Opelims,axiom,
! [X: nat,Xa2: nat,Y4: nat] :
( ( ( gcd_gcd_nat @ X @ Xa2 )
= Y4 )
=> ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2 = zero_zero_nat )
=> ( Y4 = X ) )
& ( ( Xa2 != zero_zero_nat )
=> ( Y4
= ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_9710_card__greaterThanLessThan__int,axiom,
! [L2: int,U2: int] :
( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U2 ) )
= ( nat2 @ ( minus_minus_int @ U2 @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_9711_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L2: int,U2: int] :
( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U2 )
= ( set_or5832277885323065728an_int @ L2 @ U2 ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9712_infinite__nat__iff__unbounded__le,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( member_nat @ N2 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_9713_xor__minus__numerals_I2_J,axiom,
! [K: int,N: num] :
( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_9714_xor__minus__numerals_I1_J,axiom,
! [N: num,K: int] :
( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
= ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% xor_minus_numerals(1)
thf(fact_9715_card__greaterThanLessThan,axiom,
! [L2: nat,U2: nat] :
( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U2 ) )
= ( minus_minus_nat @ U2 @ ( suc @ L2 ) ) ) ).
% card_greaterThanLessThan
thf(fact_9716_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L2: nat,U2: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U2 )
= ( set_or5834768355832116004an_nat @ L2 @ U2 ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_9717_tanh__real__bounds,axiom,
! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% tanh_real_bounds
thf(fact_9718_sub__BitM__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
= ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% sub_BitM_One_eq
thf(fact_9719_unbounded__k__infinite,axiom,
! [K: nat,S3: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K @ M4 )
=> ? [N9: nat] :
( ( ord_less_nat @ M4 @ N9 )
& ( member_nat @ N9 @ S3 ) ) )
=> ~ ( finite_finite_nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_9720_infinite__nat__iff__unbounded,axiom,
! [S3: set_nat] :
( ( ~ ( finite_finite_nat @ S3 ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ( member_nat @ N2 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_9721_finite__enumerate,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ? [R3: nat > nat] :
( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
& ! [N9: nat] :
( ( ord_less_nat @ N9 @ ( finite_card_nat @ S3 ) )
=> ( member_nat @ ( R3 @ N9 ) @ S3 ) ) ) ) ).
% finite_enumerate
thf(fact_9722_Suc__funpow,axiom,
! [N: nat] :
( ( compow_nat_nat @ N @ suc )
= ( plus_plus_nat @ N ) ) ).
% Suc_funpow
thf(fact_9723_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
@ ^ [X2: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X2 )
@ ^ [X2: nat,Y: nat] : ( ord_less_nat @ Y @ X2 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_9724_times__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) )
@ Xa2
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_9725_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X3: nat,Y3: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_9726_zero__int__def,axiom,
( zero_zero_int
= ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% zero_int_def
thf(fact_9727_int__def,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).
% int_def
thf(fact_9728_uminus__int_Oabs__eq,axiom,
! [X: product_prod_nat_nat] :
( ( uminus_uminus_int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X2 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_9729_one__int__def,axiom,
( one_one_int
= ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% one_int_def
thf(fact_9730_less__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) )
@ Xa2
@ X ) ) ).
% less_int.abs_eq
thf(fact_9731_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) )
@ Xa2
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_9732_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
@ Xa2
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_9733_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) )
@ Xa2
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_9734_Gcd__remove0__nat,axiom,
! [M7: set_nat] :
( ( finite_finite_nat @ M7 )
=> ( ( gcd_Gcd_nat @ M7 )
= ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_9735_Gcd__nat__eq__one,axiom,
! [N4: set_nat] :
( ( member_nat @ one_one_nat @ N4 )
=> ( ( gcd_Gcd_nat @ N4 )
= one_one_nat ) ) ).
% Gcd_nat_eq_one
thf(fact_9736_Gcd__in,axiom,
! [A2: set_nat] :
( ! [A3: nat,B2: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( member_nat @ B2 @ A2 )
=> ( member_nat @ ( gcd_gcd_nat @ A3 @ B2 ) @ A2 ) ) )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).
% Gcd_in
thf(fact_9737_less__eq__int_Orep__eq,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y: nat,Z4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U3 @ Z4 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_9738_less__int_Orep__eq,axiom,
( ord_less_int
= ( ^ [X2: int,Xa4: int] :
( produc8739625826339149834_nat_o
@ ^ [Y: nat,Z4: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U3 @ Z4 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_9739_uminus__int__def,axiom,
( uminus_uminus_int
= ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X2 ) ) ) ) ).
% uminus_int_def
thf(fact_9740_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one ) ) ) ).
% num_of_nat.simps(2)
thf(fact_9741_num__of__nat__numeral__eq,axiom,
! [Q2: num] :
( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
= Q2 ) ).
% num_of_nat_numeral_eq
thf(fact_9742_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ zero_zero_nat )
= one ) ).
% num_of_nat.simps(1)
thf(fact_9743_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_9744_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ one_one_nat )
=> ( ( num_of_nat @ N )
= one ) ) ).
% num_of_nat_One
thf(fact_9745_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_9746_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_9747_times__int__def,axiom,
( times_times_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_9748_minus__int__def,axiom,
( minus_minus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_9749_plus__int__def,axiom,
( plus_plus_int
= ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_9750_pow_Osimps_I3_J,axiom,
! [X: num,Y4: num] :
( ( pow @ X @ ( bit1 @ Y4 ) )
= ( times_times_num @ ( sqr @ ( pow @ X @ Y4 ) ) @ X ) ) ).
% pow.simps(3)
thf(fact_9751_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_9752_sqr_Osimps_I1_J,axiom,
( ( sqr @ one )
= one ) ).
% sqr.simps(1)
thf(fact_9753_sqr__conv__mult,axiom,
( sqr
= ( ^ [X2: num] : ( times_times_num @ X2 @ X2 ) ) ) ).
% sqr_conv_mult
thf(fact_9754_pow_Osimps_I2_J,axiom,
! [X: num,Y4: num] :
( ( pow @ X @ ( bit0 @ Y4 ) )
= ( sqr @ ( pow @ X @ Y4 ) ) ) ).
% pow.simps(2)
thf(fact_9755_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_9756_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I2: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I2 ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9757_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu3: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( Xa2 = one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_9758_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_9759_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y4 )
=> ( ( ? [Uu3: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( Y4
= ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( Y4
= ( ~ ( ( Deg2 = Xa2 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_9760_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu3: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( Xa2 != one_one_nat ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_9761_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y4: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y4 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ( Y4
= ( Xa2 = one_one_nat ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa2 ) ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( Y4
= ( ( Deg2 = Xa2 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_9762_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa2 ) )
=> ( Xa2 != one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_9763_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
=> ( ! [Uu3: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu3 @ Uv2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa2 ) )
=> ( Xa2 = one_one_nat ) ) )
=> ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
=> ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [X3: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
& ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
& ( case_o184042715313410164at_nat
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
@ ( produc6081775807080527818_nat_o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq_nat @ Mi3 @ Ma3 )
& ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
& ! [X2: nat] :
( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
=> ( ( ord_less_nat @ Mi3 @ X2 )
& ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_9764_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] :
( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( case_option_int_num @ zero_zero_int
@ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_9765_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_9766_take__bit__num__simps_I1_J,axiom,
! [M: num] :
( ( bit_take_bit_num @ zero_zero_nat @ M )
= none_num ) ).
% take_bit_num_simps(1)
thf(fact_9767_take__bit__num__simps_I2_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ ( suc @ N ) @ one )
= ( some_num @ one ) ) ).
% take_bit_num_simps(2)
thf(fact_9768_take__bit__num__simps_I5_J,axiom,
! [R2: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
= ( some_num @ one ) ) ).
% take_bit_num_simps(5)
thf(fact_9769_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_9770_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_9771_take__bit__num__simps_I6_J,axiom,
! [R2: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
= ( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_9772_take__bit__num__simps_I7_J,axiom,
! [R2: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_9773_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_9774_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_9775_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
= ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_9776_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one @ one )
= none_num ) ).
% and_not_num.simps(1)
thf(fact_9777_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
= ( case_nat_option_num @ none_num
@ ^ [N2: nat] :
( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N2 @ M ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9778_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ N @ one )
= ( case_nat_option_num @ none_num
@ ^ [N2: nat] : ( some_num @ one )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9779_and__not__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ one )
= ( some_num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(4)
thf(fact_9780_and__not__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one @ ( bit0 @ N ) )
= ( some_num @ one ) ) ).
% and_not_num.simps(2)
thf(fact_9781_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_9782_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_9783_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_9784_and__not__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one @ ( bit1 @ N ) )
= none_num ) ).
% and_not_num.simps(3)
thf(fact_9785_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
= ( case_nat_option_num @ none_num
@ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9786_and__not__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ one )
= ( some_num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(7)
thf(fact_9787_and__not__num__eq__Some__iff,axiom,
! [M: num,N: num,Q2: num] :
( ( ( bit_and_not_num @ M @ N )
= ( some_num @ Q2 ) )
= ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= ( numeral_numeral_int @ Q2 ) ) ) ).
% and_not_num_eq_Some_iff
thf(fact_9788_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_9789_and__not__num__eq__None__iff,axiom,
! [M: num,N: num] :
( ( ( bit_and_not_num @ M @ N )
= none_num )
= ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
= zero_zero_int ) ) ).
% and_not_num_eq_None_iff
thf(fact_9790_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M2: num] :
( if_option_num
@ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) )
= zero_zero_nat )
@ none_num
@ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_9791_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M2: num] :
( produc478579273971653890on_num
@ ^ [A4: nat,X2: num] :
( case_nat_option_num @ none_num
@ ^ [O: nat] :
( case_num_option_num @ ( some_num @ one )
@ ^ [P4: num] :
( case_o6005452278849405969um_num @ none_num
@ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ O @ P4 ) )
@ ^ [P4: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P4 ) ) )
@ X2 )
@ A4 )
@ ( product_Pair_nat_num @ N2 @ M2 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_9792_and__not__num_Oelims,axiom,
! [X: num,Xa2: num,Y4: option_num] :
( ( ( bit_and_not_num @ X @ Xa2 )
= Y4 )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y4 != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa2
= ( bit0 @ N3 ) )
=> ( Y4
!= ( some_num @ one ) ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa2
= ( bit1 @ N3 ) )
=> ( Y4 != none_num ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ( ( Xa2 = one )
=> ( Y4
!= ( some_num @ ( bit0 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y4
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y4
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ( ( Xa2 = one )
=> ( Y4
!= ( some_num @ ( bit0 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y4
!= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
=> ~ ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y4
!= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_9793_and__not__num_Osimps_I5_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(5)
thf(fact_9794_and__not__num_Osimps_I6_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(6)
thf(fact_9795_and__not__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(9)
thf(fact_9796_and__num_Oelims,axiom,
! [X: num,Xa2: num,Y4: option_num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
= Y4 )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y4
!= ( some_num @ one ) ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa2
= ( bit0 @ N3 ) )
=> ( Y4 != none_num ) ) )
=> ( ( ( X = one )
=> ( ? [N3: num] :
( Xa2
= ( bit1 @ N3 ) )
=> ( Y4
!= ( some_num @ one ) ) ) )
=> ( ( ? [M4: num] :
( X
= ( bit0 @ M4 ) )
=> ( ( Xa2 = one )
=> ( Y4 != none_num ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y4
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y4
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
=> ( ( ? [M4: num] :
( X
= ( bit1 @ M4 ) )
=> ( ( Xa2 = one )
=> ( Y4
!= ( some_num @ one ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y4
!= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
=> ~ ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y4
!= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_9797_xor__num_Oelims,axiom,
! [X: num,Xa2: num,Y4: option_num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
= Y4 )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( Y4 != none_num ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y4
!= ( some_num @ ( bit1 @ N3 ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y4
!= ( some_num @ ( bit0 @ N3 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ( ( Xa2 = one )
=> ( Y4
!= ( some_num @ ( bit1 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y4
!= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y4
!= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ( ( Xa2 = one )
=> ( Y4
!= ( some_num @ ( bit0 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( Y4
!= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( Y4
!= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
thf(fact_9798_and__num_Osimps_I1_J,axiom,
( ( bit_un7362597486090784418nd_num @ one @ one )
= ( some_num @ one ) ) ).
% and_num.simps(1)
thf(fact_9799_xor__num_Osimps_I1_J,axiom,
( ( bit_un2480387367778600638or_num @ one @ one )
= none_num ) ).
% xor_num.simps(1)
thf(fact_9800_xor__num_Osimps_I5_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% xor_num.simps(5)
thf(fact_9801_and__num_Osimps_I5_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(5)
thf(fact_9802_and__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
= ( some_num @ one ) ) ).
% and_num.simps(7)
thf(fact_9803_and__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
= ( some_num @ one ) ) ).
% and_num.simps(3)
thf(fact_9804_and__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
= none_num ) ).
% and_num.simps(2)
thf(fact_9805_and__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
= none_num ) ).
% and_num.simps(4)
thf(fact_9806_and__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(8)
thf(fact_9807_and__num_Osimps_I6_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(6)
thf(fact_9808_xor__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% xor_num.simps(9)
thf(fact_9809_xor__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
= ( some_num @ ( bit1 @ N ) ) ) ).
% xor_num.simps(2)
thf(fact_9810_xor__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
= ( some_num @ ( bit0 @ N ) ) ) ).
% xor_num.simps(3)
thf(fact_9811_xor__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
= ( some_num @ ( bit1 @ M ) ) ) ).
% xor_num.simps(4)
thf(fact_9812_xor__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
= ( some_num @ ( bit0 @ M ) ) ) ).
% xor_num.simps(7)
thf(fact_9813_and__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(9)
thf(fact_9814_xor__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% xor_num.simps(8)
thf(fact_9815_xor__num_Osimps_I6_J,axiom,
! [M: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% xor_num.simps(6)
thf(fact_9816_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I2: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I2 ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9817_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L2: nat,U2: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U2 )
= ( set_or6659071591806873216st_nat @ L2 @ U2 ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_9818_Rats__eq__int__div__nat,axiom,
( field_5140801741446780682s_real
= ( collect_real
@ ^ [Uu2: real] :
? [I4: int,N2: nat] :
( ( Uu2
= ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
& ( N2 != zero_zero_nat ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_9819_rat__floor__lemma,axiom,
! [A: int,B: int] :
( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
& ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% rat_floor_lemma
thf(fact_9820_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y4: nat,X: nat] :
( ( ( ord_less_nat @ C @ Y4 )
=> ( ( image_nat_nat
@ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y4 ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y4 @ C ) ) ) )
& ( ~ ( ord_less_nat @ C @ Y4 )
=> ( ( ( ord_less_nat @ X @ Y4 )
=> ( ( image_nat_nat
@ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y4 ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X @ Y4 )
=> ( ( image_nat_nat
@ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
@ ( set_or4665077453230672383an_nat @ X @ Y4 ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_9821_bij__betw__Suc,axiom,
! [M7: set_nat,N4: set_nat] :
( ( bij_betw_nat_nat @ suc @ M7 @ N4 )
= ( ( image_nat_nat @ suc @ M7 )
= N4 ) ) ).
% bij_betw_Suc
thf(fact_9822_image__Suc__atLeastAtMost,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
= ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_9823_image__Suc__atLeastLessThan,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_9824_mult__rat,axiom,
! [A: int,B: int,C: int,D: int] :
( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
% mult_rat
thf(fact_9825_divide__rat,axiom,
! [A: int,B: int,C: int,D: int] :
( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% divide_rat
thf(fact_9826_less__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% less_rat
thf(fact_9827_add__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% add_rat
thf(fact_9828_le__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% le_rat
thf(fact_9829_diff__rat,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
= ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% diff_rat
thf(fact_9830_sgn__rat,axiom,
! [A: int,B: int] :
( ( sgn_sgn_rat @ ( fract @ A @ B ) )
= ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% sgn_rat
thf(fact_9831_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_9832_eq__rat_I2_J,axiom,
! [A: int] :
( ( fract @ A @ zero_zero_int )
= ( fract @ zero_zero_int @ one_one_int ) ) ).
% eq_rat(2)
thf(fact_9833_eq__rat_I1_J,axiom,
! [B: int,D: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( D != zero_zero_int )
=> ( ( ( fract @ A @ B )
= ( fract @ C @ D ) )
= ( ( times_times_int @ A @ D )
= ( times_times_int @ C @ B ) ) ) ) ) ).
% eq_rat(1)
thf(fact_9834_mult__rat__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( fract @ A @ B ) ) ) ).
% mult_rat_cancel
thf(fact_9835_Fract__of__nat__eq,axiom,
! [K: nat] :
( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
= ( semiri681578069525770553at_rat @ K ) ) ).
% Fract_of_nat_eq
thf(fact_9836_One__rat__def,axiom,
( one_one_rat
= ( fract @ one_one_int @ one_one_int ) ) ).
% One_rat_def
thf(fact_9837_quotient__of__eq,axiom,
! [A: int,B: int,P6: int,Q2: int] :
( ( ( quotient_of @ ( fract @ A @ B ) )
= ( product_Pair_int_int @ P6 @ Q2 ) )
=> ( ( fract @ P6 @ Q2 )
= ( fract @ A @ B ) ) ) ).
% quotient_of_eq
thf(fact_9838_Fract__of__int__eq,axiom,
! [K: int] :
( ( fract @ K @ one_one_int )
= ( ring_1_of_int_rat @ K ) ) ).
% Fract_of_int_eq
thf(fact_9839_normalize__eq,axiom,
! [A: int,B: int,P6: int,Q2: int] :
( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
= ( product_Pair_int_int @ P6 @ Q2 ) )
=> ( ( fract @ P6 @ Q2 )
= ( fract @ A @ B ) ) ) ).
% normalize_eq
thf(fact_9840_Zero__rat__def,axiom,
( zero_zero_rat
= ( fract @ zero_zero_int @ one_one_int ) ) ).
% Zero_rat_def
thf(fact_9841_rat__number__expand_I3_J,axiom,
( numeral_numeral_rat
= ( ^ [K3: num] : ( fract @ ( numeral_numeral_int @ K3 ) @ one_one_int ) ) ) ).
% rat_number_expand(3)
thf(fact_9842_rat__number__collapse_I3_J,axiom,
! [W2: num] :
( ( fract @ ( numeral_numeral_int @ W2 ) @ one_one_int )
= ( numeral_numeral_rat @ W2 ) ) ).
% rat_number_collapse(3)
thf(fact_9843_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L2: int,U2: int] :
( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U2 )
= ( set_or6656581121297822940st_int @ L2 @ U2 ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9844_quotient__of__Fract,axiom,
! [A: int,B: int] :
( ( quotient_of @ ( fract @ A @ B ) )
= ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).
% quotient_of_Fract
thf(fact_9845_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% image_Suc_lessThan
thf(fact_9846_image__Suc__atMost,axiom,
! [N: nat] :
( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
= ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_9847_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9848_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9849_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_9850_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost_nat @ ( suc @ N ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_9851_one__less__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% one_less_Fract_iff
thf(fact_9852_Fract__less__one__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
= ( ord_less_int @ A @ B ) ) ) ).
% Fract_less_one_iff
thf(fact_9853_rat__number__collapse_I5_J,axiom,
( ( fract @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= ( uminus_uminus_rat @ one_one_rat ) ) ).
% rat_number_collapse(5)
thf(fact_9854_Fract__add__one,axiom,
! [N: int,M: int] :
( ( N != zero_zero_int )
=> ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
= ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% Fract_add_one
thf(fact_9855_Fract__le__one__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% Fract_le_one_iff
thf(fact_9856_one__le__Fract__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% one_le_Fract_iff
thf(fact_9857_rat__number__collapse_I4_J,axiom,
! [W2: num] :
( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ one_one_int )
= ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ).
% rat_number_collapse(4)
thf(fact_9858_rat__number__expand_I5_J,axiom,
! [K: num] :
( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
= ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% rat_number_expand(5)
thf(fact_9859_positive__rat,axiom,
! [A: int,B: int] :
( ( positive @ ( fract @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% positive_rat
thf(fact_9860_Rat_Opositive__add,axiom,
! [X: rat,Y4: rat] :
( ( positive @ X )
=> ( ( positive @ Y4 )
=> ( positive @ ( plus_plus_rat @ X @ Y4 ) ) ) ) ).
% Rat.positive_add
thf(fact_9861_Rat_Opositive__mult,axiom,
! [X: rat,Y4: rat] :
( ( positive @ X )
=> ( ( positive @ Y4 )
=> ( positive @ ( times_times_rat @ X @ Y4 ) ) ) ) ).
% Rat.positive_mult
thf(fact_9862_image__add__int__atLeastLessThan,axiom,
! [L2: int,U2: int] :
( ( image_int_int
@ ^ [X2: int] : ( plus_plus_int @ X2 @ L2 )
@ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U2 @ L2 ) ) )
= ( set_or4662586982721622107an_int @ L2 @ U2 ) ) ).
% image_add_int_atLeastLessThan
thf(fact_9863_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X2: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X2 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_9864_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_9865_Inf__nat__def1,axiom,
! [K5: set_nat] :
( ( K5 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K5 ) @ K5 ) ) ).
% Inf_nat_def1
thf(fact_9866_UNIV__nat__eq,axiom,
( top_top_set_nat
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% UNIV_nat_eq
thf(fact_9867_range__mod,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( image_nat_nat
@ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N )
@ top_top_set_nat )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% range_mod
thf(fact_9868_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_9869_range__mult,axiom,
! [A: real] :
( ( ( A = zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
& ( ( A != zero_zero_real )
=> ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
= top_top_set_real ) ) ) ).
% range_mult
thf(fact_9870_card__UNIV__bool,axiom,
( ( finite_card_o @ top_top_set_o )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% card_UNIV_bool
thf(fact_9871_root__def,axiom,
( root
= ( ^ [N2: nat,X2: real] :
( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
@ ( the_in5290026491893676941l_real @ top_top_set_real
@ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
@ X2 ) ) ) ) ).
% root_def
thf(fact_9872_card__UNIV__char,axiom,
( ( finite_card_char @ top_top_set_char )
= ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% card_UNIV_char
thf(fact_9873_UNIV__char__of__nat,axiom,
( top_top_set_char
= ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% UNIV_char_of_nat
thf(fact_9874_char_Osize_I2_J,axiom,
! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] :
( ( size_size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X8 ) )
= zero_zero_nat ) ).
% char.size(2)
thf(fact_9875_nat__of__char__less__256,axiom,
! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% nat_of_char_less_256
thf(fact_9876_range__nat__of__char,axiom,
( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
= ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% range_nat_of_char
thf(fact_9877_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_9878_String_Ochar__of__ascii__of,axiom,
! [C: char] :
( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
= ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% String.char_of_ascii_of
thf(fact_9879_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D4: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( D4
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( D4
= ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( D4
= ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_9880_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( ord_less_real @ X @ zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_9881_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_9882_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_9883_has__real__derivative__pos__inc__right,axiom,
! [F: real > real,L2: real,X: real,S3: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S3 ) )
=> ( ( ord_less_real @ zero_zero_real @ L2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S3 )
=> ( ( ord_less_real @ H4 @ D3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_9884_has__real__derivative__neg__dec__right,axiom,
! [F: real > real,L2: real,X: real,S3: set_real] :
( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S3 ) )
=> ( ( ord_less_real @ L2 @ zero_zero_real )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S3 )
=> ( ( ord_less_real @ H4 @ D3 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_9885_DERIV__pos__inc__right,axiom,
! [F: real > real,L2: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ L2 )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D3 )
=> ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_9886_DERIV__neg__dec__right,axiom,
! [F: real > real,L2: real,X: real] :
( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ L2 @ zero_zero_real )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ! [H4: real] :
( ( ord_less_real @ zero_zero_real @ H4 )
=> ( ( ord_less_real @ H4 @ D3 )
=> ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_9887_DERIV__const__ratio__const,axiom,
! [A: real,B: real,F: real > real,K: real] :
( ( A != B )
=> ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% DERIV_const_ratio_const
thf(fact_9888_MVT2,axiom,
! [A: real,B: real,F: real > real,F4: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X3: real] :
( ( ord_less_eq_real @ A @ X3 )
=> ( ( ord_less_eq_real @ X3 @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F4 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
=> ? [Z: real] :
( ( ord_less_real @ A @ Z )
& ( ord_less_real @ Z @ B )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F4 @ Z ) ) ) ) ) ) ).
% MVT2
thf(fact_9889_DERIV__const__average,axiom,
! [A: real,B: real,V: real > real,K: real] :
( ( A != B )
=> ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
= ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% DERIV_const_average
thf(fact_9890_DERIV__ln__divide,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_ln_divide
thf(fact_9891_DERIV__pow,axiom,
! [N: nat,X: real,S2: set_real] :
( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( power_power_real @ X2 @ N )
@ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ S2 ) ) ).
% DERIV_pow
thf(fact_9892_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X: real,N: nat] :
( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( power_power_real @ ( G @ X2 ) @ N )
@ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_fun_pow
thf(fact_9893_has__real__derivative__powr,axiom,
! [Z2: real,R2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( has_fi5821293074295781190e_real
@ ^ [Z4: real] : ( powr_real @ Z4 @ R2 )
@ ( times_times_real @ R2 @ ( powr_real @ Z2 @ ( minus_minus_real @ R2 @ one_one_real ) ) )
@ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).
% has_real_derivative_powr
thf(fact_9894_DERIV__log,axiom,
! [X: real,B: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_log
thf(fact_9895_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X: real,R2: real] :
( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ R2 )
@ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_fun_powr
thf(fact_9896_DERIV__powr,axiom,
! [G: real > real,M: real,X: real,F: real > real,R2: real] :
( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
=> ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) )
@ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_powr
thf(fact_9897_DERIV__real__sqrt,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_real_sqrt
thf(fact_9898_DERIV__series_H,axiom,
! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
( ! [N3: nat] :
( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( F @ X2 @ N3 )
@ ( F4 @ X0 @ N3 )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( summable_real @ ( F @ X3 ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ( summable_real @ ( F4 @ X0 ) )
=> ( ( summable_real @ L5 )
=> ( ! [N3: nat,X3: real,Y3: real] :
( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
=> ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N3 ) @ ( F @ Y3 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ) )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] : ( suminf_real @ ( F @ X2 ) )
@ ( suminf_real @ ( F4 @ X0 ) )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_9899_DERIV__arctan,axiom,
! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).
% DERIV_arctan
thf(fact_9900_arsinh__real__has__field__derivative,axiom,
! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_9901_DERIV__real__sqrt__generic,axiom,
! [X: real,D4: real] :
( ( X != zero_zero_real )
=> ( ( ( ord_less_real @ zero_zero_real @ X )
=> ( D4
= ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( ( ( ord_less_real @ X @ zero_zero_real )
=> ( D4
= ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
=> ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_9902_sorted__list__of__set__greaterThanAtMost,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ ( suc @ I2 ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) )
= ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_9903_arcosh__real__has__field__derivative,axiom,
! [X: real,A2: set_real] :
( ( ord_less_real @ one_one_real @ X )
=> ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_9904_artanh__real__has__field__derivative,axiom,
! [X: real,A2: set_real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_9905_sorted__list__of__set__greaterThanLessThan,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ J )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) )
= ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_9906_DERIV__power__series_H,axiom,
! [R: real,F: nat > real,X0: real] :
( ! [X3: real] :
( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) ) )
=> ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
=> ( ( ord_less_real @ zero_zero_real @ R )
=> ( has_fi5821293074295781190e_real
@ ^ [X2: real] :
( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) )
@ ( suminf_real
@ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
@ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_9907_DERIV__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_real_root
thf(fact_9908_DERIV__arccos,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_arccos
thf(fact_9909_DERIV__arcsin,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_arcsin
thf(fact_9910_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_9911_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
( ( ( ( Diff @ zero_zero_nat )
= F )
& ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
=> ? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_9912_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( X != zero_zero_real )
=> ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_9913_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less_nat @ M4 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ T3 )
& ( ord_less_real @ T3 @ H2 )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_9914_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
( ( ord_less_real @ zero_zero_real @ H2 )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less_nat @ M4 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ H2 )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_9915_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
( ( ord_less_real @ H2 @ zero_zero_real )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less_nat @ M4 @ N )
& ( ord_less_eq_real @ H2 @ T3 )
& ( ord_less_eq_real @ T3 @ zero_zero_real ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ? [T3: real] :
( ( ord_less_real @ H2 @ T3 )
& ( ord_less_real @ T3 @ zero_zero_real )
& ( ( F @ H2 )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_9916_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( X != zero_zero_real )
=> ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
=> ? [T3: real] :
( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
& ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_9917_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less_nat @ M4 @ N )
& ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ? [T3: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_9918_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less_nat @ M4 @ N )
& ( ord_less_eq_real @ A @ T3 )
& ( ord_less_eq_real @ T3 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ( X != C )
=> ? [T3: real] :
( ( ( ord_less_real @ X @ C )
=> ( ( ord_less_real @ X @ T3 )
& ( ord_less_real @ T3 @ C ) ) )
& ( ~ ( ord_less_real @ X @ C )
=> ( ( ord_less_real @ C @ T3 )
& ( ord_less_real @ T3 @ X ) ) )
& ( ( F @ X )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_9919_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less_nat @ M4 @ N )
& ( ord_less_eq_real @ A @ T3 )
& ( ord_less_eq_real @ T3 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_real @ C @ B )
=> ? [T3: real] :
( ( ord_less_real @ C @ T3 )
& ( ord_less_real @ T3 @ B )
& ( ( F @ B )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_9920_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( Diff @ zero_zero_nat )
= F )
=> ( ! [M4: nat,T3: real] :
( ( ( ord_less_nat @ M4 @ N )
& ( ord_less_eq_real @ A @ T3 )
& ( ord_less_eq_real @ T3 @ B ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ( ( ord_less_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ B )
=> ? [T3: real] :
( ( ord_less_real @ A @ T3 )
& ( ord_less_real @ T3 @ C )
& ( ( F @ A )
= ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M2 ) )
@ ( set_ord_lessThan_nat @ N ) )
@ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T3 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_9921_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K: nat,B5: real] :
( ! [M4: nat,T3: real] :
( ( ( ord_less_nat @ M4 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T3 )
& ( ord_less_eq_real @ T3 @ H2 ) )
=> ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M3: nat,T6: real] :
( ( ( ord_less_nat @ M3 @ N )
& ( ord_less_eq_real @ zero_zero_real @ T6 )
& ( ord_less_eq_real @ T6 @ H2 ) )
=> ( has_fi5821293074295781190e_real
@ ^ [U3: real] :
( minus_minus_real @ ( Diff @ M3 @ U3 )
@ ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M3 @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ U3 @ P4 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M3 ) ) )
@ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U3 @ ( minus_minus_nat @ N @ M3 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M3 ) ) ) ) ) )
@ ( minus_minus_real @ ( Diff @ ( suc @ M3 ) @ T6 )
@ ( plus_plus_real
@ ( groups6591440286371151544t_real
@ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M3 ) @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ T6 @ P4 ) )
@ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) )
@ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_9922_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( has_fi5821293074295781190e_real
@ ^ [X9: real] :
( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
@ ( suminf_real
@ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% DERIV_arctan_series
thf(fact_9923_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_9924_isCont__arcosh,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% isCont_arcosh
thf(fact_9925_isCont__arccos,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_9926_isCont__arcsin,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_9927_isCont__artanh,axiom,
! [X: real] :
( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% isCont_artanh
thf(fact_9928_GMVT_H,axiom,
! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ A @ Z )
=> ( ( ord_less_eq_real @ Z @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) @ F ) ) )
=> ( ! [Z: real] :
( ( ord_less_eq_real @ A @ Z )
=> ( ( ord_less_eq_real @ Z @ B )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) @ G ) ) )
=> ( ! [Z: real] :
( ( ord_less_real @ A @ Z )
=> ( ( ord_less_real @ Z @ B )
=> ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z ) @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) )
=> ( ! [Z: real] :
( ( ord_less_real @ A @ Z )
=> ( ( ord_less_real @ Z @ B )
=> ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z ) @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) )
=> ? [C3: real] :
( ( ord_less_real @ A @ C3 )
& ( ord_less_real @ C3 @ B )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
= ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F4 @ C3 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_9929_upto_Opsimps,axiom,
! [I2: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
=> ( ( ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
& ( ~ ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= nil_int ) ) ) ) ).
% upto.psimps
thf(fact_9930_upto_Opelims,axiom,
! [X: int,Xa2: int,Y4: list_int] :
( ( ( upto @ X @ Xa2 )
= Y4 )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
=> ( Y4
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa2 )
=> ( Y4 = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% upto.pelims
thf(fact_9931_nth__upto,axiom,
! [I2: int,K: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
=> ( ( nth_int @ ( upto @ I2 @ J ) @ K )
= ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% nth_upto
thf(fact_9932_length__upto,axiom,
! [I2: int,J: int] :
( ( size_size_list_int @ ( upto @ I2 @ J ) )
= ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I2 ) @ one_one_int ) ) ) ).
% length_upto
thf(fact_9933_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_9934_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(4)
thf(fact_9935_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(3)
thf(fact_9936_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(2)
thf(fact_9937_atLeastAtMost__upto,axiom,
( set_or1266510415728281911st_int
= ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ J3 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_9938_upto__split2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I2 @ K )
= ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_9939_upto__split1,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I2 @ K )
= ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_9940_atLeastLessThan__upto,axiom,
( set_or4662586982721622107an_int
= ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_9941_greaterThanAtMost__upto,axiom,
( set_or6656581121297822940st_int
= ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_9942_upto__rec1,axiom,
! [I2: int,J: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_9943_upto_Osimps,axiom,
( upto
= ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% upto.simps
thf(fact_9944_upto_Oelims,axiom,
! [X: int,Xa2: int,Y4: list_int] :
( ( ( upto @ X @ Xa2 )
= Y4 )
=> ( ( ( ord_less_eq_int @ X @ Xa2 )
=> ( Y4
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa2 )
=> ( Y4 = nil_int ) ) ) ) ).
% upto.elims
thf(fact_9945_upto__rec2,axiom,
! [I2: int,J: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_9946_greaterThanLessThan__upto,axiom,
( set_or5832277885323065728an_int
= ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_9947_upto__split3,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_eq_int @ I2 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I2 @ K )
= ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_9948_GMVT,axiom,
! [A: real,B: real,F: real > real,G: real > real] :
( ( ord_less_real @ A @ B )
=> ( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
=> ( ! [X3: real] :
( ( ( ord_less_real @ A @ X3 )
& ( ord_less_real @ X3 @ B ) )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
=> ( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_eq_real @ X3 @ B ) )
=> ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
=> ( ! [X3: real] :
( ( ( ord_less_real @ A @ X3 )
& ( ord_less_real @ X3 @ B ) )
=> ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
=> ? [G_c: real,F_c: real,C3: real] :
( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
& ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
& ( ord_less_real @ A @ C3 )
& ( ord_less_real @ C3 @ B )
& ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
= ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_9949_LIM__cos__div__sin,axiom,
( filterlim_real_real
@ ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% LIM_cos_div_sin
thf(fact_9950_summable__Leibniz_I2_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
=> ! [N9: nat] :
( member_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_9951_summable__Leibniz_I3_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
=> ! [N9: nat] :
( member_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
@ ( set_or1222579329274155063t_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_9952_filterlim__Suc,axiom,
filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% filterlim_Suc
thf(fact_9953_mult__nat__left__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% mult_nat_left_at_top
thf(fact_9954_mult__nat__right__at__top,axiom,
! [C: nat] :
( ( ord_less_nat @ zero_zero_nat @ C )
=> ( filterlim_nat_nat
@ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
@ at_top_nat
@ at_top_nat ) ) ).
% mult_nat_right_at_top
thf(fact_9955_LIMSEQ__root,axiom,
( filterlim_nat_real
@ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ).
% LIMSEQ_root
thf(fact_9956_nested__sequence__unique,axiom,
! [F: nat > real,G: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
=> ( ( filterlim_nat_real
@ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat )
=> ? [L4: real] :
( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L4 )
& ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
& ! [N9: nat] : ( ord_less_eq_real @ L4 @ ( G @ N9 ) )
& ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_9957_LIMSEQ__inverse__zero,axiom,
! [X7: nat > real] :
( ! [R3: real] :
? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ N7 @ N3 )
=> ( ord_less_real @ R3 @ ( X7 @ N3 ) ) )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( inverse_inverse_real @ ( X7 @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_zero
thf(fact_9958_lim__inverse__n_H,axiom,
( filterlim_nat_real
@ ^ [N2: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% lim_inverse_n'
thf(fact_9959_LIMSEQ__root__const,axiom,
! [C: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( root @ N2 @ C )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat ) ) ).
% LIMSEQ_root_const
thf(fact_9960_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim_nat_real
@ ^ [N2: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_9961_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_9962_increasing__LIMSEQ,axiom,
! [F: nat > real,L2: real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L2 )
=> ( ! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
=> ? [N9: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
=> ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_9963_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_9964_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_9965_LIMSEQ__abs__realpow__zero,axiom,
! [C: real] :
( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_9966_LIMSEQ__abs__realpow__zero2,axiom,
! [C: real] :
( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
=> ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_9967_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N2 ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_9968_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N2: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9969_tendsto__exp__limit__sequentially,axiom,
! [X: real] :
( filterlim_nat_real
@ ^ [N2: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ at_top_nat ) ).
% tendsto_exp_limit_sequentially
thf(fact_9970_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R2: real] :
( filterlim_nat_real
@ ^ [N2: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
@ ( topolo2815343760600316023s_real @ R2 )
@ at_top_nat ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9971_summable__Leibniz_I1_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_9972_summable,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( summable_real
@ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).
% summable
thf(fact_9973_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim_nat_real
@ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat )
=> ~ ! [K2: nat > int] :
~ ( filterlim_nat_real
@ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
@ ( topolo2815343760600316023s_real @ Theta2 )
@ at_top_nat ) ) ).
% cos_diff_limit_1
thf(fact_9974_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim_nat_real
@ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
@ ( topolo2815343760600316023s_real @ one_one_real )
@ at_top_nat )
=> ? [K2: nat > int] :
( filterlim_nat_real
@ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% cos_limit_1
thf(fact_9975_summable__Leibniz_I4_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
@ at_top_nat ) ) ) ).
% summable_Leibniz(4)
thf(fact_9976_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
=> ( filterlim_nat_real
@ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ zero_zero_real )
@ at_top_nat ) ) ).
% zeroseq_arctan_series
thf(fact_9977_summable__Leibniz_H_I2_J,axiom,
! [A: nat > real,N: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ord_less_eq_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_9978_summable__Leibniz_H_I3_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_9979_sums__alternating__upper__lower,axiom,
! [A: nat > real] :
( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ? [L4: real] :
( ! [N9: nat] :
( ord_less_eq_real
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
@ L4 )
& ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
@ ( topolo2815343760600316023s_real @ L4 )
@ at_top_nat )
& ! [N9: nat] :
( ord_less_eq_real @ L4
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
& ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real @ L4 )
@ at_top_nat ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_9980_summable__Leibniz_I5_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ( topolo6980174941875973593q_real @ A )
=> ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
@ at_top_nat ) ) ) ).
% summable_Leibniz(5)
thf(fact_9981_summable__Leibniz_H_I5_J,axiom,
! [A: nat > real] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( filterlim_nat_real
@ ^ [N2: nat] :
( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
@ ( topolo2815343760600316023s_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
@ at_top_nat ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_9982_summable__Leibniz_H_I4_J,axiom,
! [A: nat > real,N: nat] :
( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
=> ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
=> ( ord_less_eq_real
@ ( suminf_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
@ ( groups6591440286371151544t_real
@ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
@ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_9983_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually_nat
@ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_Suc
thf(fact_9984_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually_nat
@ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
@ at_top_nat )
= ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentially_seg
thf(fact_9985_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually_nat @ P @ at_top_nat )
= ( ? [N6: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( P @ N2 ) ) ) ) ).
% eventually_sequentially
thf(fact_9986_eventually__sequentiallyI,axiom,
! [C: nat,P: nat > $o] :
( ! [X3: nat] :
( ( ord_less_eq_nat @ C @ X3 )
=> ( P @ X3 ) )
=> ( eventually_nat @ P @ at_top_nat ) ) ).
% eventually_sequentiallyI
thf(fact_9987_real__bounded__linear,axiom,
( real_V5970128139526366754l_real
= ( ^ [F3: real > real] :
? [C2: real] :
( F3
= ( ^ [X2: real] : ( times_times_real @ X2 @ C2 ) ) ) ) ) ).
% real_bounded_linear
thf(fact_9988_le__sequentially,axiom,
! [F5: filter_nat] :
( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
= ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).
% le_sequentially
thf(fact_9989_sequentially__offset,axiom,
! [P: nat > $o,K: nat] :
( ( eventually_nat @ P @ at_top_nat )
=> ( eventually_nat
@ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
@ at_top_nat ) ) ).
% sequentially_offset
thf(fact_9990_tanh__real__at__top,axiom,
filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% tanh_real_at_top
thf(fact_9991_artanh__real__at__left__1,axiom,
filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% artanh_real_at_left_1
thf(fact_9992_tendsto__exp__limit__at__top,axiom,
! [X: real] :
( filterlim_real_real
@ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y ) ) @ Y )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ at_top_real ) ).
% tendsto_exp_limit_at_top
thf(fact_9993_filterlim__tan__at__left,axiom,
filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% filterlim_tan_at_left
thf(fact_9994_tendsto__arctan__at__top,axiom,
filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% tendsto_arctan_at_top
thf(fact_9995_tendsto__exp__limit__at__right,axiom,
! [X: real] :
( filterlim_real_real
@ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
@ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_9996_tendsto__arctan__at__bot,axiom,
filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% tendsto_arctan_at_bot
thf(fact_9997_artanh__real__at__right__1,axiom,
filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% artanh_real_at_right_1
thf(fact_9998_eventually__at__right__to__0,axiom,
! [P: real > $o,A: real] :
( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
= ( eventually_real
@ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A ) )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_9999_filterlim__tan__at__right,axiom,
filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% filterlim_tan_at_right
thf(fact_10000_tanh__real__at__bot,axiom,
filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% tanh_real_at_bot
thf(fact_10001_tendsto__arcosh__at__left__1,axiom,
filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% tendsto_arcosh_at_left_1
thf(fact_10002_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F: real > real,F5: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
=> ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
@ at_bot_real
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_10003_filterlim__pow__at__bot__even,axiom,
! [N: nat,F: real > real,F5: filter_real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
=> ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( filterlim_real_real
@ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
@ at_top_real
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_10004_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( set_or1210151606488870762an_nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_10005_INT__greaterThan__UNIV,axiom,
( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
= bot_bot_set_nat ) ).
% INT_greaterThan_UNIV
thf(fact_10006_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_10007_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_10008_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% atLeast_Suc
thf(fact_10009_Gcd__eq__Max,axiom,
! [M7: set_nat] :
( ( finite_finite_nat @ M7 )
=> ( ( M7 != bot_bot_set_nat )
=> ( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( gcd_Gcd_nat @ M7 )
= ( lattic8265883725875713057ax_nat
@ ( comple7806235888213564991et_nat
@ ( image_nat_set_nat
@ ^ [M2: nat] :
( collect_nat
@ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M2 ) )
@ M7 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_10010_card__le__Suc__Max,axiom,
! [S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_10011_Sup__nat__def,axiom,
( complete_Sup_Sup_nat
= ( ^ [X6: set_nat] : ( if_nat @ ( X6 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X6 ) ) ) ) ).
% Sup_nat_def
thf(fact_10012_divide__nat__def,axiom,
( divide_divide_nat
= ( ^ [M2: nat,N2: nat] :
( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M2 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_10013_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( gcd_gcd_nat @ M @ N )
= ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D2: nat] :
( ( dvd_dvd_nat @ D2 @ M )
& ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_10014_MVT,axiom,
! [A: real,B: real,F: real > real] :
( ( ord_less_real @ A @ B )
=> ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
=> ( ! [X3: real] :
( ( ord_less_real @ A @ X3 )
=> ( ( ord_less_real @ X3 @ B )
=> ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
=> ? [L4: real,Z: real] :
( ( ord_less_real @ A @ Z )
& ( ord_less_real @ Z @ B )
& ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) )
& ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
= ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% MVT
thf(fact_10015_continuous__on__arcosh,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
=> ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% continuous_on_arcosh
thf(fact_10016_continuous__on__arcosh_H,axiom,
! [A2: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ A2 @ F )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
=> ( topolo5044208981011980120l_real @ A2
@ ^ [X2: real] : ( arcosh_real @ ( F @ X2 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_10017_continuous__on__arccos_H,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% continuous_on_arccos'
thf(fact_10018_continuous__on__arcsin_H,axiom,
topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% continuous_on_arcsin'
thf(fact_10019_continuous__on__artanh,axiom,
! [A2: set_real] :
( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
=> ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% continuous_on_artanh
thf(fact_10020_continuous__on__artanh_H,axiom,
! [A2: set_real,F: real > real] :
( ( topolo5044208981011980120l_real @ A2 @ F )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
=> ( topolo5044208981011980120l_real @ A2
@ ^ [X2: real] : ( artanh_real @ ( F @ X2 ) ) ) ) ) ).
% continuous_on_artanh'
thf(fact_10021_mono__Suc,axiom,
order_mono_nat_nat @ suc ).
% mono_Suc
thf(fact_10022_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% mono_times_nat
thf(fact_10023_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( order_mono_nat_nat
@ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_10024_Bseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% Bseq_realpow
thf(fact_10025_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( inj_on_real_real
@ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
@ top_top_set_real ) ) ).
% inj_sgn_power
thf(fact_10026_log__inj,axiom,
! [B: real] :
( ( ord_less_real @ one_one_real @ B )
=> ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% log_inj
thf(fact_10027_inj__on__diff__nat,axiom,
! [N4: set_nat,K: nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ N4 )
=> ( ord_less_eq_nat @ K @ N3 ) )
=> ( inj_on_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
@ N4 ) ) ).
% inj_on_diff_nat
thf(fact_10028_inj__Suc,axiom,
! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% inj_Suc
thf(fact_10029_inj__on__char__of__nat,axiom,
inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% inj_on_char_of_nat
thf(fact_10030_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_10031_min__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_10032_min__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
= ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_10033_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
= ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_10034_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
= ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_10035_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min_nat @ ( suc @ N ) @ M )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
@ M ) ) ).
% min_Suc1
thf(fact_10036_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_10037_pred__nat__def,axiom,
( pred_nat
= ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [M2: nat,N2: nat] :
( N2
= ( suc @ M2 ) ) ) ) ) ).
% pred_nat_def
thf(fact_10038_min__enat__simps_I2_J,axiom,
! [Q2: extended_enat] :
( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
= zero_z5237406670263579293d_enat ) ).
% min_enat_simps(2)
thf(fact_10039_min__enat__simps_I3_J,axiom,
! [Q2: extended_enat] :
( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
= zero_z5237406670263579293d_enat ) ).
% min_enat_simps(3)
thf(fact_10040_inf__enat__def,axiom,
inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% inf_enat_def
thf(fact_10041_sup__enat__def,axiom,
sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% sup_enat_def
thf(fact_10042_atLeastLessThan__add__Un,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_10043_less__eq,axiom,
! [M: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
= ( ord_less_nat @ M @ N ) ) ).
% less_eq
thf(fact_10044_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_10045_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl_nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_10046_hd__upt,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( hd_nat @ ( upt @ I2 @ J ) )
= I2 ) ) ).
% hd_upt
thf(fact_10047_drop__upt,axiom,
! [M: nat,I2: nat,J: nat] :
( ( drop_nat @ M @ ( upt @ I2 @ J ) )
= ( upt @ ( plus_plus_nat @ I2 @ M ) @ J ) ) ).
% drop_upt
thf(fact_10048_length__upt,axiom,
! [I2: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I2 @ J ) )
= ( minus_minus_nat @ J @ I2 ) ) ).
% length_upt
thf(fact_10049_take__upt,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I2 @ N ) )
= ( upt @ I2 @ ( plus_plus_nat @ I2 @ M ) ) ) ) ).
% take_upt
thf(fact_10050_upt__conv__Nil,axiom,
! [J: nat,I2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( upt @ I2 @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_10051_upt__eq__Nil__conv,axiom,
! [I2: nat,J: nat] :
( ( ( upt @ I2 @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I2 ) ) ) ).
% upt_eq_Nil_conv
thf(fact_10052_nth__upt,axiom,
! [I2: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ K ) ) ) ).
% nth_upt
thf(fact_10053_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_10054_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_10055_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_10056_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_10057_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M2 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_10058_atLeastAtMost__upt,axiom,
( set_or1269000886237332187st_nat
= ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M2 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_10059_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I4: nat,J3: nat] : ( set_nat2 @ ( upt @ I4 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_10060_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M2 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_10061_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_10062_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_10063_upt__conv__Cons,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( upt @ I2 @ J )
= ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_10064_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_10065_upt__add__eq__append,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_10066_upt__eq__Cons__conv,axiom,
! [I2: nat,J: nat,X: nat,Xs2: list_nat] :
( ( ( upt @ I2 @ J )
= ( cons_nat @ X @ Xs2 ) )
= ( ( ord_less_nat @ I2 @ J )
& ( I2 = X )
& ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_10067_upt__rec,axiom,
( upt
= ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_10068_upt__Suc,axiom,
! [I2: nat,J: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_10069_upt__Suc__append,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_10070_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [X2: nat] : X2
@ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_10071_card__length__sum__list__rec,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L: list_nat] :
( ( ( size_size_list_nat @ L )
= M )
& ( ( groups4561878855575611511st_nat @ L )
= N4 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L: list_nat] :
( ( ( size_size_list_nat @ L )
= ( minus_minus_nat @ M @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L )
= N4 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L: list_nat] :
( ( ( size_size_list_nat @ L )
= M )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L ) @ one_one_nat )
= N4 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_10072_card__length__sum__list,axiom,
! [M: nat,N4: nat] :
( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L: list_nat] :
( ( ( size_size_list_nat @ L )
= M )
& ( ( groups4561878855575611511st_nat @ L )
= N4 ) ) ) )
= ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N4 @ M ) @ one_one_nat ) @ N4 ) ) ).
% card_length_sum_list
thf(fact_10073_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_10074_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_10075_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I2: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_10076_prod__encode__prod__decode__aux,axiom,
! [K: nat,M: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
= ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% prod_encode_prod_decode_aux
thf(fact_10077_le__prod__encode__1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_1
thf(fact_10078_le__prod__encode__2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% le_prod_encode_2
thf(fact_10079_prod__encode__def,axiom,
( nat_prod_encode
= ( produc6842872674320459806at_nat
@ ^ [M2: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M2 @ N2 ) ) @ M2 ) ) ) ).
% prod_encode_def
thf(fact_10080_list__encode_Oelims,axiom,
! [X: list_nat,Y4: nat] :
( ( ( nat_list_encode @ X )
= Y4 )
=> ( ( ( X = nil_nat )
=> ( Y4 != zero_zero_nat ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( X
= ( cons_nat @ X3 @ Xs3 ) )
=> ( Y4
!= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_10081_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( nat_list_encode @ ( cons_nat @ X @ Xs2 ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_10082_list__encode_Opelims,axiom,
! [X: list_nat,Y4: nat] :
( ( ( nat_list_encode @ X )
= Y4 )
=> ( ( accp_list_nat @ nat_list_encode_rel @ X )
=> ( ( ( X = nil_nat )
=> ( ( Y4 = zero_zero_nat )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( X
= ( cons_nat @ X3 @ Xs3 ) )
=> ( ( Y4
= ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) )
=> ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_10083_Gcd__int__set__eq__fold,axiom,
! [Xs2: list_int] :
( ( gcd_Gcd_int @ ( set_int2 @ Xs2 ) )
= ( fold_int_int @ gcd_gcd_int @ Xs2 @ zero_zero_int ) ) ).
% Gcd_int_set_eq_fold
thf(fact_10084_Gcd__nat__set__eq__fold,axiom,
! [Xs2: list_nat] :
( ( gcd_Gcd_nat @ ( set_nat2 @ Xs2 ) )
= ( fold_nat_nat @ gcd_gcd_nat @ Xs2 @ zero_zero_nat ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_10085_vanishes__mult__bounded,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ? [A6: rat] :
( ( ord_less_rat @ zero_zero_rat @ A6 )
& ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N3 ) ) @ A6 ) )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N2: nat] : ( times_times_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_10086_vanishes__add,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ( vanishes @ X7 )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N2: nat] : ( plus_plus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% vanishes_add
thf(fact_10087_vanishes__def,axiom,
( vanishes
= ( ^ [X6: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N2 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
thf(fact_10088_vanishesI,axiom,
! [X7: nat > rat] :
( ! [R3: rat] :
( ( ord_less_rat @ zero_zero_rat @ R3 )
=> ? [K4: nat] :
! [N3: nat] :
( ( ord_less_eq_nat @ K4 @ N3 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N3 ) ) @ R3 ) ) )
=> ( vanishes @ X7 ) ) ).
% vanishesI
thf(fact_10089_vanishesD,axiom,
! [X7: nat > rat,R2: rat] :
( ( vanishes @ X7 )
=> ( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K2: nat] :
! [N9: nat] :
( ( ord_less_eq_nat @ K2 @ N9 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N9 ) ) @ R2 ) ) ) ) ).
% vanishesD
thf(fact_10090_and__not__num_Opelims,axiom,
! [X: num,Xa2: num,Y4: option_num] :
( ( ( bit_and_not_num @ X @ Xa2 )
= Y4 )
=> ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y4 = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y4
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y4 = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ( ( Xa2 = one )
=> ( ( Y4
= ( some_num @ ( bit0 @ M4 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y4
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y4
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ( ( Xa2 = one )
=> ( ( Y4
= ( some_num @ ( bit0 @ M4 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y4
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_and_not_num @ M4 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y4
= ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_10091_and__num_Opelims,axiom,
! [X: num,Xa2: num,Y4: option_num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
= Y4 )
=> ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y4
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y4 = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y4
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ( ( Xa2 = one )
=> ( ( Y4 = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y4
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y4
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ( ( Xa2 = one )
=> ( ( Y4
= ( some_num @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y4
= ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y4
= ( case_o6005452278849405969um_num @ ( some_num @ one )
@ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
@ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_10092_xor__num_Opelims,axiom,
! [X: num,Xa2: num,Y4: option_num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
= Y4 )
=> ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y4 = none_num )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y4
= ( some_num @ ( bit1 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y4
= ( some_num @ ( bit0 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ( ( Xa2 = one )
=> ( ( Y4
= ( some_num @ ( bit1 @ M4 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y4
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y4
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ( ( Xa2 = one )
=> ( ( Y4
= ( some_num @ ( bit0 @ M4 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit0 @ N3 ) )
=> ( ( Y4
= ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa2
= ( bit1 @ N3 ) )
=> ( ( Y4
= ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_10093_or__not__num__neg_Opelims,axiom,
! [X: num,Xa2: num,Y4: num] :
( ( ( bit_or_not_num_neg @ X @ Xa2 )
= Y4 )
=> ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
=> ( ( ( X = one )
=> ( ( Xa2 = one )
=> ( ( Y4 = one )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
=> ( ( ( X = one )
=> ! [M4: num] :
( ( Xa2
= ( bit0 @ M4 ) )
=> ( ( Y4
= ( bit1 @ M4 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
=> ( ( ( X = one )
=> ! [M4: num] :
( ( Xa2
= ( bit1 @ M4 ) )
=> ( ( Y4
= ( bit1 @ M4 ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ( ( Xa2 = one )
=> ( ( Y4
= ( bit0 @ one ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ one ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M4: num] :
( ( Xa2
= ( bit0 @ M4 ) )
=> ( ( Y4
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M4: num] :
( ( Xa2
= ( bit1 @ M4 ) )
=> ( ( Y4
= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ( ( Xa2 = one )
=> ( ( Y4 = one )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ one ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M4: num] :
( ( Xa2
= ( bit0 @ M4 ) )
=> ( ( Y4
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
=> ~ ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M4: num] :
( ( Xa2
= ( bit1 @ M4 ) )
=> ( ( Y4
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
=> ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_10094_rcis__inverse,axiom,
! [R2: real,A: real] :
( ( invers8013647133539491842omplex @ ( rcis @ R2 @ A ) )
= ( rcis @ ( divide_divide_real @ one_one_real @ R2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% rcis_inverse
thf(fact_10095_Re__rcis,axiom,
! [R2: real,A: real] :
( ( re @ ( rcis @ R2 @ A ) )
= ( times_times_real @ R2 @ ( cos_real @ A ) ) ) ).
% Re_rcis
thf(fact_10096_Im__rcis,axiom,
! [R2: real,A: real] :
( ( im @ ( rcis @ R2 @ A ) )
= ( times_times_real @ R2 @ ( sin_real @ A ) ) ) ).
% Im_rcis
thf(fact_10097_cis__rcis__eq,axiom,
( cis
= ( rcis @ one_one_real ) ) ).
% cis_rcis_eq
thf(fact_10098_rcis__mult,axiom,
! [R1: real,A: real,R22: real,B: real] :
( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
= ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B ) ) ) ).
% rcis_mult
thf(fact_10099_rcis__def,axiom,
( rcis
= ( ^ [R5: real,A4: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( cis @ A4 ) ) ) ) ).
% rcis_def
thf(fact_10100_DeMoivre2,axiom,
! [R2: real,A: real,N: nat] :
( ( power_power_complex @ ( rcis @ R2 @ A ) @ N )
= ( rcis @ ( power_power_real @ R2 @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% DeMoivre2
thf(fact_10101_Field__natLeq__on,axiom,
! [N: nat] :
( ( field_nat
@ ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ N )
& ( ord_less_nat @ Y @ N )
& ( ord_less_eq_nat @ X2 @ Y ) ) ) ) )
= ( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ X2 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_10102_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% natLess_def
thf(fact_10103_wf__less,axiom,
wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% wf_less
thf(fact_10104_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( order_5726023648592871131at_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_10105_cauchy__def,axiom,
( cauchy
= ( ^ [X6: nat > rat] :
! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [M2: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% cauchy_def
thf(fact_10106_cauchy__add,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N2: nat] : ( plus_plus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% cauchy_add
thf(fact_10107_cauchy__mult,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N2: nat] : ( times_times_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% cauchy_mult
thf(fact_10108_cauchy__not__vanishes__cases,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ~ ( vanishes @ X7 )
=> ? [B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ B2 )
& ? [K2: nat] :
( ! [N9: nat] :
( ( ord_less_eq_nat @ K2 @ N9 )
=> ( ord_less_rat @ B2 @ ( uminus_uminus_rat @ ( X7 @ N9 ) ) ) )
| ! [N9: nat] :
( ( ord_less_eq_nat @ K2 @ N9 )
=> ( ord_less_rat @ B2 @ ( X7 @ N9 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_10109_cauchy__not__vanishes,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ~ ( vanishes @ X7 )
=> ? [B2: rat] :
( ( ord_less_rat @ zero_zero_rat @ B2 )
& ? [K2: nat] :
! [N9: nat] :
( ( ord_less_eq_nat @ K2 @ N9 )
=> ( ord_less_rat @ B2 @ ( abs_abs_rat @ ( X7 @ N9 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_10110_cauchyD,axiom,
! [X7: nat > rat,R2: rat] :
( ( cauchy @ X7 )
=> ( ( ord_less_rat @ zero_zero_rat @ R2 )
=> ? [K2: nat] :
! [M3: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ! [N9: nat] :
( ( ord_less_eq_nat @ K2 @ N9 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M3 ) @ ( X7 @ N9 ) ) ) @ R2 ) ) ) ) ) ).
% cauchyD
thf(fact_10111_cauchyI,axiom,
! [X7: nat > rat] :
( ! [R3: rat] :
( ( ord_less_rat @ zero_zero_rat @ R3 )
=> ? [K4: nat] :
! [M4: nat] :
( ( ord_less_eq_nat @ K4 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq_nat @ K4 @ N3 )
=> ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M4 ) @ ( X7 @ N3 ) ) ) @ R3 ) ) ) )
=> ( cauchy @ X7 ) ) ).
% cauchyI
thf(fact_10112_le__Real,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( cauchy @ Y7 )
=> ( ( ord_less_eq_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
= ( ! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_eq_rat @ ( X7 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_10113_one__real__def,axiom,
( one_one_real
= ( real2
@ ^ [N2: nat] : one_one_rat ) ) ).
% one_real_def
thf(fact_10114_add__Real,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( cauchy @ Y7 )
=> ( ( plus_plus_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N2: nat] : ( plus_plus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% add_Real
thf(fact_10115_mult__Real,axiom,
! [X7: nat > rat,Y7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( cauchy @ Y7 )
=> ( ( times_times_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N2: nat] : ( times_times_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% mult_Real
thf(fact_10116_not__positive__Real,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( ~ ( positive2 @ ( real2 @ X7 ) ) )
= ( ! [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_eq_rat @ ( X7 @ N2 ) @ R5 ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_10117_positive__Real,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( positive2 @ ( real2 @ X7 ) )
= ( ? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X7 @ N2 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_10118_Real_Opositive__mult,axiom,
! [X: real,Y4: real] :
( ( positive2 @ X )
=> ( ( positive2 @ Y4 )
=> ( positive2 @ ( times_times_real @ X @ Y4 ) ) ) ) ).
% Real.positive_mult
thf(fact_10119_Real_Opositive__add,axiom,
! [X: real,Y4: real] :
( ( positive2 @ X )
=> ( ( positive2 @ Y4 )
=> ( positive2 @ ( plus_plus_real @ X @ Y4 ) ) ) ) ).
% Real.positive_add
thf(fact_10120_Real_Opositive_Orep__eq,axiom,
( positive2
= ( ^ [X2: real] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( rep_real @ X2 @ N2 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_10121_le__enumerate,axiom,
! [S3: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S3 )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ).
% le_enumerate
thf(fact_10122_finite__le__enumerate,axiom,
! [S3: set_nat,N: nat] :
( ( finite_finite_nat @ S3 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S3 ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_10123_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ zero_zero_nat )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least_nat @ P )
= ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_10124_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_10125_divmod__nat__code,axiom,
( divmod_nat
= ( ^ [M2: nat,N2: nat] :
( produc8678311845419106900er_nat @ code_nat_of_integer @ code_nat_of_integer
@ ( if_Pro6119634080678213985nteger
@ ( ( code_integer_of_nat @ M2 )
= zero_z3403309356797280102nteger )
@ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
@ ( if_Pro6119634080678213985nteger
@ ( ( code_integer_of_nat @ N2 )
= zero_z3403309356797280102nteger )
@ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( code_integer_of_nat @ M2 ) )
@ ( code_divmod_abs @ ( code_integer_of_nat @ M2 ) @ ( code_integer_of_nat @ N2 ) ) ) ) ) ) ) ).
% divmod_nat_code
thf(fact_10126_integer__of__nat__numeral,axiom,
! [N: num] :
( ( code_integer_of_nat @ ( numeral_numeral_nat @ N ) )
= ( numera6620942414471956472nteger @ N ) ) ).
% integer_of_nat_numeral
thf(fact_10127_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ one_one_nat )
= one_one_Code_integer ) ).
% integer_of_nat_1
thf(fact_10128_eventually__prod__sequentially,axiom,
! [P: product_prod_nat_nat > $o] :
( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
= ( ? [N6: nat] :
! [M2: nat] :
( ( ord_less_eq_nat @ N6 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq_nat @ N6 @ N2 )
=> ( P @ ( product_Pair_nat_nat @ N2 @ M2 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_10129_at__right__to__0,axiom,
! [A: real] :
( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
= ( filtermap_real_real
@ ^ [X2: real] : ( plus_plus_real @ X2 @ A )
@ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% at_right_to_0
thf(fact_10130_last__upt,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( last_nat @ ( upt @ I2 @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_10131_Real_Opositive_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( positive2 @ ( real2 @ X ) )
= ( ? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X @ N2 ) ) ) ) ) ) ) ).
% Real.positive.abs_eq
thf(fact_10132_one__real_Orsp,axiom,
( realrel
@ ^ [N2: nat] : one_one_rat
@ ^ [N2: nat] : one_one_rat ) ).
% one_real.rsp
thf(fact_10133_plus__real_Oabs__eq,axiom,
! [Xa2: nat > rat,X: nat > rat] :
( ( realrel @ Xa2 @ Xa2 )
=> ( ( realrel @ X @ X )
=> ( ( plus_plus_real @ ( real2 @ Xa2 ) @ ( real2 @ X ) )
= ( real2
@ ^ [N2: nat] : ( plus_plus_rat @ ( Xa2 @ N2 ) @ ( X @ N2 ) ) ) ) ) ) ).
% plus_real.abs_eq
thf(fact_10134_times__real_Oabs__eq,axiom,
! [Xa2: nat > rat,X: nat > rat] :
( ( realrel @ Xa2 @ Xa2 )
=> ( ( realrel @ X @ X )
=> ( ( times_times_real @ ( real2 @ Xa2 ) @ ( real2 @ X ) )
= ( real2
@ ^ [N2: nat] : ( times_times_rat @ ( Xa2 @ N2 ) @ ( X @ N2 ) ) ) ) ) ) ).
% times_real.abs_eq
thf(fact_10135_Real_Opositive_Orsp,axiom,
( bNF_re728719798268516973at_o_o @ realrel
@ ^ [Y6: $o,Z3: $o] : Y6 = Z3
@ ^ [X6: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
@ ^ [X6: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ).
% Real.positive.rsp
thf(fact_10136_times__real_Orsp,axiom,
( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
@ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
@ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).
% times_real.rsp
thf(fact_10137_plus__real_Orsp,axiom,
( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
@ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
@ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).
% plus_real.rsp
thf(fact_10138_less__eq__natural_Orsp,axiom,
( bNF_re578469030762574527_nat_o
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
@ ord_less_eq_nat
@ ord_less_eq_nat ) ).
% less_eq_natural.rsp
thf(fact_10139_times__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
@ times_times_nat
@ times_times_nat ) ).
% times_natural.rsp
thf(fact_10140_times__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y6: int,Z3: int] : Y6 = Z3
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z3: int] : Y6 = Z3
@ ^ [Y6: int,Z3: int] : Y6 = Z3 )
@ times_times_int
@ times_times_int ) ).
% times_integer.rsp
thf(fact_10141_Suc_Orsp,axiom,
( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_10142_plus__integer_Orsp,axiom,
( bNF_re711492959462206631nt_int
@ ^ [Y6: int,Z3: int] : Y6 = Z3
@ ( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z3: int] : Y6 = Z3
@ ^ [Y6: int,Z3: int] : Y6 = Z3 )
@ plus_plus_int
@ plus_plus_int ) ).
% plus_integer.rsp
thf(fact_10143_plus__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
@ plus_plus_nat
@ plus_plus_nat ) ).
% plus_natural.rsp
thf(fact_10144_less__natural_Orsp,axiom,
( bNF_re578469030762574527_nat_o
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ( bNF_re4705727531993890431at_o_o
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
@ ord_less_nat
@ ord_less_nat ) ).
% less_natural.rsp
thf(fact_10145_divide__natural_Orsp,axiom,
( bNF_re1345281282404953727at_nat
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ( bNF_re5653821019739307937at_nat
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3 )
@ divide_divide_nat
@ divide_divide_nat ) ).
% divide_natural.rsp
thf(fact_10146_dup_Orsp,axiom,
( bNF_re4712519889275205905nt_int
@ ^ [Y6: int,Z3: int] : Y6 = Z3
@ ^ [Y6: int,Z3: int] : Y6 = Z3
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 )
@ ^ [K3: int] : ( plus_plus_int @ K3 @ K3 ) ) ).
% dup.rsp
thf(fact_10147_Real_Opositive_Otransfer,axiom,
( bNF_re4297313714947099218al_o_o @ pcr_real
@ ^ [Y6: $o,Z3: $o] : Y6 = Z3
@ ^ [X6: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
@ positive2 ) ).
% Real.positive.transfer
thf(fact_10148_plus__rat_Otransfer,axiom,
( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
@ plus_plus_rat ) ).
% plus_rat.transfer
thf(fact_10149_one__real_Otransfer,axiom,
( pcr_real
@ ^ [N2: nat] : one_one_rat
@ one_one_real ) ).
% one_real.transfer
thf(fact_10150_plus__real_Otransfer,axiom,
( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
@ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
@ plus_plus_real ) ).
% plus_real.transfer
thf(fact_10151_times__real_Otransfer,axiom,
( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
@ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
@ times_times_real ) ).
% times_real.transfer
thf(fact_10152_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% one_rat.transfer
thf(fact_10153_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% zero_rat.transfer
thf(fact_10154_Fract_Otransfer,axiom,
( bNF_re3461391660133120880nt_rat
@ ^ [Y6: int,Z3: int] : Y6 = Z3
@ ( bNF_re2214769303045360666nt_rat
@ ^ [Y6: int,Z3: int] : Y6 = Z3
@ pcr_rat )
@ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B3 ) )
@ fract ) ).
% Fract.transfer
thf(fact_10155_uminus__rat_Otransfer,axiom,
( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
@ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) )
@ uminus_uminus_rat ) ).
% uminus_rat.transfer
thf(fact_10156_times__rat_Otransfer,axiom,
( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
@ times_times_rat ) ).
% times_rat.transfer
thf(fact_10157_Rat_Opositive_Otransfer,axiom,
( bNF_re1494630372529172596at_o_o @ pcr_rat
@ ^ [Y6: $o,Z3: $o] : Y6 = Z3
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
@ positive ) ).
% Rat.positive.transfer
thf(fact_10158_inverse__rat_Otransfer,axiom,
( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
@ ^ [X2: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X2 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
@ inverse_inverse_rat ) ).
% inverse_rat.transfer
thf(fact_10159_times__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) ) )
@ times_times_int ) ).
% times_int.transfer
thf(fact_10160_minus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) ) )
@ minus_minus_int ) ).
% minus_int.transfer
thf(fact_10161_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% zero_int.transfer
thf(fact_10162_int__transfer,axiom,
( bNF_re6830278522597306478at_int
@ ^ [Y6: nat,Z3: nat] : Y6 = Z3
@ pcr_int
@ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
@ semiri1314217659103216013at_int ) ).
% int_transfer
thf(fact_10163_uminus__int_Otransfer,axiom,
( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X2 ) )
@ uminus_uminus_int ) ).
% uminus_int.transfer
thf(fact_10164_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% one_int.transfer
thf(fact_10165_less__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
@ ord_less_int ) ).
% less_int.transfer
thf(fact_10166_less__eq__int_Otransfer,axiom,
( bNF_re717283939379294677_int_o @ pcr_int
@ ( bNF_re6644619430987730960nt_o_o @ pcr_int
@ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
@ ord_less_eq_int ) ).
% less_eq_int.transfer
thf(fact_10167_plus__int_Otransfer,axiom,
( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
@ plus_plus_int ) ).
% plus_int.transfer
thf(fact_10168_times__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_10169_intrel__iff,axiom,
! [X: nat,Y4: nat,U2: nat,V: nat] :
( ( intrel @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( product_Pair_nat_nat @ U2 @ V ) )
= ( ( plus_plus_nat @ X @ V )
= ( plus_plus_nat @ U2 @ Y4 ) ) ) ).
% intrel_iff
thf(fact_10170_uminus__int_Orsp,axiom,
( bNF_re2241393799969408733at_nat @ intrel @ intrel
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X2 ) )
@ ( produc2626176000494625587at_nat
@ ^ [X2: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X2 ) ) ) ).
% uminus_int.rsp
thf(fact_10171_zero__int_Orsp,axiom,
intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% zero_int.rsp
thf(fact_10172_one__int_Orsp,axiom,
intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% one_int.rsp
thf(fact_10173_intrel__def,axiom,
( intrel
= ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] :
( ( plus_plus_nat @ X2 @ V4 )
= ( plus_plus_nat @ U3 @ Y ) ) ) ) ) ).
% intrel_def
thf(fact_10174_less__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) ) ) ).
% less_int.rsp
thf(fact_10175_less__eq__int_Orsp,axiom,
( bNF_re4202695980764964119_nat_o @ intrel
@ ( bNF_re3666534408544137501at_o_o @ intrel
@ ^ [Y6: $o,Z3: $o] : Y6 = Z3 )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
@ ( produc8739625826339149834_nat_o
@ ^ [X2: nat,Y: nat] :
( produc6081775807080527818_nat_o
@ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_10176_plus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_10177_minus__int_Orsp,axiom,
( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) ) )
@ ( produc27273713700761075at_nat
@ ^ [X2: nat,Y: nat] :
( produc2626176000494625587at_nat
@ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_10178_plus__rat_Orsp,axiom,
( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% plus_rat.rsp
thf(fact_10179_ratrel__iff,axiom,
( ratrel
= ( ^ [X2: product_prod_int_int,Y: product_prod_int_int] :
( ( ( product_snd_int_int @ X2 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) )
= ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_10180_zero__rat_Orsp,axiom,
ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% zero_rat.rsp
thf(fact_10181_one__rat_Orsp,axiom,
ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% one_rat.rsp
thf(fact_10182_Fract_Orsp,axiom,
( bNF_re157797125943740599nt_int
@ ^ [Y6: int,Z3: int] : Y6 = Z3
@ ( bNF_re6250860962936578807nt_int
@ ^ [Y6: int,Z3: int] : Y6 = Z3
@ ratrel )
@ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B3 ) )
@ ^ [A4: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( B3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A4 @ B3 ) ) ) ).
% Fract.rsp
thf(fact_10183_ratrel__def,axiom,
( ratrel
= ( ^ [X2: product_prod_int_int,Y: product_prod_int_int] :
( ( ( product_snd_int_int @ X2 )
!= zero_zero_int )
& ( ( product_snd_int_int @ Y )
!= zero_zero_int )
& ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) )
= ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_10184_uminus__rat_Orsp,axiom,
( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
@ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) )
@ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) ) ) ).
% uminus_rat.rsp
thf(fact_10185_times__rat_Orsp,axiom,
( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% times_rat.rsp
thf(fact_10186_Rat_Opositive_Orsp,axiom,
( bNF_re8699439704749558557nt_o_o @ ratrel
@ ^ [Y6: $o,Z3: $o] : Y6 = Z3
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ).
% Rat.positive.rsp
thf(fact_10187_inverse__rat_Orsp,axiom,
( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
@ ^ [X2: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X2 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) )
@ ^ [X2: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X2 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_10188_plus__rat_Oabs__eq,axiom,
! [Xa2: product_prod_int_int,X: product_prod_int_int] :
( ( ratrel @ Xa2 @ Xa2 )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_10189_inverse__rat_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( ( ratrel @ X @ X )
=> ( ( inverse_inverse_rat @ ( abs_Rat @ X ) )
= ( abs_Rat
@ ( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_10190_one__rat__def,axiom,
( one_one_rat
= ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% one_rat_def
thf(fact_10191_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X2: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X2 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_10192_zero__rat__def,axiom,
( zero_zero_rat
= ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% zero_rat_def
thf(fact_10193_uminus__rat_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( ( ratrel @ X @ X )
=> ( ( uminus_uminus_rat @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_10194_times__rat_Oabs__eq,axiom,
! [Xa2: product_prod_int_int,X: product_prod_int_int] :
( ( ratrel @ Xa2 @ Xa2 )
=> ( ( ratrel @ X @ X )
=> ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_10195_Rat_Opositive_Oabs__eq,axiom,
! [X: product_prod_int_int] :
( ( ratrel @ X @ X )
=> ( ( positive @ ( abs_Rat @ X ) )
= ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% Rat.positive.abs_eq
thf(fact_10196_inverse__rat__def,axiom,
( inverse_inverse_rat
= ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
@ ^ [X2: product_prod_int_int] :
( if_Pro3027730157355071871nt_int
@ ( ( product_fst_int_int @ X2 )
= zero_zero_int )
@ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
@ ( product_Pair_int_int @ ( product_snd_int_int @ X2 ) @ ( product_fst_int_int @ X2 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_10197_uminus__rat__def,axiom,
( uminus_uminus_rat
= ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
@ ^ [X2: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X2 ) ) @ ( product_snd_int_int @ X2 ) ) ) ) ).
% uminus_rat_def
thf(fact_10198_plus__rat__def,axiom,
( plus_plus_rat
= ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% plus_rat_def
thf(fact_10199_times__rat__def,axiom,
( times_times_rat
= ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
@ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% times_rat_def
thf(fact_10200_Rat_Opositive__def,axiom,
( positive
= ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
@ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).
% Rat.positive_def
thf(fact_10201_Real_Opositive__def,axiom,
( positive2
= ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
@ ^ [X6: nat > rat] :
? [R5: rat] :
( ( ord_less_rat @ zero_zero_rat @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ) ).
% Real.positive_def
thf(fact_10202_pairs__le__eq__Sigma,axiom,
! [M: nat] :
( ( collec3392354462482085612at_nat
@ ( produc6081775807080527818_nat_o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M ) ) )
= ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
@ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_10203_finite__vimage__Suc__iff,axiom,
! [F5: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F5 ) )
= ( finite_finite_nat @ F5 ) ) ).
% finite_vimage_Suc_iff
thf(fact_10204_vimage__Suc__insert__Suc,axiom,
! [N: nat,A2: set_nat] :
( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A2 ) )
= ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A2 ) ) ) ).
% vimage_Suc_insert_Suc
% Helper facts (39)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y4: int] :
( ( if_int @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y4: int] :
( ( if_int @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y4: nat] :
( ( if_nat @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y4: nat] :
( ( if_nat @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y4: num] :
( ( if_num @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y4: num] :
( ( if_num @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y4: rat] :
( ( if_rat @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
! [X: rat,Y4: rat] :
( ( if_rat @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y4: real] :
( ( if_real @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y4: real] :
( ( if_real @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y4: complex] :
( ( if_complex @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y4: complex] :
( ( if_complex @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y4: extended_enat] :
( ( if_Extended_enat @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
! [X: extended_enat,Y4: extended_enat] :
( ( if_Extended_enat @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y4: code_integer] :
( ( if_Code_integer @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
! [X: code_integer,Y4: code_integer] :
( ( if_Code_integer @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y4: set_int] :
( ( if_set_int @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X: set_int,Y4: set_int] :
( ( if_set_int @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y4: set_nat] :
( ( if_set_nat @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y4: set_nat] :
( ( if_set_nat @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y4: vEBT_VEBT] :
( ( if_VEBT_VEBT @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
! [X: vEBT_VEBT,Y4: vEBT_VEBT] :
( ( if_VEBT_VEBT @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y4: list_int] :
( ( if_list_int @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y4: list_int] :
( ( if_list_int @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y4: list_nat] :
( ( if_list_nat @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y4: list_nat] :
( ( if_list_nat @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y4: option_nat] :
( ( if_option_nat @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y4: option_nat] :
( ( if_option_nat @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y4: option_num] :
( ( if_option_num @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
! [X: option_num,Y4: option_num] :
( ( if_option_num @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y4: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
! [X: product_prod_int_int,Y4: product_prod_int_int] :
( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y4 )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X: produc6271795597528267376eger_o,Y4: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
! [X: produc6271795597528267376eger_o,Y4: produc6271795597528267376eger_o] :
( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y4 )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y4: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $false @ X @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
! [X: produc8923325533196201883nteger,Y4: produc8923325533196201883nteger] :
( ( if_Pro6119634080678213985nteger @ $true @ X @ Y4 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ ( vEBT_VEBT_low @ za @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ maxy ).
%------------------------------------------------------------------------------